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(*^ ::[automaticGrouping; magnification = 125; currentKernel; fontset = title, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M7, bold, L1, e8, 24, "Times"; ; fontset = subtitle, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M7, bold, L1, e6, 18, "Times"; ; fontset = subsubtitle, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M7, italic, L1, e6, 14, "Times"; ; fontset = section, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, grayBox, M22, bold, L1, a20, 18, "Times"; ; fontset = subsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, blackBox, M19, bold, L1, a15, 14, "Times"; ; fontset = subsubsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, whiteBox, M18, bold, L1, a12, 12, "Times"; ; fontset = text, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = smalltext, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 10, "Times"; ; fontset = input, noPageBreakBelow, nowordwrap, preserveAspect, groupLikeInput, M42, N23, bold, L1, 12, "Courier"; ; fontset = output, output, inactive, noPageBreakBelow, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L-5, 12, "Courier"; ; fontset = message, inactive, noPageBreakBelow, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L1, 12, "Courier"; ; fontset = print, inactive, noPageBreakBelow, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L1, 12, "Courier"; ; fontset = info, inactive, noPageBreakBelow, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L1, 12, "Courier"; ; fontset = postscript, PostScript, formatAsPostScript, output, inactive, noPageBreakBelow, nowordwrap, preserveAspect, groupLikeGraphics, M7, l34, w282, h287, L1, 12, "Courier"; ; fontset = name, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, italic, L1, 10, "Times"; ; fontset = header, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = Left Header, nohscroll, cellOutline, 12; fontset = footer, inactive, nohscroll, noKeepOnOnePage, preserveAspect, center, M7, L1, 12; fontset = Left Footer, cellOutline, blackBox, 12; fontset = help, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 10, "Times"; ; fontset = clipboard, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = completions, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12, "Courier"; ; fontset = special1, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = special2, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = special3, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = special4, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = special5, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; next21StandardFontEncoding; ] :[font = title; inactive; preserveAspect; startGroup; ] Fourier Series :[font = section; inactive; Cclosed; preserveAspect; startGroup; ] Graphing Fourier Series Using Mathematica ;[s] 2:0,0;31,1;42,-1; 2:1,16,12,Times,1,18,0,0,0;1,17,13,Times,3,18,0,0,0; :[font = text; inactive; preserveAspect; ] Let us find the first terms in the Fourier series of the function defined by :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 0; pictureWidth = 125; pictureHeight = 13; ] %! %%Creator: Mathematica %%AspectRatio: 0.104 200 dict begin MathPictureStart /Courier findfont 10 scalefont setfont % Scaling calculations 0 1 0 1 [ [ 0 0 0 0 ] [ 125 13 0 0 ] ] MathScale % Start of Graphics %!PS-Adobe-2.0 EPSF-2.0 %%Title: %%Creator: EquationBuilder %%CreationDate: Sun Aug 2 08:16:02 1992 %%For: gray %%DocumentFonts: (atend) %%Pages: 0 1 %%BoundingBox: 0 0 125 13 %%Translation: 1 2.46 123.585327 0 %%EndComments %%BeginDocument: /usr/lib/NextStep/printPackage.ps %!PS-Adobe-2.0 %%Title: Printing Package %%EndComments %Version: 2.0 %Copyright: 1988, NeXT, Inc. /__NXdef{1 index where{pop pop pop}{def}ifelse}bind def /__NXbdef{1 index where{pop pop pop}{bind def}ifelse}bind def /UserObjects 10 array __NXdef /defineuserobject{ exch dup 1 add dup UserObjects length gt{ array dup 0 UserObjects putinterval /UserObjects exch def }{pop}ifelse UserObjects exch 3 -1 roll put }__NXbdef /undefineuserobject{UserObjects exch null put}__NXbdef /execuserobject{UserObjects exch get exec}__NXbdef /__NXRectPath{4 2 roll moveto 1 index 0 rlineto 0 exch rlineto neg 0 rlineto closepath}__NXbdef /__NXProcessRectArgs{ 1 index type /arraytype eq{ exch 0 4 2 index length 1 sub{ dup 3 add 1 exch{1 index exch get exch}for 5 1 roll 5 index exec }for pop pop }{exec}ifelse }__NXbdef /rectfill{gsave newpath {__NXRectPath fill} __NXProcessRectArgs grestore}__NXbdef /rectclip{newpath {__NXRectPath} __NXProcessRectArgs clip newpath}__NXbdef /rectstroke{ gsave newpath dup type /arraytype eq{dup length 6 eq}{false}ifelse{ {gsave __NXRectPath null concat stroke grestore} dup length array cvx copy dup 2 4 -1 roll put __NXProcessRectArgs }{{__NXRectPath stroke} __NXProcessRectArgs}ifelse grestore }__NXbdef /xyshow{ 0 1 3 index length 1 sub{ currentpoint 4 index 3 index 1 getinterval show 3 index 3 index 2 mul 1 add get add exch 3 index 3 index 2 mul get add exch moveto pop }for pop pop }__NXbdef /xshow{ 0 1 3 index length 1 sub{ currentpoint 4 index 3 index 1 getinterval show exch 3 index 3 index get add exch moveto pop }for pop pop }__NXbdef /yshow{ 0 1 3 index length 1 sub{ currentpoint 4 index 3 index 1 getinterval show 3 index 3 index get add moveto pop }for pop pop }__NXbdef /arct{arcto pop pop pop pop}__NXbdef /setbbox{pop pop pop pop}__NXbdef /ucache{}__NXbdef /ucachestatus{mark 0 0 0 0 0}__NXbdef /setucacheparams{cleartomark}__NXbdef /uappend{systemdict begin cvx exec end}__NXbdef /ueofill{gsave newpath uappend eofill grestore}__NXbdef /ufill{gsave newpath uappend fill grestore}__NXbdef /ustroke{ gsave newpath dup length 6 eq {exch uappend concat}{uappend}ifelse stroke grestore }__NXbdef /__NXustrokepathMatrix dup where {pop pop}{matrix def}ifelse /ustrokepath{ newpath dup length 6 eq{ exch uappend __NXustrokepathMatrix currentmatrix exch concat strokepath setmatrix }{uappend strokepath}ifelse } __NXbdef /upath{ [exch {/ucache cvx}if pathbbox /setbbox cvx {/moveto cvx}{/lineto cvx}{/curveto cvx}{/closepath cvx}pathforall]cvx } __NXbdef /setstrokeadjust{pop}__NXbdef /currentstrokeadjust{false}__NXbdef /selectfont{exch findfont exch dup type /arraytype eq {makefont}{scalefont}ifelse setfont}__NXbdef /_NXCombineArrays{ counttomark dup 2 add index dup length 3 -1 roll { 2 index length sub dup 4 1 roll 1 index exch 4 -1 roll putinterval exch }repeat pop pop pop }__NXbdef /setcmykcolor{ 1.0 exch sub dup dup 6 -1 roll sub dup 0 lt{pop 0}if 5 1 roll 4 -1 roll sub dup 0 lt{pop 0}if 3 1 roll exch sub dup 0 lt{pop 0}if setrgbcolor }__NXbdef /currentcmykcolor{currentrgbcolor 3{1.0 exch sub 3 1 roll}repeat 0}__NXbdef /flushgraphics{}def /setwindowtype{pop pop}def /currentwindowtype{pop 0}def /setalpha{pop}def /currentalpha{1.0}def /hidecursor{}def /obscurecursor{}def /revealcursor{}def /setcursor{4 {pop}repeat}bind def /showcursor{}def /NextStepEncoding where not{ /NextStepEncoding StandardEncoding 256 array copy def 0 [129/Agrave/Aacute/Acircumflex/Atilde/Adieresis/Aring/Ccedilla/Egrave /Eacute/Ecircumflex/Edieresis/Igrave/Iacute/Icircumflex/Idieresis /Eth/Ntilde/Ograve/Oacute/Ocircumflex/Otilde/Odieresis/Ugrave/Uacute /Ucircumflex/Udieresis/Yacute/Thorn/mu/multiply/divide/copyright 176/registered 181/brokenbar 190/logicalnot 192/onesuperior 201/twosuperior 204/threesuperior 209/plusminus/onequarter/onehalf/threequarters/agrave /aacute/acircumflex/atilde/adieresis/aring/ccedilla/egrave/eacute /ecircumflex/edieresis/igrave 226/iacute 228/icircumflex/idieresis/eth /ntilde 236/ograve/oacute/ocircumflex/otilde/odieresis 242/ugrave/uacute /ucircumflex 246/udieresis/yacute 252/thorn/ydieresis] {dup type /nametype eq {NextStepEncoding 2 index 2 index put pop 1 add}{exch pop}ifelse }forall pop /NextStepEncoding NextStepEncoding readonly def /_NXfstr 128 string dup 0 (_NX) putinterval def /findfont{ % Because we can never let NextStepEncoding get into % SharedFontDirectory, we cannot reencode a font to NextStepEncoding % if we are in shared mode. So if currentshared is true, % we call the normal findfont and return that /currentshared where {pop currentshared} {false} ifelse {//findfont exec} {dup _NXfstr 3 125 getinterval cvs length 3 add _NXfstr 0 3 -1 roll getinterval cvn exch FontDirectory 2 index known {pop FontDirectory exch get} {//findfont exec dup /Encoding get StandardEncoding eq { dup length dict exch {1 index /FID ne {2 index 3 1 roll put}{pop pop}ifelse}forall dup /Encoding NextStepEncoding put definefont }{exch pop} ifelse }ifelse }ifelse }bind def }{pop}ifelse /_NXProcArray 5 array __NXdef /_NXChannels 0 __NXdef /_NXTotalBytes 0 __NXdef /_NXDoImageOp{ 1 index{dup}{1}ifelse /_NXChannels exch store _NXChannels 2 add 2 roll _NXProcArray 0 _NXChannels getinterval astore pop 5 index 4 index mul 2 index{1 sub 8 idiv 1 add mul}{mul 1 sub 8 idiv 1 add}ifelse 4 index mul /_NXTotalBytes exch store pop exch pop gsave matrix invertmatrix concat 0.5 setgray 0 0 4 2 roll rectfill grestore { 0 1 _NXChannels 1 sub{ _NXProcArray exch get exec length _NXTotalBytes exch sub /_NXTotalBytes exch store}for _NXTotalBytes 0 le{exit}if }loop /_NXProcArray 5 array def }__NXbdef /colorimage{_NXDoImageOp}__NXbdef /alphaimage{1 add _NXDoImageOp}def %%EndDocument % EquationBuilder(TM) Expression % +.\Jr(=:/*V[StRFQAYmdjEEA;1nuN+[co.OZ7L#cq3iK%3:</XJ@:d#6;^G;Td47dud<A3@ % 5%MGToUOR5&d`'TuEDU)B-tbR".RTMGY`!!!E7U+baR+Ae6lj@P.-O?%B4'FH)c:eX1]TM0' % o;,/#l!%\^3YU,#41L183Q(&Hf"9NTgc2h<D&k0$!hQXOnBCNsA60Psq6cY@ca,g65@$1_&! % 4;jU<*6b$7LUDFr[O:uVi;P0M&.7'0Eb)e3N3m7Jt0r$-W#0s#nWO><.IUg"/qk0!(88&QON % h05bngX1i:[fGo<*sm)QL?h%!8h/SIJC+Hs*jgCU@@^B#<5FZ^ih6fubu09s\hI6do)!s"#s % ^^i^."lLJQJcLm!$=3/q?W,$D`'?q2BZ-#PeU^%+H$l0l?NKa1_B*\[PR$'!g6JS0D]O:o8q % VdB!(@oGi)CkV[q-t[H60L2!"1>)0EiG)N)<&ef1D-2NUcD&hKkj=TU[VkRWp'@:(dUSFr2Q % _,*&NWq.,^J.[d#t<e@=4Orfd3V3+oPLh;uRY'Hh-@WJHFR;S$&-_=86\L3[hL:JUq/1Mc1) % K*3Ra,GNmGp8#KlI^]/iAF4T.$1U0,94k2_WQY&lH"U6Q*M:kR@t8!+sEO].VP\>&g@^_T44 % 7(/F_c#hE.D.e!F`8e02jPDnr6;Cln%+Wik[\PmJiEXL6)FI5$b5!UVI("`u6%+N7eu\Bu?# % gU>M3&;$%,"<Co52b&-$P^@;ZTEHHHJ,f]Q"MX_=69mqK"MkBESG6S"x % EndExpression N 620 26c E 86 S 10312 R a0f60fc3 gsave /__NXbasematrix matrix currentmatrix def grestore %%EndProlog %%BeginSetup %%EndSetup 0 0 124.585327 12.836 rectclip 0 0 124.585327 12.836 rectclip 0 setgray 1 setalpha 1 1 scale /Symbol findfont 12 scalefont 45 exch defineuserobject 45 execuserobject setfont 5.56 3.568 moveto (\() [0 0] xyshow 1 1 scale 1 1 scale 45 execuserobject setfont 17.200001 3.562 moveto (\)) [0 0] xyshow 1 1 scale 0 setgray 1 setalpha /Times-BoldItalic findfont 12 scalefont 44 exch defineuserobject 44 execuserobject setfont 97.742668 3.46 moveto (xxxf) [-62.764 0 -24.618668 0 -9.360001 0 0 0 0 0] xyshow 45 execuserobject setfont 116.997337 3.46 moveto (p<<p-=) [-9.921333 0 -19.254669 0 -9.921333 0 -9.254669 0 -43.587997 0 0 0 0 0] xyshow /Times-Roman findfont 12 scalefont 47 exch defineuserobject 47 execuserobject setfont 40.978668 3.46 moveto (,) [0 0 -1.681558e-44 1.050009e+12] xyshow %%Trailer %%DocumentFonts: Times-BoldItalic %%+ Times-Roman %%+ Symbol %End of Graphics MathPictureEnd end :[font = text; inactive; preserveAspect; ] and plot it. The Fourier series is :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 0; pictureWidth = 114; pictureHeight = 36; ] %! %%Creator: Mathematica %%AspectRatio: 0.315789 200 dict begin MathPictureStart /Courier findfont 10 scalefont setfont % Scaling calculations 0 1 0 1 [ [ 0 0 0 0 ] [ 114 36 0 0 ] ] MathScale % Start of Graphics %!PS-Adobe-2.0 EPSF-2.0 %%Title: %%Creator: EquationBuilder %%CreationDate: Mon Aug 3 19:59:17 1992 %%For: gray %%DocumentFonts: (atend) %%Pages: 0 1 %%BoundingBox: 0 0 114 36 %%Translation: 1 15.082992 112.755661 0 %%EndComments %%BeginDocument: /usr/lib/NextStep/printPackage.ps %!PS-Adobe-2.0 %%Title: Printing Package %%EndComments %Version: 2.0 %Copyright: 1988, NeXT, Inc. /__NXdef{1 index where{pop pop pop}{def}ifelse}bind def /__NXbdef{1 index where{pop pop pop}{bind def}ifelse}bind def /UserObjects 10 array __NXdef /defineuserobject{ exch dup 1 add dup UserObjects length gt{ array dup 0 UserObjects putinterval /UserObjects exch def }{pop}ifelse UserObjects exch 3 -1 roll put }__NXbdef /undefineuserobject{UserObjects exch null put}__NXbdef /execuserobject{UserObjects exch get exec}__NXbdef /__NXRectPath{4 2 roll moveto 1 index 0 rlineto 0 exch rlineto neg 0 rlineto closepath}__NXbdef /__NXProcessRectArgs{ 1 index type /arraytype eq{ exch 0 4 2 index length 1 sub{ dup 3 add 1 exch{1 index exch get exch}for 5 1 roll 5 index exec }for pop pop }{exec}ifelse }__NXbdef /rectfill{gsave newpath {__NXRectPath fill} __NXProcessRectArgs grestore}__NXbdef /rectclip{newpath {__NXRectPath} __NXProcessRectArgs clip newpath}__NXbdef /rectstroke{ gsave newpath dup type /arraytype eq{dup length 6 eq}{false}ifelse{ {gsave __NXRectPath null concat stroke grestore} dup length array cvx copy dup 2 4 -1 roll put __NXProcessRectArgs }{{__NXRectPath stroke} __NXProcessRectArgs}ifelse grestore }__NXbdef /xyshow{ 0 1 3 index length 1 sub{ currentpoint 4 index 3 index 1 getinterval show 3 index 3 index 2 mul 1 add get add exch 3 index 3 index 2 mul get add exch moveto pop }for pop pop }__NXbdef /xshow{ 0 1 3 index length 1 sub{ currentpoint 4 index 3 index 1 getinterval show exch 3 index 3 index get add exch moveto pop }for pop pop }__NXbdef /yshow{ 0 1 3 index length 1 sub{ currentpoint 4 index 3 index 1 getinterval show 3 index 3 index get add moveto pop }for pop pop }__NXbdef /arct{arcto pop pop pop pop}__NXbdef /setbbox{pop pop pop pop}__NXbdef /ucache{}__NXbdef /ucachestatus{mark 0 0 0 0 0}__NXbdef /setucacheparams{cleartomark}__NXbdef /uappend{systemdict begin cvx exec end}__NXbdef /ueofill{gsave newpath uappend eofill grestore}__NXbdef /ufill{gsave newpath uappend fill grestore}__NXbdef /ustroke{ gsave newpath dup length 6 eq {exch uappend concat}{uappend}ifelse stroke grestore }__NXbdef /__NXustrokepathMatrix dup where {pop pop}{matrix def}ifelse /ustrokepath{ newpath dup length 6 eq{ exch uappend __NXustrokepathMatrix currentmatrix exch concat strokepath setmatrix }{uappend strokepath}ifelse } __NXbdef /upath{ [exch {/ucache cvx}if pathbbox /setbbox cvx {/moveto cvx}{/lineto cvx}{/curveto cvx}{/closepath cvx}pathforall]cvx } __NXbdef /setstrokeadjust{pop}__NXbdef /currentstrokeadjust{false}__NXbdef /selectfont{exch findfont exch dup type /arraytype eq {makefont}{scalefont}ifelse setfont}__NXbdef /_NXCombineArrays{ counttomark dup 2 add index dup length 3 -1 roll { 2 index length sub dup 4 1 roll 1 index exch 4 -1 roll putinterval exch }repeat pop pop pop }__NXbdef /setcmykcolor{ 1.0 exch sub dup dup 6 -1 roll sub dup 0 lt{pop 0}if 5 1 roll 4 -1 roll sub dup 0 lt{pop 0}if 3 1 roll exch sub dup 0 lt{pop 0}if setrgbcolor }__NXbdef /currentcmykcolor{currentrgbcolor 3{1.0 exch sub 3 1 roll}repeat 0}__NXbdef /flushgraphics{}def /setwindowtype{pop pop}def /currentwindowtype{pop 0}def /setalpha{pop}def /currentalpha{1.0}def /hidecursor{}def /obscurecursor{}def /revealcursor{}def /setcursor{4 {pop}repeat}bind def /showcursor{}def /NextStepEncoding where not{ /NextStepEncoding StandardEncoding 256 array copy def 0 [129/Agrave/Aacute/Acircumflex/Atilde/Adieresis/Aring/Ccedilla/Egrave /Eacute/Ecircumflex/Edieresis/Igrave/Iacute/Icircumflex/Idieresis /Eth/Ntilde/Ograve/Oacute/Ocircumflex/Otilde/Odieresis/Ugrave/Uacute /Ucircumflex/Udieresis/Yacute/Thorn/mu/multiply/divide/copyright 176/registered 181/brokenbar 190/logicalnot 192/onesuperior 201/twosuperior 204/threesuperior 209/plusminus/onequarter/onehalf/threequarters/agrave /aacute/acircumflex/atilde/adieresis/aring/ccedilla/egrave/eacute /ecircumflex/edieresis/igrave 226/iacute 228/icircumflex/idieresis/eth /ntilde 236/ograve/oacute/ocircumflex/otilde/odieresis 242/ugrave/uacute /ucircumflex 246/udieresis/yacute 252/thorn/ydieresis] {dup type /nametype eq {NextStepEncoding 2 index 2 index put pop 1 add}{exch pop}ifelse }forall pop /NextStepEncoding NextStepEncoding readonly def /_NXfstr 128 string dup 0 (_NX) putinterval def /findfont{ % Because we can never let NextStepEncoding get into % SharedFontDirectory, we cannot reencode a font to NextStepEncoding % if we are in shared mode. So if currentshared is true, % we call the normal findfont and return that /currentshared where {pop currentshared} {false} ifelse {//findfont exec} {dup _NXfstr 3 125 getinterval cvs length 3 add _NXfstr 0 3 -1 roll getinterval cvn exch FontDirectory 2 index known {pop FontDirectory exch get} {//findfont exec dup /Encoding get StandardEncoding eq { dup length dict exch {1 index /FID ne {2 index 3 1 roll put}{pop pop}ifelse}forall dup /Encoding NextStepEncoding put definefont }{exch pop} ifelse }ifelse }ifelse }bind def }{pop}ifelse /_NXProcArray 5 array __NXdef /_NXChannels 0 __NXdef /_NXTotalBytes 0 __NXdef /_NXDoImageOp{ 1 index{dup}{1}ifelse /_NXChannels exch store _NXChannels 2 add 2 roll _NXProcArray 0 _NXChannels getinterval astore pop 5 index 4 index mul 2 index{1 sub 8 idiv 1 add mul}{mul 1 sub 8 idiv 1 add}ifelse 4 index mul /_NXTotalBytes exch store pop exch pop gsave matrix invertmatrix concat 0.5 setgray 0 0 4 2 roll rectfill grestore { 0 1 _NXChannels 1 sub{ _NXProcArray exch get exec length _NXTotalBytes exch sub /_NXTotalBytes exch store}for _NXTotalBytes 0 le{exit}if }loop /_NXProcArray 5 array def }__NXbdef /colorimage{_NXDoImageOp}__NXbdef /alphaimage{1 add _NXDoImageOp}def %%EndDocument % EquationBuilder(TM) Expression % +.\Jr(=:/*V[StRFQAYmdjEEA;1nuN+[co.OZ7L#cq3iK%3:</XJ@:d#6;^G;Td47dud<A3@ % 5%MGToUOR5&d`'TuEDU)B-tbR".R5Rn)o!!!E767usRSid;k!&A0sO@"L$!/a>R!L,$-1s/C % B#Cn*e"h)@2@AXE`Ui2FuJ3`%0`&Rf+8YXkFaW!,slN$o42$fa3D<P)b`0`8d!_h#b$ue*i) % ?NgGciDLJYC\3Kg5,Q?(boG%6dULK;`aHG`gQOD!gT+6!22tM"h8(uJ-q/75a2=&_[p/Z:C> % 7lpkJHMf<Ah=&s-6#B0r4/PEMN(<:0h#LRT2'Z'96&g8,>$4QCa2`iHTgnd[p/GXCGAY;&e7 % 5tX$RETBT)FF_eAI"]_ph08VT(Jt^9E+%IJOs$^]I(AVKJ9^,Y&;),Ceu)q*\:F,\M"KB0+Y % JEVON*1[,fLiK$lH5h:')(')Ip#u3*e"eTZ<EPlE,*^'691sL)[2TTjqTG'ZjDY7$PMW)Ju9 % 9a%K2`p5="nA/<(*6:rG\F92G6OTIj#fd)+Q100?].>##PIN>jN@:OBT\=eWe%eg<f)2@=5M % 1brAiLp;9#)XN(6s:\.!;li<UB+>?W?;2:O?sbE=t4$DNIC'O!(Kil%@r40R3*s!+k,)DIYJ % ,cSK%^?_Oc_6H=;[Ui^q4a[Z8E<U3n7`6lU(R"D=^8eq^<1VLX7`U+2HANgZ'3l4%SdI[8?D % B65RoPdLQ3-q+6^Z(uBW5RAbhUb[[bPSC,2D77<uRd:`-KJ>q6k3AFRX5\[c"0[k)r%#Y$7Y % D#T,MIc`Z/Jj@BVUY]MQ*Hp>P`$2T8'pW?5RTonVPmU7c_CCjpGnHV<FZn_`^1'F2S*h5]M& % J+M-0n6e!:"*6jn`'?pBCU/r\mU:&$<I5$b5!TYgt"dUXG+9c4_\3V/=>IW"f&:]h)"<CoU3 % <ja9P^R`/TELGBa$B]V2\;<`(;LbQDup"erYP\:x % EndExpression N 779 30b E e2 S 14f3c R fad5c0ae gsave /__NXbasematrix matrix currentmatrix def grestore %%EndProlog %%BeginSetup /EqBFont 18 dict dup begin /FontType 3 def /PaintType 0 def /FontMatrix [0.001 0 0 0.001 0 0] def /FontBBox [15 -2950 1360 1045] def /Encoding [/.notdef /rad1 /rad2 /rad3 /rad4 /rad5 /radext /int /pathint /sum /prod /cdot /vdot /ddot /hbar /bigcap /bigcup /parallel] def /Metrics 18 dict dup begin /.notdef 0 def /rad1 1000 def /rad2 1000 def /rad3 1000 def /rad4 1000 def /rad5 1048 def /radext 1048 def /int 500 def /pathint 500 def /sum 700 def /prod 700 def /cdot 1000 def /vdot 550 def /ddot 1000 def /hbar 500 def /bigcap 700 def /bigcup 700 def /parallel 500 def end def /BBox 18 dict dup begin /.notdef [0 0 0 0] def /rad1 [110 -1150 1020 0] def /rad2 [110 -1750 1020 0] def /rad3 [110 -2350 1020 0] def /rad4 [110 -2950 1020 0] def /rad5 [128 -2890 740 0] def /radext [700 -600 740 0] def /int [20 0 425 1045] def /pathint [20 0 425 1045] def /sum [15 5 640 715] def /prod [15 0 670 715] def /cdot [110 215 890 325] def /vdot [165 -5 280 770] def /ddot [110 -5 890 770] def /hbar [20 -10 480 685] def /bigcap [0 0 680 720] def /bigcup [0 0 680 720] def /parallel [65 -175 400 675] def end def /CharacterDefs 18 dict dup begin /.notdef { } def /rad1 { 980 0 moveto 464 -1048 lineto 248 -560 lineto 110 -656 lineto 125 -675 lineto 200 -624 lineto 420 -1150 lineto 460 -1150 lineto 1020 0 lineto 980 0 lineto closepath } def /rad2 { 980 0 moveto 460 -1600 lineto 245 -864 lineto 110 -1000 lineto 128 -1023 lineto 200 -950 lineto 424 -1750 lineto 456 -1750 lineto 1020 0 lineto 980 0 lineto closepath } def /rad3 { 980 0 moveto 464 -2144 lineto 248 -1160 lineto 110 -1368 lineto 130 -1390 lineto 192 -1288 lineto 424 -2350 lineto 464 -2350 lineto 1020 0 lineto 980 0 lineto closepath } def /rad4 { 980 0 moveto 880 -488 lineto 672 -1600 lineto 464 -2696 lineto 248 -1464 lineto 110 -1728 lineto 130 -1750 lineto 192 -1616 lineto 424 -2950 lineto 464 -2950 lineto 768 -1312 lineto 1020 0 lineto 980 0 lineto closepath } def /rad5 { 700 0 moveto 700 -2628 lineto 248 -1284 lineto 128 -1564 lineto 150 -1580 lineto 215 -1425 lineto 688 -2890 lineto 740 -2890 lineto 740 0 lineto 700 0 lineto closepath } def /radext { 700 0 moveto 700 -600 lineto 740 -600 lineto 740 0 lineto 700 0 lineto closepath } def /int { 245 390 moveto 225 235 195 145 180 100 curveto 165 60 150 50 140 40 curveto 135 30 85 0 50 20 curveto 35 25 25 45 25 55 curveto 20 80 30 100 55 110 curveto 80 125 93 110 110 90 curveto 135 60 155 135 160 195 curveto 165 270 170 340 190 510 curveto 205 650 lineto 220 810 230 890 265 960 curveto 275 980 295 1015 320 1030 curveto 350 1045 380 1030 395 1020 curveto 410 1010 425 980 400 955 curveto 385 940 347 925 330 950 curveto 310 970 290 955 290 940 curveto 280 900 285 745 270 610 curveto 245 390 lineto closepath } def /pathint { 245 390 moveto 225 235 195 145 180 100 curveto 165 60 150 50 140 40 curveto 135 30 85 0 50 20 curveto 35 25 25 45 25 55 curveto 20 80 30 100 55 110 curveto 80 125 93 110 110 90 curveto 135 60 155 135 160 195 curveto 165 270 170 340 190 510 curveto 205 650 lineto 220 810 230 890 265 960 curveto 275 980 295 1015 320 1030 curveto 350 1045 380 1030 395 1020 curveto 410 1010 425 980 400 955 curveto 385 940 347 925 330 950 curveto 310 970 290 955 290 940 curveto 280 900 285 745 270 610 curveto 245 390 lineto fill 20 setlinewidth 220 400 moveto 290 400 340 455 340 500 curveto 340 585 290 640 220 640 curveto 155 640 100 585 100 500 curveto 100 455 155 400 220 400 curveto stroke } def /sum { 330 335 moveto 20 715 lineto 575 715 lineto 600 545 lineto 575 535 lineto 550 625 525 685 395 685 curveto 160 685 lineto 411 370 lineto 160 90 lineto 470 90 lineto 565 90 590 140 610 210 curveto 640 200 lineto 585 5 lineto 15 5 lineto 330 335 lineto closepath } def /prod { 590 90 moveto 590 30 620 20 670 20 curveto 670 0 lineto 410 0 lineto 410 20 lineto 460 20 495 30 495 90 curveto 495 670 lineto 190 670 lineto 190 90 lineto 190 30 225 20 275 20 curveto 275 0 lineto 15 0 lineto 15 20 lineto 65 20 100 30 100 90 curveto 100 615 lineto 100 690 40 700 20 700 curveto 20 715 lineto 670 715 lineto 670 700 lineto 645 700 590 690 590 615 curveto 590 90 lineto closepath } def /cdot { 110 270 moveto 110 300 135 325 165 325 curveto 195 325 220 300 220 270 curveto 220 235 185 215 165 215 curveto 145 215 110 235 110 270 curveto fill 445 270 moveto 445 300 470 325 500 325 curveto 530 325 555 300 555 270 curveto 555 235 520 215 500 215 curveto 480 215 445 235 445 270 curveto fill 775 270 moveto 775 300 800 325 830 325 curveto 860 325 890 300 890 270 curveto 890 235 850 215 830 215 curveto 810 215 775 235 775 270 curveto fill } def /vdot { 220 770 moveto 250 770 280 750 280 720 curveto 280 680 250 660 220 660 curveto 185 660 165 695 165 720 curveto 165 740 185 770 220 770 curveto fill 220 440 moveto 250 440 280 415 280 385 curveto 280 355 250 330 220 330 curveto 185 330 165 365 165 385 curveto 165 405 185 440 220 440 curveto fill 220 105 moveto 250 105 280 80 280 50 curveto 280 20 250 -5 220 -5 curveto 185 -5 165 30 165 50 curveto 165 70 185 105 220 105 curveto fill } def /ddot { 165 720 moveto 205 720 220 700 220 665 curveto 220 630 195 610 165 610 curveto 130 620 110 645 110 665 curveto 110 690 130 720 165 720 curveto fill 500 440 moveto 530 440 555 415 555 385 curveto 555 355 530 330 500 330 curveto 465 330 445 365 445 385 curveto 445 405 465 440 500 440 curveto fill 830 155 moveto 860 155 890 130 890 100 curveto 890 70 860 45 830 45 curveto 795 45 775 80 775 100 curveto 775 120 795 155 830 155 curveto fill } def /hbar { 480 105 moveto 440 55 400 -10 335 -10 curveto 300 -10 290 10 290 35 curveto 290 65 370 350 370 365 curveto 370 380 355 390 340 390 curveto 290 390 205 275 185 250 curveto 140 180 125 125 95 0 curveto 20 0 lineto 150 500 lineto 160 545 180 600 180 615 curveto 180 625 170 640 135 640 curveto 110 640 lineto 100 655 lineto 165 660 215 670 270 685 curveto 275 680 lineto 155 240 lineto 155 235 lineto 190 290 290 440 380 440 curveto 400 440 450 435 450 370 curveto 450 335 365 80 365 55 curveto 365 45 375 40 385 40 curveto 405 40 425 70 465 115 curveto 480 105 lineto fill 55 545 moveto 55 520 lineto 310 560 lineto 310 585 lineto 55 545 lineto fill } def /bigcap { 615 400 moveto 615 575 500 655 340 655 curveto 180 655 70 575 70 400 curveto 70 0 lineto 5 0 lineto 5 400 lineto 5 610 140 720 340 720 curveto 545 720 680 610 680 400 curveto 680 0 lineto 615 0 lineto 615 400 lineto closepath } def /bigcup { 65 320 moveto 65 145 180 65 340 65 curveto 500 65 615 145 615 320 curveto 615 720 lineto 680 720 lineto 680 320 lineto 680 110 540 0 340 0 curveto 140 0 0 110 0 320 curveto 0 720 lineto 65 720 lineto 65 320 lineto closepath } def /parallel { 135 675 moveto 135 -175 lineto 65 -175 lineto 65 675 lineto 135 675 lineto fill 335 675 moveto 335 -175 lineto 265 -175 lineto 265 675 lineto 335 675 lineto fill 400 675 moveto } def end def /BuildChar { 0 begin /char exch def /fontdict exch def /charname fontdict /Encoding get char get def fontdict begin Metrics charname get 0 BBox charname get aload pop setcachedevice CharacterDefs charname get exec eofill end end } def /BuildChar load 0 3 dict put end definefont pop %%EndSetup 0 0 113.755661 35.889019 rectclip 0 0 113.755661 35.889019 rectclip 0 setgray 1 setalpha /EqBFont 24 selectfont 8.64 10.442991 moveto <09> [0 0] xyshow 0 setgray 1 setalpha 1 1 scale /Symbol findfont 12 scalefont 48 exch defineuserobject 48 execuserobject setfont 27.063999 23.990992 moveto (\() [0 0] xyshow 1 1 scale 1 1 scale 48 execuserobject setfont 47.958668 23.984991 moveto (\)) [0 0] xyshow 1 1 scale 0 setgray 1 setalpha 1 1 scale 48 execuserobject setfont 83.091667 23.990992 moveto (\() [0 0] xyshow 1 1 scale 1 1 scale 48 execuserobject setfont 102.731667 23.984991 moveto (\)) [0 0] xyshow 1 1 scale 0 setgray 1 setalpha 0.48 setlinewidth 26 19.082993 moveto 107.755661 19.082993 lineto stroke 0 setgray 1 setalpha /Times-Roman findfont 12 scalefont 50 exch defineuserobject 50 execuserobject setfont 68.523666 23.882992 moveto (sin12) [4.667999 0 3.335999 0 -35.408997 0 -40.118668 -7.799999 0 0 -38.509338 0] xyshow /Symbol findfont 9 scalefont 60 exch defineuserobject 60 execuserobject setfont 57.432667 28.598021 moveto (+=\245) [-43.403168 -26.326021 -0.738 27.730968 0 0 -2.526191e+30 2.056568e-38] xyshow /Times-BoldItalic findfont 9 scalefont 67 exch defineuserobject 67 execuserobject setfont 52.482666 28.598021 moveto (kk) [-43.403168 -26.326021 0 0 -1.51935e-36 -3.104157e-36] xyshow /Times-Roman findfont 9 scalefont 51 exch defineuserobject 51 execuserobject setfont 62.823666 28.598021 moveto (11) [-43.403168 -26.326021 0 0 6.0336 13.733974] xyshow 48 execuserobject setfont 31.864 23.882992 moveto (-) [0 0 19.254662 14.999975] xyshow /Times-BoldItalic findfont 12 scalefont 49 exch defineuserobject 49 execuserobject setfont 109.755661 16.082991 moveto (.kxk) [-45.877831 -8.231411 32.013832 16.031412 -8 0 0 0 0 0] xyshow %%Trailer %%DocumentFonts: Times-BoldItalic %%+ Times-Roman %%+ Symbol %End of Graphics MathPictureEnd end :[font = text; inactive; preserveAspect; ] Enter the following commands in a cell: :[font = input; preserveAspect; ] f[n_,x_]:=Sum[(2/k)(-1)^(k+1)Sin[k x],{k,1,n}] :[font = text; inactive; preserveAspect; ] Then do "Enter". (You won't see anything immediately.) This defines in Mathematica a function f of two variables. You can get a quick check of whether f is defined correctly by typing ?f. Here f[n,x] is the sum of the first n terms of the Fourier series for f(x). For example, you can see the first three terms of the series by using the mouse to move the cursor to a new cell and typing ;[s] 17:0,0;73,1;84,2;97,3;99,4;156,5;158,6;191,7;193,8;201,9;208,10;233,11;234,12;269,13;273,14;274,15;276,16;400,-1; 17:1,11,8,Times,0,12,0,0,0;1,10,8,Times,2,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Courier,0,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Courier,0,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Courier,0,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Courier,0,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Times,2,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Times,2,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Courier,0,12,0,0,0;1,11,8,Times,0,12,0,0,0; :[font = input; preserveAspect; startGroup; ] f[3,x] :[font = output; output; inactive; preserveAspect; endGroup; ] 2*Sin[x] - Sin[2*x] + (2*Sin[3*x])/3 ;[o] 2 Sin[3 x] 2 Sin[x] - Sin[2 x] + ---------- 3 :[font = text; inactive; preserveAspect; ] This is Mathematica's way of printing out ;[s] 3:0,0;9,1;22,2;43,-1; 3:1,11,8,Times,0,12,0,0,0;1,10,8,Times,2,12,0,0,0;1,11,8,Times,0,12,0,0,0; :[font = postscript; PostScript; formatAsPostScript; output; inactive; pageBreakBelow; dontNoPageBreakBelow; preserveAspect; pictureLeft = 0; pictureWidth = 117; pictureHeight = 27; ] %! %%Creator: Mathematica %%AspectRatio: 0.230769 200 dict begin MathPictureStart /Courier findfont 10 scalefont setfont % Scaling calculations 0 1 0 1 [ [ 0 0 0 0 ] [ 117 27 0 0 ] ] MathScale % Start of Graphics %!PS-Adobe-2.0 EPSF-2.0 %%Title: %%Creator: EquationBuilder %%CreationDate: Sun Aug 2 08:27:43 1992 %%For: gray %%DocumentFonts: (atend) %%Pages: 0 1 %%BoundingBox: 0 0 117 27 %%Translation: 1 8.399412 115.85466 0 %%EndComments %%BeginDocument: /usr/lib/NextStep/printPackage.ps %!PS-Adobe-2.0 %%Title: Printing Package %%EndComments %Version: 2.0 %Copyright: 1988, NeXT, Inc. /__NXdef{1 index where{pop pop pop}{def}ifelse}bind def /__NXbdef{1 index where{pop pop pop}{bind def}ifelse}bind def /UserObjects 10 array __NXdef /defineuserobject{ exch dup 1 add dup UserObjects length gt{ array dup 0 UserObjects putinterval /UserObjects exch def }{pop}ifelse UserObjects exch 3 -1 roll put }__NXbdef /undefineuserobject{UserObjects exch null put}__NXbdef /execuserobject{UserObjects exch get exec}__NXbdef /__NXRectPath{4 2 roll moveto 1 index 0 rlineto 0 exch rlineto neg 0 rlineto closepath}__NXbdef /__NXProcessRectArgs{ 1 index type /arraytype eq{ exch 0 4 2 index length 1 sub{ dup 3 add 1 exch{1 index exch get exch}for 5 1 roll 5 index exec }for pop pop }{exec}ifelse }__NXbdef /rectfill{gsave newpath {__NXRectPath fill} __NXProcessRectArgs grestore}__NXbdef /rectclip{newpath {__NXRectPath} __NXProcessRectArgs clip newpath}__NXbdef /rectstroke{ gsave newpath dup type /arraytype eq{dup length 6 eq}{false}ifelse{ {gsave __NXRectPath null concat stroke grestore} dup length array cvx copy dup 2 4 -1 roll put __NXProcessRectArgs }{{__NXRectPath stroke} __NXProcessRectArgs}ifelse grestore }__NXbdef /xyshow{ 0 1 3 index length 1 sub{ currentpoint 4 index 3 index 1 getinterval show 3 index 3 index 2 mul 1 add get add exch 3 index 3 index 2 mul get add exch moveto pop }for pop pop }__NXbdef /xshow{ 0 1 3 index length 1 sub{ currentpoint 4 index 3 index 1 getinterval show exch 3 index 3 index get add exch moveto pop }for pop pop }__NXbdef /yshow{ 0 1 3 index length 1 sub{ currentpoint 4 index 3 index 1 getinterval show 3 index 3 index get add moveto pop }for pop pop }__NXbdef /arct{arcto pop pop pop pop}__NXbdef /setbbox{pop pop pop pop}__NXbdef /ucache{}__NXbdef /ucachestatus{mark 0 0 0 0 0}__NXbdef /setucacheparams{cleartomark}__NXbdef /uappend{systemdict begin cvx exec end}__NXbdef /ueofill{gsave newpath uappend eofill grestore}__NXbdef /ufill{gsave newpath uappend fill grestore}__NXbdef /ustroke{ gsave newpath dup length 6 eq {exch uappend concat}{uappend}ifelse stroke grestore }__NXbdef /__NXustrokepathMatrix dup where {pop pop}{matrix def}ifelse /ustrokepath{ newpath dup length 6 eq{ exch uappend __NXustrokepathMatrix currentmatrix exch concat strokepath setmatrix }{uappend strokepath}ifelse } __NXbdef /upath{ [exch {/ucache cvx}if pathbbox /setbbox cvx {/moveto cvx}{/lineto cvx}{/curveto cvx}{/closepath cvx}pathforall]cvx } __NXbdef /setstrokeadjust{pop}__NXbdef /currentstrokeadjust{false}__NXbdef /selectfont{exch findfont exch dup type /arraytype eq {makefont}{scalefont}ifelse setfont}__NXbdef /_NXCombineArrays{ counttomark dup 2 add index dup length 3 -1 roll { 2 index length sub dup 4 1 roll 1 index exch 4 -1 roll putinterval exch }repeat pop pop pop }__NXbdef /setcmykcolor{ 1.0 exch sub dup dup 6 -1 roll sub dup 0 lt{pop 0}if 5 1 roll 4 -1 roll sub dup 0 lt{pop 0}if 3 1 roll exch sub dup 0 lt{pop 0}if setrgbcolor }__NXbdef /currentcmykcolor{currentrgbcolor 3{1.0 exch sub 3 1 roll}repeat 0}__NXbdef /flushgraphics{}def /setwindowtype{pop pop}def /currentwindowtype{pop 0}def /setalpha{pop}def /currentalpha{1.0}def /hidecursor{}def /obscurecursor{}def /revealcursor{}def /setcursor{4 {pop}repeat}bind def /showcursor{}def /NextStepEncoding where not{ /NextStepEncoding StandardEncoding 256 array copy def 0 [129/Agrave/Aacute/Acircumflex/Atilde/Adieresis/Aring/Ccedilla/Egrave /Eacute/Ecircumflex/Edieresis/Igrave/Iacute/Icircumflex/Idieresis /Eth/Ntilde/Ograve/Oacute/Ocircumflex/Otilde/Odieresis/Ugrave/Uacute /Ucircumflex/Udieresis/Yacute/Thorn/mu/multiply/divide/copyright 176/registered 181/brokenbar 190/logicalnot 192/onesuperior 201/twosuperior 204/threesuperior 209/plusminus/onequarter/onehalf/threequarters/agrave /aacute/acircumflex/atilde/adieresis/aring/ccedilla/egrave/eacute /ecircumflex/edieresis/igrave 226/iacute 228/icircumflex/idieresis/eth /ntilde 236/ograve/oacute/ocircumflex/otilde/odieresis 242/ugrave/uacute /ucircumflex 246/udieresis/yacute 252/thorn/ydieresis] {dup type /nametype eq {NextStepEncoding 2 index 2 index put pop 1 add}{exch pop}ifelse }forall pop /NextStepEncoding NextStepEncoding readonly def /_NXfstr 128 string dup 0 (_NX) putinterval def /findfont{ % Because we can never let NextStepEncoding get into % SharedFontDirectory, we cannot reencode a font to NextStepEncoding % if we are in shared mode. So if currentshared is true, % we call the normal findfont and return that /currentshared where {pop currentshared} {false} ifelse {//findfont exec} {dup _NXfstr 3 125 getinterval cvs length 3 add _NXfstr 0 3 -1 roll getinterval cvn exch FontDirectory 2 index known {pop FontDirectory exch get} {//findfont exec dup /Encoding get StandardEncoding eq { dup length dict exch {1 index /FID ne {2 index 3 1 roll put}{pop pop}ifelse}forall dup /Encoding NextStepEncoding put definefont }{exch pop} ifelse }ifelse }ifelse }bind def }{pop}ifelse /_NXProcArray 5 array __NXdef /_NXChannels 0 __NXdef /_NXTotalBytes 0 __NXdef /_NXDoImageOp{ 1 index{dup}{1}ifelse /_NXChannels exch store _NXChannels 2 add 2 roll _NXProcArray 0 _NXChannels getinterval astore pop 5 index 4 index mul 2 index{1 sub 8 idiv 1 add mul}{mul 1 sub 8 idiv 1 add}ifelse 4 index mul /_NXTotalBytes exch store pop exch pop gsave matrix invertmatrix concat 0.5 setgray 0 0 4 2 roll rectfill grestore { 0 1 _NXChannels 1 sub{ _NXProcArray exch get exec length _NXTotalBytes exch sub /_NXTotalBytes exch store}for _NXTotalBytes 0 le{exit}if }loop /_NXProcArray 5 array def }__NXbdef /colorimage{_NXDoImageOp}__NXbdef /alphaimage{1 add _NXDoImageOp}def %%EndDocument % EquationBuilder(TM) Expression % +.\Jr(=:/*V[StRFQAYmdjEEA;1nuN+[co.OZ7L#cq3iK%3:</XJ@:d#6;^G;Td47dud<A3@ % 5%MGToUOR5&d`'TuEDU)B-tbR".RT[!a8!!!E7U+Y[Q+P;kJj@P.-O?%B4'FH)c:eX1]TM0' % o;,/#l!%\^3YU,#41L183Q(&Hf"9NTgc2h<D&k0$!hQXOnBCNsA60Psq6cY@ca,g65@$1_&! % 4;jU/6LXb#*:He'hl`u7Z?9DM[q$LJgQEre;Q=PcK)GJ.)(?;#YbXU^a50F<)<u'U==-"L-e % [-R)[AgfQ:XgVG5,i?W+Ob"#+W832$@oRO:<["\tF>AY;\_7,>Etli;A3jpl,qnkZT-hgV2G % 5,%D^i@uAs$4,%LCO&/M8q,=X#9P>tp]k1p/L4C)KC1n$*I#N"P+4!=3nW>5s4)=#Lt@X-+X % *bR31YLI`jaoV`iYDo/1C'*?\B\"(r-I\115bTkAA_2H6EVleJ9of7YiCQ-)4fJ9@i#HIZH2 % cUQ4p<BO$IK>aXW=)G8Z@LQH6'@#SC>B-/:!8topP.ag<3RrQ_&U5W[s@O%3YIKR_W/kR>LR % Xk)\M]mpp2bn:MQKdV+I>'RI*HKBo`bUIN#2)b-9$1FHR=Kn)>'g\.#6P8Li8"H6UsXoh/P* % -K(iFR'6*.E<d=Mp]Oc]be@Bbm6(u?^W/$6Um5mA`g";%1JdKT^T.+A<+m3Mm'1e&%H!5OFU % L&_2Rx % EndExpression N 578 242 E c4 S f1aa R fada18bd gsave /__NXbasematrix matrix currentmatrix def grestore %%EndProlog %%BeginSetup %%EndSetup 0 0 116.85466 26.395412 rectclip 0 0 116.85466 26.395412 rectclip 0 setgray 1 setalpha 0.48 setlinewidth 80.85067 12.399412 moveto 115.854668 12.399412 lineto stroke 0 setgray 1 setalpha /Times-Roman findfont 12 scalefont 47 exch defineuserobject 47 execuserobject setfont 95.352669 1.168 moveto (33sin22sinsin2) [8.001999 16.031412 -16.003998 1.907349e-06 4.667999 0 3.335999 0 -14.003998 -1.907349e-06 -24.421337 -7.799999 -16.003998 0 4.667999 0 3.336002 0 -41.929337 0 4.668 0 3.335999 0 -14.004 0 0 0 0 0] xyshow /Times-BoldItalic findfont 12 scalefont 44 exch defineuserobject 44 execuserobject setfont 109.354668 17.199411 moveto (xxx) [-46.425335 -7.799999 -39.925331 0 0 0 0 0] xyshow /Symbol findfont 12 scalefont 45 exch defineuserobject 45 execuserobject setfont 71.596001 9.399412 moveto (+-) [-39.925335 0 0 0 0 0] xyshow %%Trailer %%DocumentFonts: Times-BoldItalic %%+ Times-Roman %%+ Symbol %End of Graphics MathPictureEnd end :[font = text; inactive; preserveAspect; ] To graph the function f[3,x] over the interval -3p < x < 3p use ;[s] 9:0,0;31,1;37,2;59,3;60,4;63,5;64,6;68,7;69,8;75,-1; 9:1,11,8,Times,0,12,0,0,0;1,10,8,Courier,0,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,0,0,Symbol,0,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Times,2,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,0,0,Symbol,0,12,0,0,0;1,11,8,Times,0,12,0,0,0; :[font = input; preserveAspect; startGroup; ] Plot[f[3,x],{x,-3Pi,3Pi}]; :[font = postscript; PostScript; formatAsPostScript; output; inactive; pageBreakBelow; dontNoPageBreakBelow; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174; endGroup; ] %! %%Creator: Mathematica %%AspectRatio: 0.61803 MathPictureStart /Courier findfont 10 scalefont setfont % Scaling calculations 0.5 0.050525 0.309017 0.101989 [ [(-7.5)] 0.12106 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UserObjects exch 3 -1 roll put }__NXbdef /undefineuserobject{UserObjects exch null put}__NXbdef /execuserobject{UserObjects exch get exec}__NXbdef /__NXRectPath{4 2 roll moveto 1 index 0 rlineto 0 exch rlineto neg 0 rlineto closepath}__NXbdef /__NXProcessRectArgs{ 1 index type /arraytype eq{ exch 0 4 2 index length 1 sub{ dup 3 add 1 exch{1 index exch get exch}for 5 1 roll 5 index exec }for pop pop }{exec}ifelse }__NXbdef /rectfill{gsave newpath {__NXRectPath fill} __NXProcessRectArgs grestore}__NXbdef /rectclip{newpath {__NXRectPath} __NXProcessRectArgs clip newpath}__NXbdef /rectstroke{ gsave newpath dup type /arraytype eq{dup length 6 eq}{false}ifelse{ {gsave __NXRectPath null concat stroke grestore} dup length array cvx copy dup 2 4 -1 roll put __NXProcessRectArgs }{{__NXRectPath stroke} __NXProcessRectArgs}ifelse grestore }__NXbdef /xyshow{ 0 1 3 index length 1 sub{ currentpoint 4 index 3 index 1 getinterval show 3 index 3 index 2 mul 1 add get add exch 3 index 3 index 2 mul get add exch moveto pop }for pop pop }__NXbdef /xshow{ 0 1 3 index length 1 sub{ currentpoint 4 index 3 index 1 getinterval show exch 3 index 3 index get add exch moveto pop }for pop pop }__NXbdef /yshow{ 0 1 3 index length 1 sub{ currentpoint 4 index 3 index 1 getinterval show 3 index 3 index get add moveto pop }for pop pop }__NXbdef /arct{arcto pop pop pop pop}__NXbdef /setbbox{pop pop pop pop}__NXbdef /ucache{}__NXbdef /ucachestatus{mark 0 0 0 0 0}__NXbdef /setucacheparams{cleartomark}__NXbdef /uappend{systemdict begin cvx exec end}__NXbdef /ueofill{gsave newpath uappend eofill grestore}__NXbdef /ufill{gsave newpath uappend fill grestore}__NXbdef /ustroke{ gsave newpath dup length 6 eq {exch uappend concat}{uappend}ifelse stroke grestore }__NXbdef /__NXustrokepathMatrix dup where {pop pop}{matrix def}ifelse /ustrokepath{ newpath dup length 6 eq{ exch uappend __NXustrokepathMatrix currentmatrix exch concat strokepath setmatrix }{uappend strokepath}ifelse } __NXbdef /upath{ [exch {/ucache cvx}if pathbbox /setbbox cvx {/moveto cvx}{/lineto cvx}{/curveto cvx}{/closepath cvx}pathforall]cvx } __NXbdef /setstrokeadjust{pop}__NXbdef /currentstrokeadjust{false}__NXbdef /selectfont{exch findfont exch dup type /arraytype eq {makefont}{scalefont}ifelse setfont}__NXbdef /_NXCombineArrays{ counttomark dup 2 add index dup length 3 -1 roll { 2 index length sub dup 4 1 roll 1 index exch 4 -1 roll putinterval exch }repeat pop pop pop }__NXbdef /setcmykcolor{ 1.0 exch sub dup dup 6 -1 roll sub dup 0 lt{pop 0}if 5 1 roll 4 -1 roll sub dup 0 lt{pop 0}if 3 1 roll exch sub dup 0 lt{pop 0}if setrgbcolor }__NXbdef /currentcmykcolor{currentrgbcolor 3{1.0 exch sub 3 1 roll}repeat 0}__NXbdef /flushgraphics{}def /setwindowtype{pop pop}def /currentwindowtype{pop 0}def /setalpha{pop}def /currentalpha{1.0}def /hidecursor{}def /obscurecursor{}def /revealcursor{}def /setcursor{4 {pop}repeat}bind def /showcursor{}def /NextStepEncoding where not{ /NextStepEncoding StandardEncoding 256 array copy def 0 [129/Agrave/Aacute/Acircumflex/Atilde/Adieresis/Aring/Ccedilla/Egrave /Eacute/Ecircumflex/Edieresis/Igrave/Iacute/Icircumflex/Idieresis /Eth/Ntilde/Ograve/Oacute/Ocircumflex/Otilde/Odieresis/Ugrave/Uacute /Ucircumflex/Udieresis/Yacute/Thorn/mu/multiply/divide/copyright 176/registered 181/brokenbar 190/logicalnot 192/onesuperior 201/twosuperior 204/threesuperior 209/plusminus/onequarter/onehalf/threequarters/agrave /aacute/acircumflex/atilde/adieresis/aring/ccedilla/egrave/eacute /ecircumflex/edieresis/igrave 226/iacute 228/icircumflex/idieresis/eth /ntilde 236/ograve/oacute/ocircumflex/otilde/odieresis 242/ugrave/uacute /ucircumflex 246/udieresis/yacute 252/thorn/ydieresis] {dup type /nametype eq {NextStepEncoding 2 index 2 index put pop 1 add}{exch pop}ifelse }forall pop /NextStepEncoding NextStepEncoding readonly def /_NXfstr 128 string dup 0 (_NX) putinterval def /findfont{ % Because we can never let NextStepEncoding get into % SharedFontDirectory, we cannot reencode a font to 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So if currentshared is true, % we call the normal findfont and return that /currentshared where {pop currentshared} {false} ifelse {//findfont exec} {dup _NXfstr 3 125 getinterval cvs length 3 add _NXfstr 0 3 -1 roll getinterval cvn exch FontDirectory 2 index known {pop FontDirectory exch get} {//findfont exec dup /Encoding get StandardEncoding eq { dup length dict exch {1 index /FID ne {2 index 3 1 roll put}{pop pop}ifelse}forall dup /Encoding NextStepEncoding put definefont }{exch pop} ifelse }ifelse }ifelse }bind def }{pop}ifelse /_NXProcArray 5 array __NXdef /_NXChannels 0 __NXdef /_NXTotalBytes 0 __NXdef /_NXDoImageOp{ 1 index{dup}{1}ifelse /_NXChannels exch store _NXChannels 2 add 2 roll _NXProcArray 0 _NXChannels getinterval astore pop 5 index 4 index mul 2 index{1 sub 8 idiv 1 add mul}{mul 1 sub 8 idiv 1 add}ifelse 4 index mul /_NXTotalBytes exch store pop exch pop gsave matrix invertmatrix concat 0.5 setgray 0 0 4 2 roll rectfill grestore { 0 1 _NXChannels 1 sub{ _NXProcArray exch get exec length _NXTotalBytes exch sub /_NXTotalBytes exch store}for _NXTotalBytes 0 le{exit}if }loop /_NXProcArray 5 array def }__NXbdef /colorimage{_NXDoImageOp}__NXbdef /alphaimage{1 add _NXDoImageOp}def %%EndDocument % EquationBuilder(TM) Expression % +.\Jr(=:/*V[StRFQAYmdjEEA;1nuN+[co.OZ7L#cq3iK%3:</XJ@:d#6;^G;Td47dud<A3@ % 5%MGToUOR5&d`'TuEDU)B-tbR".R+:nNM,VT6!0M%"UV$VD;,X_XRS2ihA"VLe`-6WLgVDQ[ % K-Oh*sQ3\5X&jqDA/d.K[>m^Q5J[c<T!&,[Tin#Mqf\EPQp`n<3FPaBu@nDhna*n=K8V8]^g % &Ph(W+@I]8/K#JVW;G9-88oJL(Oh['1U:3`L@o]lIuWp?:"^H/d3!JP0!\R-jBZKMU3Y:KKd % MTD;,W$h7Wh'2cVtL%-/*A&1PLD9q*t]?G<9Mg[T)H[0'6:lT+cN<6Y)##Jf*go]cJH(ig,B % IOj7T$51.)oEbf=%tK1>+G'q%O?s:[.#?eZfFbtc8o/1ZOW3>3Z5e28bm-iAp(9W^E'8`g6% % o^a+\*A/D)bjIP+5UC!:+e$)+=rr+U8m!\C"J$1Hn89Es&L&6[Qs9G"[5\d79pQ-K578E_T0 % s,`)$a8*,R>a\ZCI7C3*95TN%sASk^F=lP:8J.fpBGO-\dFJTZrM(Xtd,)J"u\`_V->XhlRQ % h\g5VjF5*_cqg%#SH-ZBpo:'Lf<.Xg^5Sr;:(5X\X"(K+5Z4=TTVQ]l#)ni.9n=CC%@Xk+:\ % AZWn4=E+.fW)Zj(QT4I0cVR(c]C.tM:m/=VCbHK$B3mV:W=T_oJjTr!]8a8fpr.fb9VUf&[\ % W=VtNA*#Bl^o=V#G)j/3Asn[^f_-3`%ERN!fY"WK.^!:M0Gd!bm7J!%]S&!1fKAf7$>'8i@( % @u)c5$Y%ZhEcbmBq1R*'f#V##Lu6$ndp?+:^^d@99pL))/=S\sa"I_`[WJZc+,'@WNAZ2h8` % -oHVJuHhCXd!#lTGYId?^RUlL6)b\8rW'Us"ZoZAso4(h]bc904(4E*/&P-qm,R[06!7,.J( % Vuk9^]!]C!YVFSaX?-)i.DUK!lPJo9EFAl#mGTDOd9<Z0:2j#n\uPY:&tTYAKY;he6dg%.&o % 1[)"WS`BA%R"M\i8ua7b@oEF0D/0LQq%bjgFdW6r,=im.+[#q-(X[Du*l0T^eceLa%?G:L!C % KjnK#;&oe[[un<7JWZ&t*FX%K7Wf&+5CN3FML7>_5l`Wgq3MX(d25/g!WXVI(]j[;x % EndExpression N 914 392 E 63 S 18793 R 3619c044 gsave /__NXbasematrix matrix currentmatrix def grestore %%EndProlog %%BeginSetup /EqBFont 18 dict dup begin /FontType 3 def /PaintType 0 def /FontMatrix [0.001 0 0 0.001 0 0] def /FontBBox [15 -2950 1360 1045] def /Encoding [/.notdef /rad1 /rad2 /rad3 /rad4 /rad5 /radext /int /pathint /sum /prod /cdot /vdot /ddot /hbar /bigcap /bigcup /parallel] def /Metrics 18 dict dup begin /.notdef 0 def /rad1 1000 def /rad2 1000 def /rad3 1000 def /rad4 1000 def /rad5 1048 def /radext 1048 def /int 500 def /pathint 500 def /sum 700 def /prod 700 def /cdot 1000 def /vdot 550 def /ddot 1000 def /hbar 500 def /bigcap 700 def /bigcup 700 def /parallel 500 def end def /BBox 18 dict dup begin /.notdef [0 0 0 0] def /rad1 [110 -1150 1020 0] def /rad2 [110 -1750 1020 0] def /rad3 [110 -2350 1020 0] def /rad4 [110 -2950 1020 0] def /rad5 [128 -2890 740 0] def /radext [700 -600 740 0] def /int [20 0 425 1045] def /pathint [20 0 425 1045] def /sum [15 5 640 715] def /prod [15 0 670 715] def /cdot [110 215 890 325] def /vdot [165 -5 280 770] def /ddot [110 -5 890 770] def /hbar [20 -10 480 685] def /bigcap [0 0 680 720] def /bigcup [0 0 680 720] def /parallel [65 -175 400 675] def end def /CharacterDefs 18 dict dup begin /.notdef { } def /rad1 { 980 0 moveto 464 -1048 lineto 248 -560 lineto 110 -656 lineto 125 -675 lineto 200 -624 lineto 420 -1150 lineto 460 -1150 lineto 1020 0 lineto 980 0 lineto closepath } def /rad2 { 980 0 moveto 460 -1600 lineto 245 -864 lineto 110 -1000 lineto 128 -1023 lineto 200 -950 lineto 424 -1750 lineto 456 -1750 lineto 1020 0 lineto 980 0 lineto closepath } def /rad3 { 980 0 moveto 464 -2144 lineto 248 -1160 lineto 110 -1368 lineto 130 -1390 lineto 192 -1288 lineto 424 -2350 lineto 464 -2350 lineto 1020 0 lineto 980 0 lineto closepath } def /rad4 { 980 0 moveto 880 -488 lineto 672 -1600 lineto 464 -2696 lineto 248 -1464 lineto 110 -1728 lineto 130 -1750 lineto 192 -1616 lineto 424 -2950 lineto 464 -2950 lineto 768 -1312 lineto 1020 0 lineto 980 0 lineto closepath } def /rad5 { 700 0 moveto 700 -2628 lineto 248 -1284 lineto 128 -1564 lineto 150 -1580 lineto 215 -1425 lineto 688 -2890 lineto 740 -2890 lineto 740 0 lineto 700 0 lineto closepath } def /radext { 700 0 moveto 700 -600 lineto 740 -600 lineto 740 0 lineto 700 0 lineto closepath } def /int { 245 390 moveto 225 235 195 145 180 100 curveto 165 60 150 50 140 40 curveto 135 30 85 0 50 20 curveto 35 25 25 45 25 55 curveto 20 80 30 100 55 110 curveto 80 125 93 110 110 90 curveto 135 60 155 135 160 195 curveto 165 270 170 340 190 510 curveto 205 650 lineto 220 810 230 890 265 960 curveto 275 980 295 1015 320 1030 curveto 350 1045 380 1030 395 1020 curveto 410 1010 425 980 400 955 curveto 385 940 347 925 330 950 curveto 310 970 290 955 290 940 curveto 280 900 285 745 270 610 curveto 245 390 lineto closepath } def /pathint { 245 390 moveto 225 235 195 145 180 100 curveto 165 60 150 50 140 40 curveto 135 30 85 0 50 20 curveto 35 25 25 45 25 55 curveto 20 80 30 100 55 110 curveto 80 125 93 110 110 90 curveto 135 60 155 135 160 195 curveto 165 270 170 340 190 510 curveto 205 650 lineto 220 810 230 890 265 960 curveto 275 980 295 1015 320 1030 curveto 350 1045 380 1030 395 1020 curveto 410 1010 425 980 400 955 curveto 385 940 347 925 330 950 curveto 310 970 290 955 290 940 curveto 280 900 285 745 270 610 curveto 245 390 lineto fill 20 setlinewidth 220 400 moveto 290 400 340 455 340 500 curveto 340 585 290 640 220 640 curveto 155 640 100 585 100 500 curveto 100 455 155 400 220 400 curveto stroke } def /sum { 330 335 moveto 20 715 lineto 575 715 lineto 600 545 lineto 575 535 lineto 550 625 525 685 395 685 curveto 160 685 lineto 411 370 lineto 160 90 lineto 470 90 lineto 565 90 590 140 610 210 curveto 640 200 lineto 585 5 lineto 15 5 lineto 330 335 lineto closepath } def /prod { 590 90 moveto 590 30 620 20 670 20 curveto 670 0 lineto 410 0 lineto 410 20 lineto 460 20 495 30 495 90 curveto 495 670 lineto 190 670 lineto 190 90 lineto 190 30 225 20 275 20 curveto 275 0 lineto 15 0 lineto 15 20 lineto 65 20 100 30 100 90 curveto 100 615 lineto 100 690 40 700 20 700 curveto 20 715 lineto 670 715 lineto 670 700 lineto 645 700 590 690 590 615 curveto 590 90 lineto closepath } def /cdot { 110 270 moveto 110 300 135 325 165 325 curveto 195 325 220 300 220 270 curveto 220 235 185 215 165 215 curveto 145 215 110 235 110 270 curveto fill 445 270 moveto 445 300 470 325 500 325 curveto 530 325 555 300 555 270 curveto 555 235 520 215 500 215 curveto 480 215 445 235 445 270 curveto fill 775 270 moveto 775 300 800 325 830 325 curveto 860 325 890 300 890 270 curveto 890 235 850 215 830 215 curveto 810 215 775 235 775 270 curveto fill } def /vdot { 220 770 moveto 250 770 280 750 280 720 curveto 280 680 250 660 220 660 curveto 185 660 165 695 165 720 curveto 165 740 185 770 220 770 curveto fill 220 440 moveto 250 440 280 415 280 385 curveto 280 355 250 330 220 330 curveto 185 330 165 365 165 385 curveto 165 405 185 440 220 440 curveto fill 220 105 moveto 250 105 280 80 280 50 curveto 280 20 250 -5 220 -5 curveto 185 -5 165 30 165 50 curveto 165 70 185 105 220 105 curveto fill } def /ddot { 165 720 moveto 205 720 220 700 220 665 curveto 220 630 195 610 165 610 curveto 130 620 110 645 110 665 curveto 110 690 130 720 165 720 curveto fill 500 440 moveto 530 440 555 415 555 385 curveto 555 355 530 330 500 330 curveto 465 330 445 365 445 385 curveto 445 405 465 440 500 440 curveto fill 830 155 moveto 860 155 890 130 890 100 curveto 890 70 860 45 830 45 curveto 795 45 775 80 775 100 curveto 775 120 795 155 830 155 curveto fill } def /hbar { 480 105 moveto 440 55 400 -10 335 -10 curveto 300 -10 290 10 290 35 curveto 290 65 370 350 370 365 curveto 370 380 355 390 340 390 curveto 290 390 205 275 185 250 curveto 140 180 125 125 95 0 curveto 20 0 lineto 150 500 lineto 160 545 180 600 180 615 curveto 180 625 170 640 135 640 curveto 110 640 lineto 100 655 lineto 165 660 215 670 270 685 curveto 275 680 lineto 155 240 lineto 155 235 lineto 190 290 290 440 380 440 curveto 400 440 450 435 450 370 curveto 450 335 365 80 365 55 curveto 365 45 375 40 385 40 curveto 405 40 425 70 465 115 curveto 480 105 lineto fill 55 545 moveto 55 520 lineto 310 560 lineto 310 585 lineto 55 545 lineto fill } def /bigcap { 615 400 moveto 615 575 500 655 340 655 curveto 180 655 70 575 70 400 curveto 70 0 lineto 5 0 lineto 5 400 lineto 5 610 140 720 340 720 curveto 545 720 680 610 680 400 curveto 680 0 lineto 615 0 lineto 615 400 lineto closepath } def /bigcup { 65 320 moveto 65 145 180 65 340 65 curveto 500 65 615 145 615 320 curveto 615 720 lineto 680 720 lineto 680 320 lineto 680 110 540 0 340 0 curveto 140 0 0 110 0 320 curveto 0 720 lineto 65 720 lineto 65 320 lineto closepath } def /parallel { 135 675 moveto 135 -175 lineto 65 -175 lineto 65 675 lineto 135 675 lineto fill 335 675 moveto 335 -175 lineto 265 -175 lineto 265 675 lineto 335 675 lineto fill 400 675 moveto } def end def /BuildChar { 0 begin /char exch def /fontdict exch def /charname fontdict /Encoding get char get def fontdict begin Metrics charname get 0 BBox charname get aload pop setcachedevice CharacterDefs charname get exec eofill end end } def /BuildChar load 0 3 dict put end definefont pop %%EndSetup 0 0 161.594681 35.889019 rectclip 0 0 161.594681 35.889019 rectclip 0 setgray 1 setalpha 1 1 scale /Symbol findfont 12 scalefont 45 exch defineuserobject 45 execuserobject setfont 11.56 16.190992 moveto (\() [0 0] xyshow 1 1 scale 1 1 scale 45 execuserobject setfont 23.200001 16.184992 moveto (\)) [0 0] xyshow 1 1 scale 0 setgray 1 setalpha 0.48 setlinewidth 40.978668 19.082993 moveto 54.26667 19.082993 lineto stroke 0 setgray 1 setalpha /EqBFont 24 selectfont 65.828003 10.442991 moveto <09> [0 0] xyshow 0 setgray 1 setalpha 1 1 scale 45 execuserobject setfont 93.616005 23.990992 moveto (\() [0 0] xyshow 1 1 scale 1 1 scale 45 execuserobject setfont 114.510666 23.984991 moveto (\)) [0 0] xyshow 1 1 scale 0 setgray 1 setalpha 0.48 setlinewidth 86.552002 19.082993 moveto 125.234673 19.082993 lineto stroke 0 setgray 1 setalpha 1 2.747151 scale 45 execuserobject setfont 81.752007 4.054466 moveto (\() [0 0] xyshow 1 0.364013 scale 1 2.743975 scale 45 execuserobject setfont 126.074677 4.056506 moveto (\)) [0 0] xyshow 1 0.364435 scale 0 setgray 1 setalpha /Times-BoldItalic findfont 12 scalefont 44 exch defineuserobject 44 execuserobject setfont 154.594681 16.082991 moveto (xkkxf) [-6 1.907349e-06 -48.551346 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grestore gsave 0.001 setlinewidth 0.5 0.52821 moveto 0.50375 0.52821 lineto stroke grestore gsave 0.001 setlinewidth 0.5 0.58118 moveto 0.50375 0.58118 lineto stroke grestore gsave 0.001 setlinewidth 0.5 0.60767 moveto 0.50375 0.60767 lineto stroke grestore gsave 0.002 setlinewidth 0.5 0 moveto 0.5 0.61803 lineto stroke grestore grestore 0 0 moveto 1 0 lineto 1 0.61803 lineto 0 0.61803 lineto closepath clip newpath gsave gsave 0.004 setlinewidth 0.02381 0.60332 moveto 0.06349 0.40928 lineto 0.08333 0.27015 lineto 0.09325 0.21695 lineto 0.10317 0.17658 lineto 0.1131 0.1463 lineto 0.12302 0.1218 lineto 0.14286 0.07632 lineto 0.15278 0.05365 lineto 0.1627 0.03365 lineto 0.16766 0.02572 lineto 0.17014 0.02246 lineto 0.17262 0.01972 lineto 0.1751 0.01756 lineto 0.17634 0.01669 lineto 0.17758 0.01599 lineto 0.17882 0.01543 lineto 0.18006 0.01503 lineto 0.1813 0.01479 lineto 0.18254 0.01472 lineto 0.18378 0.01479 lineto 0.18502 0.01503 lineto 0.18626 0.01543 lineto 0.1875 0.01599 lineto 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0.02246 lineto 0.49008 0.01972 lineto 0.49256 0.01756 lineto 0.4938 0.01669 lineto 0.49504 0.01599 lineto 0.49628 0.01543 lineto 0.49752 0.01503 lineto 0.49876 0.01479 lineto 0.5 0.01472 lineto 0.50124 0.01479 lineto 0.50248 0.01503 lineto 0.50372 0.01543 lineto 0.50496 0.01599 lineto 0.50744 0.01756 lineto 0.50992 0.01972 lineto 0.51488 0.02572 lineto 0.51984 0.03365 lineto 0.53968 0.07632 lineto 0.55952 0.1218 lineto 0.56944 0.1463 lineto 0.57937 0.17658 lineto 0.58929 0.21695 lineto 0.59921 0.27015 lineto 0.61905 0.40928 lineto 0.62897 0.48263 lineto 0.63393 0.51607 lineto 0.63889 0.54559 lineto 0.64385 0.56999 lineto 0.64881 0.58823 lineto 0.65129 0.59478 lineto 0.65253 0.59737 lineto 0.65377 0.5995 lineto 0.65501 0.60117 lineto Mistroke 0.65625 0.60236 lineto 0.65749 0.60308 lineto 0.65873 0.60332 lineto 0.65997 0.60308 lineto 0.66121 0.60236 lineto 0.66245 0.60117 lineto 0.66369 0.5995 lineto 0.66617 0.59478 lineto 0.66865 0.58823 lineto 0.67361 0.56999 lineto 0.67857 0.54559 lineto 0.69841 0.40928 lineto 0.71825 0.27015 lineto 0.72817 0.21695 lineto 0.7381 0.17658 lineto 0.74802 0.1463 lineto 0.75794 0.1218 lineto 0.77778 0.07632 lineto 0.7877 0.05365 lineto 0.79762 0.03365 lineto 0.80258 0.02572 lineto 0.80506 0.02246 lineto 0.80754 0.01972 lineto 0.81002 0.01756 lineto 0.81126 0.01669 lineto 0.8125 0.01599 lineto 0.81374 0.01543 lineto 0.81498 0.01503 lineto 0.81622 0.01479 lineto 0.81746 0.01472 lineto 0.8187 0.01479 lineto 0.81994 0.01503 lineto 0.82118 0.01543 lineto 0.82242 0.01599 lineto 0.8249 0.01756 lineto 0.82738 0.01972 lineto 0.83234 0.02572 lineto 0.8373 0.03365 lineto 0.85714 0.07632 lineto 0.87698 0.1218 lineto 0.8869 0.1463 lineto 0.89683 0.17658 lineto 0.90675 0.21695 lineto 0.91667 0.27015 lineto 0.93651 0.40928 lineto 0.94643 0.48263 lineto 0.95139 0.51607 lineto 0.95635 0.54559 lineto 0.96131 0.56999 lineto 0.96379 0.57994 lineto Mistroke 0.96627 0.58823 lineto 0.96875 0.59478 lineto 0.96999 0.59737 lineto 0.97123 0.5995 lineto 0.97247 0.60117 lineto 0.97371 0.60236 lineto 0.97495 0.60308 lineto 0.97619 0.60332 lineto Mfstroke grestore grestore % End of Graphics MathPictureEnd :[font = subsubsection; inactive; preserveAspect; startGroup; ] #2 :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 0; pictureWidth = 185; pictureHeight = 37; endGroup; ] %! %%Creator: Mathematica %%AspectRatio: 0.2 200 dict begin MathPictureStart /Courier findfont 10 scalefont setfont % Scaling calculations 0 1 0 1 [ [ 0 0 0 0 ] [ 185 37 0 0 ] ] MathScale % Start of Graphics %!PS-Adobe-2.0 EPSF-2.0 %%Title: %%Creator: EquationBuilder %%CreationDate: Sun Aug 2 08:40:50 1992 %%For: gray %%DocumentFonts: (atend) %%Pages: 0 1 %%BoundingBox: 0 0 185 37 %%Translation: 1 15.082992 183.544006 0 %%EndComments %%BeginDocument: /usr/lib/NextStep/printPackage.ps %!PS-Adobe-2.0 %%Title: Printing Package %%EndComments %Version: 2.0 %Copyright: 1988, NeXT, Inc. /__NXdef{1 index where{pop pop pop}{def}ifelse}bind def /__NXbdef{1 index where{pop pop pop}{bind def}ifelse}bind def /UserObjects 10 array __NXdef /defineuserobject{ exch dup 1 add dup UserObjects length gt{ array dup 0 UserObjects putinterval /UserObjects exch def }{pop}ifelse UserObjects exch 3 -1 roll put }__NXbdef /undefineuserobject{UserObjects exch null put}__NXbdef /execuserobject{UserObjects exch get exec}__NXbdef /__NXRectPath{4 2 roll moveto 1 index 0 rlineto 0 exch rlineto neg 0 rlineto closepath}__NXbdef /__NXProcessRectArgs{ 1 index type /arraytype eq{ exch 0 4 2 index length 1 sub{ dup 3 add 1 exch{1 index exch get exch}for 5 1 roll 5 index exec }for pop pop }{exec}ifelse }__NXbdef /rectfill{gsave newpath {__NXRectPath fill} __NXProcessRectArgs grestore}__NXbdef /rectclip{newpath {__NXRectPath} __NXProcessRectArgs clip newpath}__NXbdef /rectstroke{ gsave newpath dup type /arraytype eq{dup length 6 eq}{false}ifelse{ {gsave __NXRectPath null concat stroke grestore} dup length array cvx copy dup 2 4 -1 roll put __NXProcessRectArgs }{{__NXRectPath stroke} __NXProcessRectArgs}ifelse grestore }__NXbdef /xyshow{ 0 1 3 index length 1 sub{ currentpoint 4 index 3 index 1 getinterval show 3 index 3 index 2 mul 1 add get add exch 3 index 3 index 2 mul get add exch moveto pop }for pop pop }__NXbdef /xshow{ 0 1 3 index length 1 sub{ currentpoint 4 index 3 index 1 getinterval show exch 3 index 3 index get add exch moveto pop }for pop pop }__NXbdef /yshow{ 0 1 3 index length 1 sub{ currentpoint 4 index 3 index 1 getinterval show 3 index 3 index get add moveto pop }for pop pop }__NXbdef /arct{arcto pop pop pop pop}__NXbdef /setbbox{pop pop pop pop}__NXbdef /ucache{}__NXbdef /ucachestatus{mark 0 0 0 0 0}__NXbdef /setucacheparams{cleartomark}__NXbdef /uappend{systemdict begin cvx exec end}__NXbdef /ueofill{gsave newpath uappend eofill grestore}__NXbdef /ufill{gsave newpath uappend fill grestore}__NXbdef /ustroke{ gsave newpath dup length 6 eq {exch uappend concat}{uappend}ifelse stroke grestore }__NXbdef /__NXustrokepathMatrix dup where {pop pop}{matrix def}ifelse /ustrokepath{ newpath dup length 6 eq{ exch uappend __NXustrokepathMatrix currentmatrix exch concat strokepath setmatrix }{uappend strokepath}ifelse } __NXbdef /upath{ [exch {/ucache cvx}if pathbbox /setbbox cvx {/moveto cvx}{/lineto cvx}{/curveto cvx}{/closepath cvx}pathforall]cvx } __NXbdef /setstrokeadjust{pop}__NXbdef /currentstrokeadjust{false}__NXbdef /selectfont{exch findfont exch dup type /arraytype eq {makefont}{scalefont}ifelse setfont}__NXbdef /_NXCombineArrays{ counttomark dup 2 add index dup length 3 -1 roll { 2 index length sub dup 4 1 roll 1 index exch 4 -1 roll putinterval exch }repeat pop pop pop }__NXbdef /setcmykcolor{ 1.0 exch sub dup dup 6 -1 roll sub dup 0 lt{pop 0}if 5 1 roll 4 -1 roll sub dup 0 lt{pop 0}if 3 1 roll exch sub dup 0 lt{pop 0}if setrgbcolor }__NXbdef /currentcmykcolor{currentrgbcolor 3{1.0 exch sub 3 1 roll}repeat 0}__NXbdef /flushgraphics{}def /setwindowtype{pop pop}def /currentwindowtype{pop 0}def /setalpha{pop}def /currentalpha{1.0}def /hidecursor{}def /obscurecursor{}def /revealcursor{}def /setcursor{4 {pop}repeat}bind def /showcursor{}def /NextStepEncoding where not{ /NextStepEncoding StandardEncoding 256 array copy def 0 [129/Agrave/Aacute/Acircumflex/Atilde/Adieresis/Aring/Ccedilla/Egrave /Eacute/Ecircumflex/Edieresis/Igrave/Iacute/Icircumflex/Idieresis /Eth/Ntilde/Ograve/Oacute/Ocircumflex/Otilde/Odieresis/Ugrave/Uacute /Ucircumflex/Udieresis/Yacute/Thorn/mu/multiply/divide/copyright 176/registered 181/brokenbar 190/logicalnot 192/onesuperior 201/twosuperior 204/threesuperior 209/plusminus/onequarter/onehalf/threequarters/agrave /aacute/acircumflex/atilde/adieresis/aring/ccedilla/egrave/eacute /ecircumflex/edieresis/igrave 226/iacute 228/icircumflex/idieresis/eth /ntilde 236/ograve/oacute/ocircumflex/otilde/odieresis 242/ugrave/uacute /ucircumflex 246/udieresis/yacute 252/thorn/ydieresis] {dup type /nametype eq {NextStepEncoding 2 index 2 index put pop 1 add}{exch pop}ifelse }forall pop /NextStepEncoding NextStepEncoding readonly def /_NXfstr 128 string dup 0 (_NX) putinterval def /findfont{ % Because we can never let NextStepEncoding get into % SharedFontDirectory, we cannot reencode a font to NextStepEncoding % if we are in shared mode. So if currentshared is true, % we call the normal findfont and return that /currentshared where {pop currentshared} {false} ifelse {//findfont exec} {dup _NXfstr 3 125 getinterval cvs length 3 add _NXfstr 0 3 -1 roll getinterval cvn exch FontDirectory 2 index known {pop FontDirectory exch get} {//findfont exec dup /Encoding get StandardEncoding eq { dup length dict exch {1 index /FID ne {2 index 3 1 roll put}{pop pop}ifelse}forall dup /Encoding NextStepEncoding put definefont }{exch pop} ifelse }ifelse }ifelse }bind def }{pop}ifelse /_NXProcArray 5 array __NXdef /_NXChannels 0 __NXdef /_NXTotalBytes 0 __NXdef /_NXDoImageOp{ 1 index{dup}{1}ifelse /_NXChannels exch store _NXChannels 2 add 2 roll _NXProcArray 0 _NXChannels getinterval astore pop 5 index 4 index mul 2 index{1 sub 8 idiv 1 add mul}{mul 1 sub 8 idiv 1 add}ifelse 4 index mul /_NXTotalBytes exch store pop exch pop gsave matrix invertmatrix concat 0.5 setgray 0 0 4 2 roll rectfill grestore { 0 1 _NXChannels 1 sub{ _NXProcArray exch get exec length _NXTotalBytes exch sub /_NXTotalBytes exch store}for _NXTotalBytes 0 le{exit}if }loop /_NXProcArray 5 array def }__NXbdef /colorimage{_NXDoImageOp}__NXbdef /alphaimage{1 add _NXDoImageOp}def %%EndDocument % EquationBuilder(TM) Expression % +.\Jr(=:/*V[StRFQAYmdjEEA;1nuN+[co.OZ7L#cq3iK%3:</XJ@:d#6;^G;Td47dud<A3@ % 5%MGToUOR5&d`'TuEDU)B-tbR".R+:s'.,qo?"0M"`pV1X@r=NZgT+TapkbQ1TiO8qO##pFg % SV27Z%C%),,7&[ne(c'<W+g@4!;?.76d#C!iJ0@Ju`AW"Zk%:1$+0PmBo;2s;99LJh%SVCE) % MA.D#"ZY)m9:Gn\cc%f,u5I7KGQRBN*lp[0QL,ShBY)pi(fsC&eZ=/.B!659En<Z0`[#-KJ" % ,doVqQOgkJAq"qCXD.HD(P/>q-N8/X="g8(G6;DoQLO_kcK*T65;d**f2HL=iqJD\2F@h4+1 % ;"X#J[IW(&JcoK>I0B]A"!F6^\6sphUl[nZ%5,0?G(_Q2&fVa/jG%rQ34BrNp46+,IY:s->S % !KHJ0>"aa:SVO*h%kU,jO>8J:!rGem)amd.cFG-WZSnIW-k4UEi-c=<E&_Wd'DN,uRf:0>Cq % oJ.QG)!F$TfMMU#N7&$Vh6n>qF6pWR`()pbiM$]%>Y/D+o0g_1.Ujg:JZCE8)iAoltm`bahP % 0;nm-(bdOYoG<X-0jQQ-CTc2/'uruM54WZC?:Ss`B`OZf7<G<W:sG?%3YhLiMlko&sA-B!Mi % %Q/Ws]:*1_`_)VI$\f"#<MMPb,;+]HN5)6kF;g8ecE4?!6lJMHYI*+X0eV2^Y',;$=07X!'u % cMB#I3l)F1N+D8h>A9?![@eeWmXD11/Z@Z1*Bg:^CF-^r&85GO#SR'.K+&)"iIhqtV;%0n:0 % Qba[:#@e&T,<<:mg7R]-jPS5j)j1rL>Bp4d,#ml+3"&/$X"7X-h)C/_>>#^*?]\5ef=GZ?*e % ^(O0>?"t*Y4KG%02a:GRq5_-WFm<gB,kXYpET[YT>JBdT?9!]dI^MI`<amLBVfU_;QCrh(u4 % Fnaa8CE#6&VrPhFVA<Mp4,F>>d/fL"-,4UJcN'.55f.T77(:O_/4L'1HN;gDp2Y['QD8u`oB % $JSg#R=I%;L=Mh5-;VPB?S0;#70>oa7M!_/:R6e!:"*,iL#g5S1N]m>QcAHE2?D='&W*t0E& % !-S`H&Y;TH8C:NVDI<^#o5-`G"CYgd5t;dMNIdW%fPAer."Mq'1]TfAZ$uS:&.m5+6#Zl*-i % Oqax % EndExpression N 924 39c E dc S 192f4 R 36b61aa4 gsave /__NXbasematrix matrix currentmatrix def grestore %%EndProlog %%BeginSetup /EqBFont 18 dict dup begin /FontType 3 def /PaintType 0 def /FontMatrix [0.001 0 0 0.001 0 0] def /FontBBox [15 -2950 1360 1045] def /Encoding [/.notdef /rad1 /rad2 /rad3 /rad4 /rad5 /radext /int /pathint /sum /prod /cdot /vdot /ddot /hbar /bigcap /bigcup /parallel] def /Metrics 18 dict dup begin /.notdef 0 def /rad1 1000 def /rad2 1000 def /rad3 1000 def /rad4 1000 def /rad5 1048 def /radext 1048 def /int 500 def /pathint 500 def /sum 700 def /prod 700 def /cdot 1000 def /vdot 550 def /ddot 1000 def /hbar 500 def /bigcap 700 def /bigcup 700 def /parallel 500 def end def /BBox 18 dict dup begin /.notdef [0 0 0 0] def /rad1 [110 -1150 1020 0] def /rad2 [110 -1750 1020 0] def /rad3 [110 -2350 1020 0] def /rad4 [110 -2950 1020 0] def /rad5 [128 -2890 740 0] def /radext [700 -600 740 0] def /int [20 0 425 1045] def /pathint [20 0 425 1045] def /sum [15 5 640 715] def /prod [15 0 670 715] def /cdot [110 215 890 325] def /vdot [165 -5 280 770] def /ddot [110 -5 890 770] def /hbar [20 -10 480 685] def /bigcap [0 0 680 720] def /bigcup [0 0 680 720] def /parallel [65 -175 400 675] def end def /CharacterDefs 18 dict dup begin /.notdef { } def /rad1 { 980 0 moveto 464 -1048 lineto 248 -560 lineto 110 -656 lineto 125 -675 lineto 200 -624 lineto 420 -1150 lineto 460 -1150 lineto 1020 0 lineto 980 0 lineto closepath } def /rad2 { 980 0 moveto 460 -1600 lineto 245 -864 lineto 110 -1000 lineto 128 -1023 lineto 200 -950 lineto 424 -1750 lineto 456 -1750 lineto 1020 0 lineto 980 0 lineto closepath } def /rad3 { 980 0 moveto 464 -2144 lineto 248 -1160 lineto 110 -1368 lineto 130 -1390 lineto 192 -1288 lineto 424 -2350 lineto 464 -2350 lineto 1020 0 lineto 980 0 lineto closepath } def /rad4 { 980 0 moveto 880 -488 lineto 672 -1600 lineto 464 -2696 lineto 248 -1464 lineto 110 -1728 lineto 130 -1750 lineto 192 -1616 lineto 424 -2950 lineto 464 -2950 lineto 768 -1312 lineto 1020 0 lineto 980 0 lineto closepath } def /rad5 { 700 0 moveto 700 -2628 lineto 248 -1284 lineto 128 -1564 lineto 150 -1580 lineto 215 -1425 lineto 688 -2890 lineto 740 -2890 lineto 740 0 lineto 700 0 lineto closepath } def /radext { 700 0 moveto 700 -600 lineto 740 -600 lineto 740 0 lineto 700 0 lineto closepath } def /int { 245 390 moveto 225 235 195 145 180 100 curveto 165 60 150 50 140 40 curveto 135 30 85 0 50 20 curveto 35 25 25 45 25 55 curveto 20 80 30 100 55 110 curveto 80 125 93 110 110 90 curveto 135 60 155 135 160 195 curveto 165 270 170 340 190 510 curveto 205 650 lineto 220 810 230 890 265 960 curveto 275 980 295 1015 320 1030 curveto 350 1045 380 1030 395 1020 curveto 410 1010 425 980 400 955 curveto 385 940 347 925 330 950 curveto 310 970 290 955 290 940 curveto 280 900 285 745 270 610 curveto 245 390 lineto closepath } def /pathint { 245 390 moveto 225 235 195 145 180 100 curveto 165 60 150 50 140 40 curveto 135 30 85 0 50 20 curveto 35 25 25 45 25 55 curveto 20 80 30 100 55 110 curveto 80 125 93 110 110 90 curveto 135 60 155 135 160 195 curveto 165 270 170 340 190 510 curveto 205 650 lineto 220 810 230 890 265 960 curveto 275 980 295 1015 320 1030 curveto 350 1045 380 1030 395 1020 curveto 410 1010 425 980 400 955 curveto 385 940 347 925 330 950 curveto 310 970 290 955 290 940 curveto 280 900 285 745 270 610 curveto 245 390 lineto fill 20 setlinewidth 220 400 moveto 290 400 340 455 340 500 curveto 340 585 290 640 220 640 curveto 155 640 100 585 100 500 curveto 100 455 155 400 220 400 curveto stroke } def /sum { 330 335 moveto 20 715 lineto 575 715 lineto 600 545 lineto 575 535 lineto 550 625 525 685 395 685 curveto 160 685 lineto 411 370 lineto 160 90 lineto 470 90 lineto 565 90 590 140 610 210 curveto 640 200 lineto 585 5 lineto 15 5 lineto 330 335 lineto closepath } def /prod { 590 90 moveto 590 30 620 20 670 20 curveto 670 0 lineto 410 0 lineto 410 20 lineto 460 20 495 30 495 90 curveto 495 670 lineto 190 670 lineto 190 90 lineto 190 30 225 20 275 20 curveto 275 0 lineto 15 0 lineto 15 20 lineto 65 20 100 30 100 90 curveto 100 615 lineto 100 690 40 700 20 700 curveto 20 715 lineto 670 715 lineto 670 700 lineto 645 700 590 690 590 615 curveto 590 90 lineto closepath } def /cdot { 110 270 moveto 110 300 135 325 165 325 curveto 195 325 220 300 220 270 curveto 220 235 185 215 165 215 curveto 145 215 110 235 110 270 curveto fill 445 270 moveto 445 300 470 325 500 325 curveto 530 325 555 300 555 270 curveto 555 235 520 215 500 215 curveto 480 215 445 235 445 270 curveto fill 775 270 moveto 775 300 800 325 830 325 curveto 860 325 890 300 890 270 curveto 890 235 850 215 830 215 curveto 810 215 775 235 775 270 curveto fill } def /vdot { 220 770 moveto 250 770 280 750 280 720 curveto 280 680 250 660 220 660 curveto 185 660 165 695 165 720 curveto 165 740 185 770 220 770 curveto fill 220 440 moveto 250 440 280 415 280 385 curveto 280 355 250 330 220 330 curveto 185 330 165 365 165 385 curveto 165 405 185 440 220 440 curveto fill 220 105 moveto 250 105 280 80 280 50 curveto 280 20 250 -5 220 -5 curveto 185 -5 165 30 165 50 curveto 165 70 185 105 220 105 curveto fill } def /ddot { 165 720 moveto 205 720 220 700 220 665 curveto 220 630 195 610 165 610 curveto 130 620 110 645 110 665 curveto 110 690 130 720 165 720 curveto fill 500 440 moveto 530 440 555 415 555 385 curveto 555 355 530 330 500 330 curveto 465 330 445 365 445 385 curveto 445 405 465 440 500 440 curveto fill 830 155 moveto 860 155 890 130 890 100 curveto 890 70 860 45 830 45 curveto 795 45 775 80 775 100 curveto 775 120 795 155 830 155 curveto fill } def /hbar { 480 105 moveto 440 55 400 -10 335 -10 curveto 300 -10 290 10 290 35 curveto 290 65 370 350 370 365 curveto 370 380 355 390 340 390 curveto 290 390 205 275 185 250 curveto 140 180 125 125 95 0 curveto 20 0 lineto 150 500 lineto 160 545 180 600 180 615 curveto 180 625 170 640 135 640 curveto 110 640 lineto 100 655 lineto 165 660 215 670 270 685 curveto 275 680 lineto 155 240 lineto 155 235 lineto 190 290 290 440 380 440 curveto 400 440 450 435 450 370 curveto 450 335 365 80 365 55 curveto 365 45 375 40 385 40 curveto 405 40 425 70 465 115 curveto 480 105 lineto fill 55 545 moveto 55 520 lineto 310 560 lineto 310 585 lineto 55 545 lineto fill } def /bigcap { 615 400 moveto 615 575 500 655 340 655 curveto 180 655 70 575 70 400 curveto 70 0 lineto 5 0 lineto 5 400 lineto 5 610 140 720 340 720 curveto 545 720 680 610 680 400 curveto 680 0 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/uappend{systemdict begin cvx exec end}__NXbdef /ueofill{gsave newpath uappend eofill grestore}__NXbdef /ufill{gsave newpath uappend fill grestore}__NXbdef /ustroke{ gsave newpath dup length 6 eq {exch uappend concat}{uappend}ifelse stroke grestore }__NXbdef /__NXustrokepathMatrix dup where {pop pop}{matrix def}ifelse /ustrokepath{ newpath dup length 6 eq{ exch uappend __NXustrokepathMatrix currentmatrix exch concat strokepath setmatrix }{uappend strokepath}ifelse } __NXbdef /upath{ [exch {/ucache cvx}if pathbbox /setbbox cvx {/moveto cvx}{/lineto cvx}{/curveto cvx}{/closepath cvx}pathforall]cvx } __NXbdef /setstrokeadjust{pop}__NXbdef /currentstrokeadjust{false}__NXbdef /selectfont{exch findfont exch dup type /arraytype eq {makefont}{scalefont}ifelse setfont}__NXbdef /_NXCombineArrays{ counttomark dup 2 add index dup length 3 -1 roll { 2 index length sub dup 4 1 roll 1 index exch 4 -1 roll putinterval exch }repeat pop pop pop }__NXbdef /setcmykcolor{ 1.0 exch sub dup dup 6 -1 roll sub dup 0 lt{pop 0}if 5 1 roll 4 -1 roll sub dup 0 lt{pop 0}if 3 1 roll exch sub dup 0 lt{pop 0}if setrgbcolor }__NXbdef /currentcmykcolor{currentrgbcolor 3{1.0 exch sub 3 1 roll}repeat 0}__NXbdef /flushgraphics{}def /setwindowtype{pop pop}def /currentwindowtype{pop 0}def /setalpha{pop}def /currentalpha{1.0}def /hidecursor{}def /obscurecursor{}def /revealcursor{}def /setcursor{4 {pop}repeat}bind def /showcursor{}def /NextStepEncoding where not{ /NextStepEncoding StandardEncoding 256 array copy def 0 [129/Agrave/Aacute/Acircumflex/Atilde/Adieresis/Aring/Ccedilla/Egrave /Eacute/Ecircumflex/Edieresis/Igrave/Iacute/Icircumflex/Idieresis /Eth/Ntilde/Ograve/Oacute/Ocircumflex/Otilde/Odieresis/Ugrave/Uacute /Ucircumflex/Udieresis/Yacute/Thorn/mu/multiply/divide/copyright 176/registered 181/brokenbar 190/logicalnot 192/onesuperior 201/twosuperior 204/threesuperior 209/plusminus/onequarter/onehalf/threequarters/agrave /aacute/acircumflex/atilde/adieresis/aring/ccedilla/egrave/eacute 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So if currentshared is true, % we call the normal findfont and return that /currentshared where {pop currentshared} {false} ifelse {//findfont exec} {dup _NXfstr 3 125 getinterval cvs length 3 add _NXfstr 0 3 -1 roll getinterval cvn exch FontDirectory 2 index known {pop FontDirectory exch get} {//findfont exec dup /Encoding get StandardEncoding eq { dup length dict exch {1 index /FID ne {2 index 3 1 roll put}{pop pop}ifelse}forall dup /Encoding NextStepEncoding put definefont }{exch pop} ifelse }ifelse }ifelse }bind def }{pop}ifelse /_NXProcArray 5 array __NXdef /_NXChannels 0 __NXdef /_NXTotalBytes 0 __NXdef /_NXDoImageOp{ 1 index{dup}{1}ifelse /_NXChannels exch store _NXChannels 2 add 2 roll _NXProcArray 0 _NXChannels getinterval astore pop 5 index 4 index mul 2 index{1 sub 8 idiv 1 add mul}{mul 1 sub 8 idiv 1 add}ifelse 4 index mul /_NXTotalBytes exch store pop exch pop gsave matrix invertmatrix concat 0.5 setgray 0 0 4 2 roll rectfill grestore { 0 1 _NXChannels 1 sub{ _NXProcArray 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U4+gjk,c#[t>!bc-!4?#MWKE*6"2i<oHl$b30H#0e]0oDL"ZT^IFXhIDNNj-H"`5!%,OO<0V % 2G08F\(*sVc?%Y;.Z$7"enJr2n3FU`MRYW=0=7B1)F+_17U$3`VZpBO_#Y*\?iUHfr1JZN$5 % 2<N'nW9OP3Z"\jg'QlJme^LZiS34@B+)ra/2,rBmp)q.u?HQpnR)dP+8TCP8qc?DpA3#XGap % j,uF*ReHqdc_?D"5TYh?t%AAFh$M&g#*"W*ld04.jCqQ5iE3cj(^p4r^Oa2BBXmSZ//j#4jJ % ?G%MoF/1KS&Tl_WaJd&O(<E\W[Gq=Dd6QRP$B-O#*O?fF5(%@RI@"jh^pOS58.<p$:>2Mq1) % !hgouB)KDm^H;O>7U5$)Z_Wo]TQOF$4".h2$sU.Yo;K0\dr:Hr#XO\hBWPae9"B+?W8)fpBh % "WT=a+Zte'V[/0'S*!6cWI3-rV"J7nY+dDFlVK9C'Ib_(*HKBo`ga$b\<H"Prfmd<#dh"[:j % k$ZJ-":S5XfoIBn6s:'*^C'#6Op@6NDg*(c!@f%Uf@`(pKcc">6p0TEk\<BGUdoE"E;A+@&? % 7&-P`.J.qo9"98E%x % EndExpression N 989 3dd E 6d S 1b2e9 R c8a7dde6 gsave /__NXbasematrix matrix currentmatrix def grestore %%EndProlog %%BeginSetup /EqBFont 18 dict dup begin /FontType 3 def /PaintType 0 def /FontMatrix [0.001 0 0 0.001 0 0] def /FontBBox [15 -2950 1360 1045] def /Encoding [/.notdef /rad1 /rad2 /rad3 /rad4 /rad5 /radext /int /pathint /sum /prod /cdot /vdot /ddot /hbar /bigcap /bigcup /parallel] def /Metrics 18 dict dup begin /.notdef 0 def /rad1 1000 def /rad2 1000 def /rad3 1000 def /rad4 1000 def /rad5 1048 def /radext 1048 def /int 500 def /pathint 500 def /sum 700 def /prod 700 def /cdot 1000 def /vdot 550 def /ddot 1000 def /hbar 500 def /bigcap 700 def /bigcup 700 def /parallel 500 def end def /BBox 18 dict dup begin /.notdef [0 0 0 0] def /rad1 [110 -1150 1020 0] def /rad2 [110 -1750 1020 0] def /rad3 [110 -2350 1020 0] def /rad4 [110 -2950 1020 0] def /rad5 [128 -2890 740 0] def /radext [700 -600 740 0] def /int [20 0 425 1045] def /pathint [20 0 425 1045] def /sum [15 5 640 715] def /prod [15 0 670 715] def /cdot [110 215 890 325] def /vdot [165 -5 280 770] def /ddot [110 -5 890 770] def /hbar [20 -10 480 685] def /bigcap [0 0 680 720] def /bigcup [0 0 680 720] def /parallel [65 -175 400 675] def end def /CharacterDefs 18 dict dup begin /.notdef { } def /rad1 { 980 0 moveto 464 -1048 lineto 248 -560 lineto 110 -656 lineto 125 -675 lineto 200 -624 lineto 420 -1150 lineto 460 -1150 lineto 1020 0 lineto 980 0 lineto closepath } def /rad2 { 980 0 moveto 460 -1600 lineto 245 -864 lineto 110 -1000 lineto 128 -1023 lineto 200 -950 lineto 424 -1750 lineto 456 -1750 lineto 1020 0 lineto 980 0 lineto closepath } def /rad3 { 980 0 moveto 464 -2144 lineto 248 -1160 lineto 110 -1368 lineto 130 -1390 lineto 192 -1288 lineto 424 -2350 lineto 464 -2350 lineto 1020 0 lineto 980 0 lineto closepath } def /rad4 { 980 0 moveto 880 -488 lineto 672 -1600 lineto 464 -2696 lineto 248 -1464 lineto 110 -1728 lineto 130 -1750 lineto 192 -1616 lineto 424 -2950 lineto 464 -2950 lineto 768 -1312 lineto 1020 0 lineto 980 0 lineto closepath } def /rad5 { 700 0 moveto 700 -2628 lineto 248 -1284 lineto 128 -1564 lineto 150 -1580 lineto 215 -1425 lineto 688 -2890 lineto 740 -2890 lineto 740 0 lineto 700 0 lineto closepath } def /radext { 700 0 moveto 700 -600 lineto 740 -600 lineto 740 0 lineto 700 0 lineto closepath } def /int { 245 390 moveto 225 235 195 145 180 100 curveto 165 60 150 50 140 40 curveto 135 30 85 0 50 20 curveto 35 25 25 45 25 55 curveto 20 80 30 100 55 110 curveto 80 125 93 110 110 90 curveto 135 60 155 135 160 195 curveto 165 270 170 340 190 510 curveto 205 650 lineto 220 810 230 890 265 960 curveto 275 980 295 1015 320 1030 curveto 350 1045 380 1030 395 1020 curveto 410 1010 425 980 400 955 curveto 385 940 347 925 330 950 curveto 310 970 290 955 290 940 curveto 280 900 285 745 270 610 curveto 245 390 lineto closepath } def /pathint { 245 390 moveto 225 235 195 145 180 100 curveto 165 60 150 50 140 40 curveto 135 30 85 0 50 20 curveto 35 25 25 45 25 55 curveto 20 80 30 100 55 110 curveto 80 125 93 110 110 90 curveto 135 60 155 135 160 195 curveto 165 270 170 340 190 510 curveto 205 650 lineto 220 810 230 890 265 960 curveto 275 980 295 1015 320 1030 curveto 350 1045 380 1030 395 1020 curveto 410 1010 425 980 400 955 curveto 385 940 347 925 330 950 curveto 310 970 290 955 290 940 curveto 280 900 285 745 270 610 curveto 245 390 lineto fill 20 setlinewidth 220 400 moveto 290 400 340 455 340 500 curveto 340 585 290 640 220 640 curveto 155 640 100 585 100 500 curveto 100 455 155 400 220 400 curveto stroke } def /sum { 330 335 moveto 20 715 lineto 575 715 lineto 600 545 lineto 575 535 lineto 550 625 525 685 395 685 curveto 160 685 lineto 411 370 lineto 160 90 lineto 470 90 lineto 565 90 590 140 610 210 curveto 640 200 lineto 585 5 lineto 15 5 lineto 330 335 lineto closepath } def /prod { 590 90 moveto 590 30 620 20 670 20 curveto 670 0 lineto 410 0 lineto 410 20 lineto 460 20 495 30 495 90 curveto 495 670 lineto 190 670 lineto 190 90 lineto 190 30 225 20 275 20 curveto 275 0 lineto 15 0 lineto 15 20 lineto 65 20 100 30 100 90 curveto 100 615 lineto 100 690 40 700 20 700 curveto 20 715 lineto 670 715 lineto 670 700 lineto 645 700 590 690 590 615 curveto 590 90 lineto closepath } def /cdot { 110 270 moveto 110 300 135 325 165 325 curveto 195 325 220 300 220 270 curveto 220 235 185 215 165 215 curveto 145 215 110 235 110 270 curveto fill 445 270 moveto 445 300 470 325 500 325 curveto 530 325 555 300 555 270 curveto 555 235 520 215 500 215 curveto 480 215 445 235 445 270 curveto fill 775 270 moveto 775 300 800 325 830 325 curveto 860 325 890 300 890 270 curveto 890 235 850 215 830 215 curveto 810 215 775 235 775 270 curveto fill } def /vdot { 220 770 moveto 250 770 280 750 280 720 curveto 280 680 250 660 220 660 curveto 185 660 165 695 165 720 curveto 165 740 185 770 220 770 curveto fill 220 440 moveto 250 440 280 415 280 385 curveto 280 355 250 330 220 330 curveto 185 330 165 365 165 385 curveto 165 405 185 440 220 440 curveto fill 220 105 moveto 250 105 280 80 280 50 curveto 280 20 250 -5 220 -5 curveto 185 -5 165 30 165 50 curveto 165 70 185 105 220 105 curveto fill } def /ddot { 165 720 moveto 205 720 220 700 220 665 curveto 220 630 195 610 165 610 curveto 130 620 110 645 110 665 curveto 110 690 130 720 165 720 curveto fill 500 440 moveto 530 440 555 415 555 385 curveto 555 355 530 330 500 330 curveto 465 330 445 365 445 385 curveto 445 405 465 440 500 440 curveto fill 830 155 moveto 860 155 890 130 890 100 curveto 890 70 860 45 830 45 curveto 795 45 775 80 775 100 curveto 775 120 795 155 830 155 curveto fill } def /hbar { 480 105 moveto 440 55 400 -10 335 -10 curveto 300 -10 290 10 290 35 curveto 290 65 370 350 370 365 curveto 370 380 355 390 340 390 curveto 290 390 205 275 185 250 curveto 140 180 125 125 95 0 curveto 20 0 lineto 150 500 lineto 160 545 180 600 180 615 curveto 180 625 170 640 135 640 curveto 110 640 lineto 100 655 lineto 165 660 215 670 270 685 curveto 275 680 lineto 155 240 lineto 155 235 lineto 190 290 290 440 380 440 curveto 400 440 450 435 450 370 curveto 450 335 365 80 365 55 curveto 365 45 375 40 385 40 curveto 405 40 425 70 465 115 curveto 480 105 lineto fill 55 545 moveto 55 520 lineto 310 560 lineto 310 585 lineto 55 545 lineto fill } def /bigcap { 615 400 moveto 615 575 500 655 340 655 curveto 180 655 70 575 70 400 curveto 70 0 lineto 5 0 lineto 5 400 lineto 5 610 140 720 340 720 curveto 545 720 680 610 680 400 curveto 680 0 lineto 615 0 lineto 615 400 lineto closepath } def /bigcup { 65 320 moveto 65 145 180 65 340 65 curveto 500 65 615 145 615 320 curveto 615 720 lineto 680 720 lineto 680 320 lineto 680 110 540 0 340 0 curveto 140 0 0 110 0 320 curveto 0 720 lineto 65 720 lineto 65 320 lineto closepath } def /parallel { 135 675 moveto 135 -175 lineto 65 -175 lineto 65 675 lineto 135 675 lineto fill 335 675 moveto 335 -175 lineto 265 -175 lineto 265 675 lineto 335 675 lineto fill 400 675 moveto } def end def /BuildChar { 0 begin /char exch def /fontdict exch def /charname fontdict /Encoding get char get def fontdict begin Metrics charname get 0 BBox charname get aload pop setcachedevice CharacterDefs charname get exec eofill end end } def /BuildChar load 0 3 dict put end definefont pop %%EndSetup 0 0 193.798676 36.982452 rectclip 0 0 193.798676 36.982452 rectclip 0 setgray 1 setalpha 1 1 scale /Symbol findfont 12 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So if currentshared is true, % we call the normal findfont and return that /currentshared where {pop currentshared} {false} ifelse {//findfont exec} {dup _NXfstr 3 125 getinterval cvs length 3 add _NXfstr 0 3 -1 roll getinterval cvn exch FontDirectory 2 index known {pop FontDirectory exch get} {//findfont exec dup /Encoding get StandardEncoding eq { dup length dict exch {1 index /FID ne {2 index 3 1 roll put}{pop pop}ifelse}forall dup /Encoding NextStepEncoding put definefont }{exch pop} ifelse }ifelse }ifelse }bind def }{pop}ifelse /_NXProcArray 5 array __NXdef /_NXChannels 0 __NXdef /_NXTotalBytes 0 __NXdef /_NXDoImageOp{ 1 index{dup}{1}ifelse /_NXChannels exch store _NXChannels 2 add 2 roll _NXProcArray 0 _NXChannels getinterval astore pop 5 index 4 index mul 2 index{1 sub 8 idiv 1 add mul}{mul 1 sub 8 idiv 1 add}ifelse 4 index mul /_NXTotalBytes exch store pop exch pop gsave matrix invertmatrix concat 0.5 setgray 0 0 4 2 roll rectfill grestore { 0 1 _NXChannels 1 sub{ _NXProcArray exch get exec length _NXTotalBytes exch sub /_NXTotalBytes exch store}for _NXTotalBytes 0 le{exit}if }loop /_NXProcArray 5 array def }__NXbdef /colorimage{_NXDoImageOp}__NXbdef /alphaimage{1 add _NXDoImageOp}def %%EndDocument % EquationBuilder(TM) Expression % +.\Jr(=:/*V[StRFQAYmdjEEA;1nuN+[co.OZ7L#cq3iK%3:</XJ@:d#6;^G;Td47dud<A3@ % 5%MGToUOR5&d`'TuEDU)B-tbR".R+:nNN,qo?"OCD#t$;5[8,X_XRS2ihA"VLe`-6WLgVDQ[ % K-Oh*sQ3\5X&jqDA/d.K[>m^Q5J[c<T!&,[Tin#Mqf\EPQp`n<3FPaBu@nDhna*n=K8V8]^g % &Ph$W+?>=8/K#JVW;G9-88oJL(Oh['1U:3`L@o]lIuWp?:"^H/d3!JP0!\R-jBZKMU3Y:KKd % MTD;,W$h7Wh'2cVsa%HJ3B&1PLD9q*t]?G<9Mg[T)H[0'6:lT+cN<6Y)##Jf*go]cJH(ig,B % IOj7T$51.)oEbf=%tK1>+G'qUPsQ8o,.J>1KJ8K#)]E_W`d".-LL,&nJPq#HKMX%$&"ml,+T % kLS,_nj-\<[tJ*nD[RH8;k*S7AqE`uf^S8lm3cV')\X@m,9#^j7g#pt.2_L)A`p_QAA//5Ja % G>EDgid[YZE62@4%r<[1I6o"$UV1$C7nrK5NQ?!BW"/KA'!#$!.8r^K[iYM\d]SMc+j-:u`1 % <^'_$6_o#[S3ERmO3oJ>1,@kOM]+ign'O'qI'AQ2C]"jbRFO[R$(QMBmjS/-OYV8#a'i\Xl[ % CXA#bRbUPL1e)*eQ[4s:.KZdr;.Y,.`4W?gFV#:cL0OTR$Uh)o;pWa'"Z$Z_D:X6Z4V'(Y=j % ;nD!9S<hlAD?'\I(dNZ@1:U_DJiF%0+;3NXBsZ^:M.nLT-;[)LE"$ihA,OnI-BPo9f,V?ph' % 7c`n45ml_r-i$caaT[[kYM]$:phAn^:W^K./o"Te%/pJ1dl@T*qcraeG47[LETL8&/'=&]PZ % !*:)8%rMF?W-o5i>(sr\79I7JL(Vk%1;?u^^PZ>1So"64i8?]]6!jRAArP1H_6F37,Zm=flS % M@!`c7phi!:[NQbU"I8[!A]YC*LHl[LJtG-E;*oI2/:,(I\Z>;26-4`45M97H84l?^ZFrJ6I % :.eHZ0_(f9#i<-Y<=$W8AgL)\Z/:jO(Y$P=3n8^J&=G@OUM_!A8O%S@l4Fr^\s_?4ca!>1;' % "=F1gif>5_x % EndExpression N 871 367 E 31 S 16c11 R 934724f0 gsave /__NXbasematrix matrix currentmatrix def grestore %%EndProlog %%BeginSetup /EqBFont 18 dict dup begin /FontType 3 def /PaintType 0 def /FontMatrix [0.001 0 0 0.001 0 0] def /FontBBox [15 -2950 1360 1045] def /Encoding [/.notdef /rad1 /rad2 /rad3 /rad4 /rad5 /radext /int /pathint /sum /prod /cdot /vdot /ddot /hbar /bigcap /bigcup /parallel] def /Metrics 18 dict dup begin /.notdef 0 def /rad1 1000 def /rad2 1000 def /rad3 1000 def /rad4 1000 def /rad5 1048 def /radext 1048 def /int 500 def /pathint 500 def /sum 700 def /prod 700 def /cdot 1000 def /vdot 550 def /ddot 1000 def /hbar 500 def /bigcap 700 def /bigcup 700 def /parallel 500 def end def /BBox 18 dict dup begin /.notdef [0 0 0 0] def /rad1 [110 -1150 1020 0] def /rad2 [110 -1750 1020 0] def /rad3 [110 -2350 1020 0] def /rad4 [110 -2950 1020 0] def /rad5 [128 -2890 740 0] def /radext [700 -600 740 0] def /int [20 0 425 1045] def /pathint [20 0 425 1045] def /sum [15 5 640 715] def /prod [15 0 670 715] def /cdot [110 215 890 325] def /vdot [165 -5 280 770] def /ddot [110 -5 890 770] def /hbar [20 -10 480 685] def /bigcap [0 0 680 720] def /bigcup [0 0 680 720] def /parallel [65 -175 400 675] def end def /CharacterDefs 18 dict dup begin /.notdef { } def /rad1 { 980 0 moveto 464 -1048 lineto 248 -560 lineto 110 -656 lineto 125 -675 lineto 200 -624 lineto 420 -1150 lineto 460 -1150 lineto 1020 0 lineto 980 0 lineto closepath } def /rad2 { 980 0 moveto 460 -1600 lineto 245 -864 lineto 110 -1000 lineto 128 -1023 lineto 200 -950 lineto 424 -1750 lineto 456 -1750 lineto 1020 0 lineto 980 0 lineto closepath } def /rad3 { 980 0 moveto 464 -2144 lineto 248 -1160 lineto 110 -1368 lineto 130 -1390 lineto 192 -1288 lineto 424 -2350 lineto 464 -2350 lineto 1020 0 lineto 980 0 lineto closepath } def /rad4 { 980 0 moveto 880 -488 lineto 672 -1600 lineto 464 -2696 lineto 248 -1464 lineto 110 -1728 lineto 130 -1750 lineto 192 -1616 lineto 424 -2950 lineto 464 -2950 lineto 768 -1312 lineto 1020 0 lineto 980 0 lineto closepath } def /rad5 { 700 0 moveto 700 -2628 lineto 248 -1284 lineto 128 -1564 lineto 150 -1580 lineto 215 -1425 lineto 688 -2890 lineto 740 -2890 lineto 740 0 lineto 700 0 lineto closepath } def /radext { 700 0 moveto 700 -600 lineto 740 -600 lineto 740 0 lineto 700 0 lineto closepath } def /int { 245 390 moveto 225 235 195 145 180 100 curveto 165 60 150 50 140 40 curveto 135 30 85 0 50 20 curveto 35 25 25 45 25 55 curveto 20 80 30 100 55 110 curveto 80 125 93 110 110 90 curveto 135 60 155 135 160 195 curveto 165 270 170 340 190 510 curveto 205 650 lineto 220 810 230 890 265 960 curveto 275 980 295 1015 320 1030 curveto 350 1045 380 1030 395 1020 curveto 410 1010 425 980 400 955 curveto 385 940 347 925 330 950 curveto 310 970 290 955 290 940 curveto 280 900 285 745 270 610 curveto 245 390 lineto closepath } def /pathint { 245 390 moveto 225 235 195 145 180 100 curveto 165 60 150 50 140 40 curveto 135 30 85 0 50 20 curveto 35 25 25 45 25 55 curveto 20 80 30 100 55 110 curveto 80 125 93 110 110 90 curveto 135 60 155 135 160 195 curveto 165 270 170 340 190 510 curveto 205 650 lineto 220 810 230 890 265 960 curveto 275 980 295 1015 320 1030 curveto 350 1045 380 1030 395 1020 curveto 410 1010 425 980 400 955 curveto 385 940 347 925 330 950 curveto 310 970 290 955 290 940 curveto 280 900 285 745 270 610 curveto 245 390 lineto fill 20 setlinewidth 220 400 moveto 290 400 340 455 340 500 curveto 340 585 290 640 220 640 curveto 155 640 100 585 100 500 curveto 100 455 155 400 220 400 curveto stroke } def /sum { 330 335 moveto 20 715 lineto 575 715 lineto 600 545 lineto 575 535 lineto 550 625 525 685 395 685 curveto 160 685 lineto 411 370 lineto 160 90 lineto 470 90 lineto 565 90 590 140 610 210 curveto 640 200 lineto 585 5 lineto 15 5 lineto 330 335 lineto closepath } def /prod { 590 90 moveto 590 30 620 20 670 20 curveto 670 0 lineto 410 0 lineto 410 20 lineto 460 20 495 30 495 90 curveto 495 670 lineto 190 670 lineto 190 90 lineto 190 30 225 20 275 20 curveto 275 0 lineto 15 0 lineto 15 20 lineto 65 20 100 30 100 90 curveto 100 615 lineto 100 690 40 700 20 700 curveto 20 715 lineto 670 715 lineto 670 700 lineto 645 700 590 690 590 615 curveto 590 90 lineto closepath } def /cdot { 110 270 moveto 110 300 135 325 165 325 curveto 195 325 220 300 220 270 curveto 220 235 185 215 165 215 curveto 145 215 110 235 110 270 curveto fill 445 270 moveto 445 300 470 325 500 325 curveto 530 325 555 300 555 270 curveto 555 235 520 215 500 215 curveto 480 215 445 235 445 270 curveto fill 775 270 moveto 775 300 800 325 830 325 curveto 860 325 890 300 890 270 curveto 890 235 850 215 830 215 curveto 810 215 775 235 775 270 curveto fill } def /vdot { 220 770 moveto 250 770 280 750 280 720 curveto 280 680 250 660 220 660 curveto 185 660 165 695 165 720 curveto 165 740 185 770 220 770 curveto fill 220 440 moveto 250 440 280 415 280 385 curveto 280 355 250 330 220 330 curveto 185 330 165 365 165 385 curveto 165 405 185 440 220 440 curveto fill 220 105 moveto 250 105 280 80 280 50 curveto 280 20 250 -5 220 -5 curveto 185 -5 165 30 165 50 curveto 165 70 185 105 220 105 curveto fill } def /ddot { 165 720 moveto 205 720 220 700 220 665 curveto 220 630 195 610 165 610 curveto 130 620 110 645 110 665 curveto 110 690 130 720 165 720 curveto fill 500 440 moveto 530 440 555 415 555 385 curveto 555 355 530 330 500 330 curveto 465 330 445 365 445 385 curveto 445 405 465 440 500 440 curveto fill 830 155 moveto 860 155 890 130 890 100 curveto 890 70 860 45 830 45 curveto 795 45 775 80 775 100 curveto 775 120 795 155 830 155 curveto fill } def /hbar { 480 105 moveto 440 55 400 -10 335 -10 curveto 300 -10 290 10 290 35 curveto 290 65 370 350 370 365 curveto 370 380 355 390 340 390 curveto 290 390 205 275 185 250 curveto 140 180 125 125 95 0 curveto 20 0 lineto 150 500 lineto 160 545 180 600 180 615 curveto 180 625 170 640 135 640 curveto 110 640 lineto 100 655 lineto 165 660 215 670 270 685 curveto 275 680 lineto 155 240 lineto 155 235 lineto 190 290 290 440 380 440 curveto 400 440 450 435 450 370 curveto 450 335 365 80 365 55 curveto 365 45 375 40 385 40 curveto 405 40 425 70 465 115 curveto 480 105 lineto fill 55 545 moveto 55 520 lineto 310 560 lineto 310 585 lineto 55 545 lineto fill } def /bigcap { 615 400 moveto 615 575 500 655 340 655 curveto 180 655 70 575 70 400 curveto 70 0 lineto 5 0 lineto 5 400 lineto 5 610 140 720 340 720 curveto 545 720 680 610 680 400 curveto 680 0 lineto 615 0 lineto 615 400 lineto closepath } def /bigcup { 65 320 moveto 65 145 180 65 340 65 curveto 500 65 615 145 615 320 curveto 615 720 lineto 680 720 lineto 680 320 lineto 680 110 540 0 340 0 curveto 140 0 0 110 0 320 curveto 0 720 lineto 65 720 lineto 65 320 lineto closepath } def /parallel { 135 675 moveto 135 -175 lineto 65 -175 lineto 65 675 lineto 135 675 lineto fill 335 675 moveto 335 -175 lineto 265 -175 lineto 265 675 lineto 335 675 lineto fill 400 675 moveto } def end def /BuildChar { 0 begin /char exch def /fontdict exch def /charname fontdict /Encoding get char get def fontdict begin Metrics charname get 0 BBox charname get aload pop setcachedevice CharacterDefs charname get exec eofill end end } def /BuildChar load 0 3 dict put end definefont pop %%EndSetup 0 0 163.042664 36.982452 rectclip 0 0 163.042664 36.982452 rectclip 0 setgray 1 setalpha 1 1 scale /Symbol findfont 12 scalefont 45 exch defineuserobject 45 execuserobject setfont 11.56 16.190992 moveto (\() [0 0] xyshow 1 1 scale 1 1 scale 45 execuserobject setfont 23.200001 16.184992 moveto (\)) [0 0] xyshow 1 1 scale 0 setgray 1 setalpha /EqBFont 24 selectfont 40.618668 10.442991 moveto <09> [0 0] xyshow 0 setgray 1 setalpha 1 1 scale 45 execuserobject setfont 91.692001 25.084427 moveto (\() [0 0] xyshow 1 1 scale 1 1 scale 45 execuserobject setfont 112.586662 25.078426 moveto (\)) [0 0] xyshow 1 1 scale 0 setgray 1 setalpha 1 1.39148 scale 45 execuserobject setfont 68.40667 17.213518 moveto (\() [0 0] xyshow 1 0.718659 scale 1 1.389871 scale 45 execuserobject setfont 123.650673 17.230787 moveto (\)) [0 0] xyshow 1 0.719491 scale 0 setgray 1 setalpha 0.48 setlinewidth 61.342667 19.082993 moveto 128.674667 19.082993 lineto stroke 0 setgray 1 setalpha 1 2.937194 scale 45 execuserobject setfont 56.542667 3.605014 moveto (\() [0 0] xyshow 1 0.340461 scale 1 2.933799 scale 45 execuserobject setfont 129.514664 3.606534 moveto (\)) [0 0] xyshow 1 0.340855 scale 0 setgray 1 setalpha /Times-BoldItalic findfont 12 scalefont 44 exch defineuserobject 44 execuserobject setfont 156.042664 16.082991 moveto (xkkxf) [-6 1.907349e-06 -53.739998 -8.231413 -79.942665 8.231413 -15.360001 0 0 0 0 0] xyshow /Times-Roman findfont 12 scalefont 47 exch defineuserobject 47 execuserobject setfont 134.038666 16.082993 moveto (sin1124) [4.667999 0 3.335999 0 -36.295998 8.893435 -32.540001 0 -11.363998 -1.907349e-06 -56.846664 -8.893433 0 0 0 0] xyshow 45 execuserobject setfont 87.714668 7.85158 moveto (p--=) [8.777328 17.124847 -14.618668 0 -50.815994 -8.893435 0 0 0 0] xyshow /Symbol findfont 9 scalefont 56 exch defineuserobject 56 execuserobject setfont 46.008171 2.272 moveto (=\245) [-0.738003 27.730968 0 0 0 0] xyshow /Times-BoldItalic findfont 9 scalefont 55 exch defineuserobject 55 execuserobject setfont 117.110664 29.691456 moveto (kk) [-76.05249 -27.419456 0 0 0 0] xyshow /Times-Roman findfont 9 scalefont 54 exch defineuserobject 54 execuserobject setfont 51.39917 2.272 moveto (1) [0 0 0 0] xyshow %%Trailer %%DocumentFonts: Times-BoldItalic %%+ Times-Roman %%+ Symbol %End of Graphics MathPictureEnd end :[font = text; inactive; preserveAspect; ] The Fourier series of g(x), -p < x < p , where ;[s] 11:0,0;23,1;24,2;25,3;26,4;31,5;32,6;34,7;36,8;39,9;40,10;49,-1; 11:1,11,8,Times,0,12,0,0,0;1,10,8,Times,2,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Times,2,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,0,0,Symbol,0,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Times,2,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,0,0,Symbol,0,12,0,0,0;1,11,8,Times,0,12,0,0,0; :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 0; pictureWidth = 114; pictureHeight = 27; endGroup; ] %! %%Creator: Mathematica %%AspectRatio: 0.236842 200 dict begin MathPictureStart /Courier findfont 10 scalefont setfont % Scaling calculations 0 1 0 1 [ [ 0 0 0 0 ] [ 114 27 0 0 ] ] MathScale % Start of Graphics %!PS-Adobe-2.0 EPSF-2.0 %%Title: %%Creator: EquationBuilder %%CreationDate: Sun Aug 2 08:57:52 1992 %%For: gray %%DocumentFonts: (atend) %%Pages: 0 1 %%BoundingBox: 0 0 114 27 %%Translation: 1 8.709397 112.437332 0 %%EndComments %%BeginDocument: /usr/lib/NextStep/printPackage.ps %!PS-Adobe-2.0 %%Title: Printing Package %%EndComments %Version: 2.0 %Copyright: 1988, NeXT, Inc. /__NXdef{1 index where{pop pop pop}{def}ifelse}bind def /__NXbdef{1 index where{pop pop pop}{bind def}ifelse}bind def /UserObjects 10 array __NXdef /defineuserobject{ exch dup 1 add dup UserObjects length gt{ array dup 0 UserObjects putinterval /UserObjects exch def }{pop}ifelse UserObjects exch 3 -1 roll put }__NXbdef /undefineuserobject{UserObjects exch null put}__NXbdef /execuserobject{UserObjects exch get exec}__NXbdef /__NXRectPath{4 2 roll moveto 1 index 0 rlineto 0 exch rlineto neg 0 rlineto closepath}__NXbdef /__NXProcessRectArgs{ 1 index type /arraytype eq{ exch 0 4 2 index length 1 sub{ dup 3 add 1 exch{1 index exch get exch}for 5 1 roll 5 index exec }for pop pop }{exec}ifelse }__NXbdef /rectfill{gsave newpath {__NXRectPath fill} __NXProcessRectArgs grestore}__NXbdef /rectclip{newpath {__NXRectPath} __NXProcessRectArgs clip newpath}__NXbdef /rectstroke{ gsave newpath dup type /arraytype eq{dup length 6 eq}{false}ifelse{ {gsave __NXRectPath null concat stroke grestore} dup length array cvx copy dup 2 4 -1 roll put __NXProcessRectArgs }{{__NXRectPath stroke} __NXProcessRectArgs}ifelse grestore }__NXbdef /xyshow{ 0 1 3 index length 1 sub{ currentpoint 4 index 3 index 1 getinterval show 3 index 3 index 2 mul 1 add get add exch 3 index 3 index 2 mul get add exch moveto pop }for pop pop }__NXbdef /xshow{ 0 1 3 index length 1 sub{ currentpoint 4 index 3 index 1 getinterval show exch 3 index 3 index get add exch moveto pop }for pop pop }__NXbdef /yshow{ 0 1 3 index length 1 sub{ currentpoint 4 index 3 index 1 getinterval show 3 index 3 index get add moveto pop }for pop pop }__NXbdef /arct{arcto pop pop pop pop}__NXbdef /setbbox{pop pop pop pop}__NXbdef /ucache{}__NXbdef /ucachestatus{mark 0 0 0 0 0}__NXbdef /setucacheparams{cleartomark}__NXbdef /uappend{systemdict begin cvx exec end}__NXbdef /ueofill{gsave newpath uappend eofill grestore}__NXbdef /ufill{gsave newpath uappend fill grestore}__NXbdef /ustroke{ gsave newpath dup length 6 eq {exch uappend concat}{uappend}ifelse stroke grestore }__NXbdef /__NXustrokepathMatrix dup where {pop pop}{matrix def}ifelse /ustrokepath{ newpath dup length 6 eq{ exch uappend __NXustrokepathMatrix currentmatrix exch concat strokepath setmatrix }{uappend strokepath}ifelse } __NXbdef /upath{ [exch {/ucache cvx}if pathbbox /setbbox cvx {/moveto cvx}{/lineto cvx}{/curveto cvx}{/closepath cvx}pathforall]cvx } __NXbdef /setstrokeadjust{pop}__NXbdef /currentstrokeadjust{false}__NXbdef /selectfont{exch findfont exch dup type /arraytype eq {makefont}{scalefont}ifelse setfont}__NXbdef /_NXCombineArrays{ counttomark dup 2 add index dup length 3 -1 roll { 2 index length sub dup 4 1 roll 1 index exch 4 -1 roll putinterval exch }repeat pop pop pop }__NXbdef /setcmykcolor{ 1.0 exch sub dup dup 6 -1 roll sub dup 0 lt{pop 0}if 5 1 roll 4 -1 roll sub dup 0 lt{pop 0}if 3 1 roll exch sub dup 0 lt{pop 0}if setrgbcolor }__NXbdef /currentcmykcolor{currentrgbcolor 3{1.0 exch sub 3 1 roll}repeat 0}__NXbdef /flushgraphics{}def /setwindowtype{pop pop}def /currentwindowtype{pop 0}def /setalpha{pop}def /currentalpha{1.0}def /hidecursor{}def /obscurecursor{}def /revealcursor{}def /setcursor{4 {pop}repeat}bind def /showcursor{}def /NextStepEncoding where not{ /NextStepEncoding StandardEncoding 256 array copy def 0 [129/Agrave/Aacute/Acircumflex/Atilde/Adieresis/Aring/Ccedilla/Egrave /Eacute/Ecircumflex/Edieresis/Igrave/Iacute/Icircumflex/Idieresis /Eth/Ntilde/Ograve/Oacute/Ocircumflex/Otilde/Odieresis/Ugrave/Uacute /Ucircumflex/Udieresis/Yacute/Thorn/mu/multiply/divide/copyright 176/registered 181/brokenbar 190/logicalnot 192/onesuperior 201/twosuperior 204/threesuperior 209/plusminus/onequarter/onehalf/threequarters/agrave /aacute/acircumflex/atilde/adieresis/aring/ccedilla/egrave/eacute /ecircumflex/edieresis/igrave 226/iacute 228/icircumflex/idieresis/eth /ntilde 236/ograve/oacute/ocircumflex/otilde/odieresis 242/ugrave/uacute /ucircumflex 246/udieresis/yacute 252/thorn/ydieresis] {dup type /nametype eq {NextStepEncoding 2 index 2 index put pop 1 add}{exch pop}ifelse }forall pop /NextStepEncoding NextStepEncoding readonly def /_NXfstr 128 string dup 0 (_NX) putinterval def /findfont{ % Because we can never let NextStepEncoding get into % SharedFontDirectory, we cannot reencode a font to NextStepEncoding % if we are in shared mode. So if currentshared is true, % we call the normal findfont and return that /currentshared where {pop currentshared} {false} ifelse {//findfont exec} {dup _NXfstr 3 125 getinterval cvs length 3 add _NXfstr 0 3 -1 roll getinterval cvn exch FontDirectory 2 index known {pop FontDirectory exch get} {//findfont exec dup /Encoding get StandardEncoding eq { dup length dict exch {1 index /FID ne {2 index 3 1 roll put}{pop pop}ifelse}forall dup /Encoding NextStepEncoding put definefont }{exch pop} ifelse }ifelse }ifelse }bind def }{pop}ifelse /_NXProcArray 5 array __NXdef /_NXChannels 0 __NXdef /_NXTotalBytes 0 __NXdef /_NXDoImageOp{ 1 index{dup}{1}ifelse /_NXChannels exch store _NXChannels 2 add 2 roll _NXProcArray 0 _NXChannels getinterval astore pop 5 index 4 index mul 2 index{1 sub 8 idiv 1 add mul}{mul 1 sub 8 idiv 1 add}ifelse 4 index mul /_NXTotalBytes exch store pop exch pop gsave matrix invertmatrix concat 0.5 setgray 0 0 4 2 roll rectfill grestore { 0 1 _NXChannels 1 sub{ _NXProcArray 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For example, let us find the the Fourier series of the function defined by ;[s] 4:0,0;13,1;23,2;32,3;154,-1; 4:1,10,8,Times,2,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Courier,0,12,0,0,0;1,11,8,Times,0,12,0,0,0; :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 0; pictureWidth = 128; pictureHeight = 13; ] %! %%Creator: Mathematica %%AspectRatio: 0.101562 200 dict begin MathPictureStart /Courier findfont 10 scalefont setfont % Scaling calculations 0 1 0 1 [ [ 0 0 0 0 ] [ 128 13 0 0 ] ] MathScale % Start of Graphics %!PS-Adobe-2.0 EPSF-2.0 %%Title: %%Creator: EquationBuilder %%CreationDate: Mon Aug 3 20:00:30 1992 %%For: gray %%DocumentFonts: (atend) %%Pages: 0 1 %%BoundingBox: 0 0 128 13 %%Translation: 1 2.46 126.585327 0 %%EndComments %%BeginDocument: /usr/lib/NextStep/printPackage.ps %!PS-Adobe-2.0 %%Title: Printing Package %%EndComments %Version: 2.0 %Copyright: 1988, NeXT, Inc. /__NXdef{1 index where{pop pop pop}{def}ifelse}bind def /__NXbdef{1 index where{pop pop pop}{bind def}ifelse}bind def /UserObjects 10 array __NXdef /defineuserobject{ exch dup 1 add dup UserObjects length gt{ array dup 0 UserObjects putinterval /UserObjects exch def }{pop}ifelse UserObjects exch 3 -1 roll put }__NXbdef /undefineuserobject{UserObjects exch null put}__NXbdef /execuserobject{UserObjects exch get exec}__NXbdef /__NXRectPath{4 2 roll moveto 1 index 0 rlineto 0 exch rlineto neg 0 rlineto closepath}__NXbdef /__NXProcessRectArgs{ 1 index type /arraytype eq{ exch 0 4 2 index length 1 sub{ dup 3 add 1 exch{1 index exch get exch}for 5 1 roll 5 index exec }for pop pop }{exec}ifelse }__NXbdef /rectfill{gsave newpath {__NXRectPath fill} __NXProcessRectArgs grestore}__NXbdef /rectclip{newpath {__NXRectPath} __NXProcessRectArgs clip newpath}__NXbdef /rectstroke{ gsave newpath dup type /arraytype eq{dup length 6 eq}{false}ifelse{ {gsave __NXRectPath null concat stroke grestore} dup length array cvx copy dup 2 4 -1 roll put __NXProcessRectArgs }{{__NXRectPath stroke} __NXProcessRectArgs}ifelse grestore }__NXbdef /xyshow{ 0 1 3 index length 1 sub{ currentpoint 4 index 3 index 1 getinterval show 3 index 3 index 2 mul 1 add get add exch 3 index 3 index 2 mul get add exch moveto pop }for pop pop }__NXbdef /xshow{ 0 1 3 index length 1 sub{ currentpoint 4 index 3 index 1 getinterval show exch 3 index 3 index get add exch moveto pop }for pop pop }__NXbdef /yshow{ 0 1 3 index length 1 sub{ currentpoint 4 index 3 index 1 getinterval show 3 index 3 index get add moveto pop }for pop pop }__NXbdef /arct{arcto pop pop pop pop}__NXbdef /setbbox{pop pop pop pop}__NXbdef /ucache{}__NXbdef /ucachestatus{mark 0 0 0 0 0}__NXbdef /setucacheparams{cleartomark}__NXbdef /uappend{systemdict begin cvx exec end}__NXbdef /ueofill{gsave newpath uappend eofill grestore}__NXbdef /ufill{gsave newpath uappend fill grestore}__NXbdef /ustroke{ gsave newpath dup length 6 eq {exch uappend concat}{uappend}ifelse stroke grestore }__NXbdef /__NXustrokepathMatrix dup where {pop pop}{matrix def}ifelse /ustrokepath{ newpath dup length 6 eq{ exch uappend __NXustrokepathMatrix currentmatrix exch concat strokepath setmatrix }{uappend strokepath}ifelse } __NXbdef /upath{ [exch {/ucache cvx}if pathbbox /setbbox cvx {/moveto cvx}{/lineto cvx}{/curveto cvx}{/closepath cvx}pathforall]cvx } __NXbdef /setstrokeadjust{pop}__NXbdef /currentstrokeadjust{false}__NXbdef /selectfont{exch findfont exch dup type /arraytype eq {makefont}{scalefont}ifelse setfont}__NXbdef /_NXCombineArrays{ counttomark dup 2 add index dup length 3 -1 roll { 2 index length sub dup 4 1 roll 1 index exch 4 -1 roll putinterval exch }repeat pop pop pop }__NXbdef /setcmykcolor{ 1.0 exch sub dup dup 6 -1 roll sub dup 0 lt{pop 0}if 5 1 roll 4 -1 roll sub dup 0 lt{pop 0}if 3 1 roll exch sub dup 0 lt{pop 0}if setrgbcolor }__NXbdef /currentcmykcolor{currentrgbcolor 3{1.0 exch sub 3 1 roll}repeat 0}__NXbdef /flushgraphics{}def /setwindowtype{pop pop}def /currentwindowtype{pop 0}def /setalpha{pop}def /currentalpha{1.0}def /hidecursor{}def /obscurecursor{}def /revealcursor{}def /setcursor{4 {pop}repeat}bind def /showcursor{}def /NextStepEncoding where not{ /NextStepEncoding StandardEncoding 256 array copy def 0 [129/Agrave/Aacute/Acircumflex/Atilde/Adieresis/Aring/Ccedilla/Egrave /Eacute/Ecircumflex/Edieresis/Igrave/Iacute/Icircumflex/Idieresis /Eth/Ntilde/Ograve/Oacute/Ocircumflex/Otilde/Odieresis/Ugrave/Uacute /Ucircumflex/Udieresis/Yacute/Thorn/mu/multiply/divide/copyright 176/registered 181/brokenbar 190/logicalnot 192/onesuperior 201/twosuperior 204/threesuperior 209/plusminus/onequarter/onehalf/threequarters/agrave /aacute/acircumflex/atilde/adieresis/aring/ccedilla/egrave/eacute /ecircumflex/edieresis/igrave 226/iacute 228/icircumflex/idieresis/eth /ntilde 236/ograve/oacute/ocircumflex/otilde/odieresis 242/ugrave/uacute /ucircumflex 246/udieresis/yacute 252/thorn/ydieresis] {dup type /nametype eq {NextStepEncoding 2 index 2 index put pop 1 add}{exch pop}ifelse }forall pop /NextStepEncoding NextStepEncoding readonly def /_NXfstr 128 string dup 0 (_NX) putinterval def /findfont{ % Because we can never let NextStepEncoding get into % SharedFontDirectory, we cannot reencode a font to NextStepEncoding % if we are in shared mode. So if currentshared is true, % we call the normal findfont and return that /currentshared where {pop currentshared} {false} ifelse {//findfont exec} {dup _NXfstr 3 125 getinterval cvs length 3 add _NXfstr 0 3 -1 roll getinterval cvn exch FontDirectory 2 index known {pop FontDirectory exch get} {//findfont exec dup /Encoding get StandardEncoding eq { dup length dict exch {1 index /FID ne {2 index 3 1 roll put}{pop pop}ifelse}forall dup /Encoding NextStepEncoding put definefont }{exch pop} ifelse }ifelse }ifelse }bind def }{pop}ifelse /_NXProcArray 5 array __NXdef /_NXChannels 0 __NXdef /_NXTotalBytes 0 __NXdef /_NXDoImageOp{ 1 index{dup}{1}ifelse /_NXChannels exch store _NXChannels 2 add 2 roll _NXProcArray 0 _NXChannels getinterval astore pop 5 index 4 index mul 2 index{1 sub 8 idiv 1 add mul}{mul 1 sub 8 idiv 1 add}ifelse 4 index mul /_NXTotalBytes exch store pop exch pop gsave matrix invertmatrix concat 0.5 setgray 0 0 4 2 roll rectfill grestore { 0 1 _NXChannels 1 sub{ _NXProcArray exch get exec length _NXTotalBytes exch sub /_NXTotalBytes exch store}for _NXTotalBytes 0 le{exit}if }loop /_NXProcArray 5 array def }__NXbdef /colorimage{_NXDoImageOp}__NXbdef /alphaimage{1 add _NXDoImageOp}def %%EndDocument % EquationBuilder(TM) Expression % +.\Jr(=:/*V[StRFQAYmdjEEA;1nuN+[co.OZ7L#cq3iK%3:</XJ@:d#6;^G;Td47dud<A3@ % 5%MGToUOR5&d`'TuEDU)B-tbR".R+:nN<%57e_!+HKN-qejV,X_XRS2ihA"VLe`-6WLgVDQ[ % K-Oh*sQ3\5X&jqDA/d.K[>m^Q5J[c<T!&,[Tin#Mqf\EPQp`n<3FPaBu@nDhna*n=K8V8]^g % &Ph<W8/Xb#*:Hd'hl`u7SI4.M@CdIJgQEre;H7KcIB<:W4die#[Ice^a50F<)<u'U(j:oLAW % qP;MIo72JMBfKM=tFA(lTP#GB651SIU*8_(OlL_Odb!0!C&*YFI,o&kLh9%CPfTE+`fJ@Z9L % #GD"ro_4t#!2P]C(]4$;<,r[-O:iKNET@=HTZf=N?KUKRJ:*$O2d?ldN0#$(f3JZSId!bD>U % g-pOoQ2=\:G<I(1p'CL:qTj@72ZRR(sDTKJ>q6e6+.*G.(fr?[<`E1.-)ne<o<R3AP<+P>Fk % sj-&\98P#'lg8H9S3dM+pOXB(T@ZsT:WTBLIG1l#G88_"rQ"Ia%jdK8rWMc1rqAJF[JE#6/7 % 1'<_L)[;W60"V]K!M)8likbQLk"tRWJ\U0OW5m>C8bX=4HT;CW,[!jihr_mW[]0tQqAQk@T) % HjLTt4o.@bcdlG@n%Q8/t!s6pcW&9Z&K`uD3.(>E=\eJ`7ZHt<?d!c%*W<$)Mo?iYN];":&J % /P)R;(iHP_J[fQY.n!cU,XFp;cAfZ3!pmK+0.;AS#Ts+5!:b!CQOj_U(dW*B!%7a_!WW3#x % EndExpression N 629 275 E 3 S 10a79 R 99133853 gsave /__NXbasematrix matrix currentmatrix def grestore %%EndProlog %%BeginSetup %%EndSetup 0 0 127.585327 12.836 rectclip 0 0 127.585327 12.836 rectclip 0 setgray 1 setalpha 1 1 scale /Symbol findfont 12 scalefont 48 exch defineuserobject 48 execuserobject setfont 5.56 3.568 moveto (\() [0 0] xyshow 1 1 scale 1 1 scale 48 execuserobject setfont 17.200001 3.562 moveto (\)) [0 0] xyshow 1 1 scale 0 setgray 1 setalpha /Times-BoldItalic findfont 12 scalefont 49 exch defineuserobject 49 execuserobject setfont 123.585335 3.46 moveto (.xxxf) [-25.842667 0 -62.764 0 -24.618668 0 -9.360001 0 0 0 0 0] xyshow 48 execuserobject setfont 116.997337 3.46 moveto (p<<p-=) [-9.921333 0 -19.254669 0 -9.921333 0 -9.254669 0 -43.587997 0 0 0 43.842667 13.806] xyshow /Times-Roman findfont 12 scalefont 50 exch defineuserobject 50 execuserobject setfont 40.978668 3.46 moveto (,) [0 0 3.335999 0] xyshow %%Trailer %%DocumentFonts: Times-BoldItalic %%+ Times-Roman %%+ Symbol %End of Graphics MathPictureEnd end :[font = text; inactive; preserveAspect; ] We know that the Fourier series has the form :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 0; pictureWidth = 165; pictureHeight = 36; ] %! %%Creator: Mathematica %%AspectRatio: 0.218182 200 dict begin MathPictureStart /Courier findfont 10 scalefont setfont % Scaling calculations 0 1 0 1 [ [ 0 0 0 0 ] [ 165 36 0 0 ] ] MathScale % Start of Graphics %!PS-Adobe-2.0 EPSF-2.0 %%Title: %%Creator: EquationBuilder %%CreationDate: Mon Aug 3 20:00:55 1992 %%For: gray %%DocumentFonts: (atend) %%Pages: 0 1 %%BoundingBox: 0 0 165 36 %%Translation: 1 14.884993 163.996643 0 %%EndComments %%BeginDocument: /usr/lib/NextStep/printPackage.ps %!PS-Adobe-2.0 %%Title: Printing Package %%EndComments %Version: 2.0 %Copyright: 1988, NeXT, Inc. /__NXdef{1 index where{pop pop pop}{def}ifelse}bind def /__NXbdef{1 index where{pop pop pop}{bind def}ifelse}bind def /UserObjects 10 array __NXdef /defineuserobject{ exch dup 1 add dup UserObjects length gt{ array dup 0 UserObjects putinterval /UserObjects exch def }{pop}ifelse UserObjects exch 3 -1 roll put }__NXbdef /undefineuserobject{UserObjects exch null put}__NXbdef /execuserobject{UserObjects exch get exec}__NXbdef /__NXRectPath{4 2 roll moveto 1 index 0 rlineto 0 exch rlineto neg 0 rlineto closepath}__NXbdef /__NXProcessRectArgs{ 1 index type /arraytype eq{ exch 0 4 2 index length 1 sub{ dup 3 add 1 exch{1 index exch get exch}for 5 1 roll 5 index exec }for pop pop }{exec}ifelse }__NXbdef /rectfill{gsave newpath {__NXRectPath fill} __NXProcessRectArgs grestore}__NXbdef /rectclip{newpath {__NXRectPath} __NXProcessRectArgs clip newpath}__NXbdef /rectstroke{ gsave newpath dup type /arraytype eq{dup length 6 eq}{false}ifelse{ {gsave __NXRectPath null concat stroke grestore} dup length array cvx copy dup 2 4 -1 roll put __NXProcessRectArgs }{{__NXRectPath stroke} __NXProcessRectArgs}ifelse grestore }__NXbdef /xyshow{ 0 1 3 index length 1 sub{ currentpoint 4 index 3 index 1 getinterval show 3 index 3 index 2 mul 1 add get add exch 3 index 3 index 2 mul get add exch moveto pop }for pop pop }__NXbdef /xshow{ 0 1 3 index length 1 sub{ currentpoint 4 index 3 index 1 getinterval show exch 3 index 3 index get add exch moveto pop }for pop pop }__NXbdef /yshow{ 0 1 3 index length 1 sub{ currentpoint 4 index 3 index 1 getinterval show 3 index 3 index get add moveto pop }for pop pop }__NXbdef /arct{arcto pop pop pop pop}__NXbdef /setbbox{pop pop pop pop}__NXbdef /ucache{}__NXbdef /ucachestatus{mark 0 0 0 0 0}__NXbdef /setucacheparams{cleartomark}__NXbdef /uappend{systemdict begin cvx exec end}__NXbdef /ueofill{gsave newpath uappend eofill grestore}__NXbdef /ufill{gsave newpath uappend fill grestore}__NXbdef /ustroke{ gsave newpath dup length 6 eq {exch uappend concat}{uappend}ifelse stroke grestore }__NXbdef /__NXustrokepathMatrix dup where {pop pop}{matrix def}ifelse /ustrokepath{ newpath dup length 6 eq{ exch uappend __NXustrokepathMatrix currentmatrix exch concat strokepath setmatrix }{uappend strokepath}ifelse } __NXbdef /upath{ [exch {/ucache cvx}if pathbbox /setbbox cvx {/moveto cvx}{/lineto cvx}{/curveto cvx}{/closepath cvx}pathforall]cvx } __NXbdef /setstrokeadjust{pop}__NXbdef /currentstrokeadjust{false}__NXbdef /selectfont{exch findfont exch dup type /arraytype eq {makefont}{scalefont}ifelse setfont}__NXbdef /_NXCombineArrays{ counttomark dup 2 add index dup length 3 -1 roll { 2 index length sub dup 4 1 roll 1 index exch 4 -1 roll putinterval exch }repeat pop pop pop }__NXbdef /setcmykcolor{ 1.0 exch sub dup dup 6 -1 roll sub dup 0 lt{pop 0}if 5 1 roll 4 -1 roll sub dup 0 lt{pop 0}if 3 1 roll exch sub dup 0 lt{pop 0}if setrgbcolor }__NXbdef /currentcmykcolor{currentrgbcolor 3{1.0 exch sub 3 1 roll}repeat 0}__NXbdef /flushgraphics{}def /setwindowtype{pop pop}def /currentwindowtype{pop 0}def /setalpha{pop}def /currentalpha{1.0}def /hidecursor{}def /obscurecursor{}def /revealcursor{}def /setcursor{4 {pop}repeat}bind def /showcursor{}def /NextStepEncoding where not{ /NextStepEncoding StandardEncoding 256 array copy def 0 [129/Agrave/Aacute/Acircumflex/Atilde/Adieresis/Aring/Ccedilla/Egrave /Eacute/Ecircumflex/Edieresis/Igrave/Iacute/Icircumflex/Idieresis /Eth/Ntilde/Ograve/Oacute/Ocircumflex/Otilde/Odieresis/Ugrave/Uacute /Ucircumflex/Udieresis/Yacute/Thorn/mu/multiply/divide/copyright 176/registered 181/brokenbar 190/logicalnot 192/onesuperior 201/twosuperior 204/threesuperior 209/plusminus/onequarter/onehalf/threequarters/agrave /aacute/acircumflex/atilde/adieresis/aring/ccedilla/egrave/eacute /ecircumflex/edieresis/igrave 226/iacute 228/icircumflex/idieresis/eth /ntilde 236/ograve/oacute/ocircumflex/otilde/odieresis 242/ugrave/uacute /ucircumflex 246/udieresis/yacute 252/thorn/ydieresis] {dup type /nametype eq {NextStepEncoding 2 index 2 index put pop 1 add}{exch pop}ifelse }forall pop /NextStepEncoding NextStepEncoding readonly def /_NXfstr 128 string dup 0 (_NX) putinterval def /findfont{ % Because we can never let NextStepEncoding get into % SharedFontDirectory, we cannot reencode a font to NextStepEncoding % if we are in shared mode. So if currentshared is true, % we call the normal findfont and return that /currentshared where {pop currentshared} {false} ifelse {//findfont exec} {dup _NXfstr 3 125 getinterval cvs length 3 add _NXfstr 0 3 -1 roll getinterval cvn exch FontDirectory 2 index known {pop FontDirectory exch get} {//findfont exec dup /Encoding get StandardEncoding eq { dup length dict exch {1 index /FID ne {2 index 3 1 roll put}{pop pop}ifelse}forall dup /Encoding NextStepEncoding put definefont }{exch pop} ifelse }ifelse }ifelse }bind def }{pop}ifelse /_NXProcArray 5 array __NXdef /_NXChannels 0 __NXdef /_NXTotalBytes 0 __NXdef /_NXDoImageOp{ 1 index{dup}{1}ifelse /_NXChannels exch store _NXChannels 2 add 2 roll _NXProcArray 0 _NXChannels getinterval astore pop 5 index 4 index mul 2 index{1 sub 8 idiv 1 add mul}{mul 1 sub 8 idiv 1 add}ifelse 4 index mul /_NXTotalBytes exch store pop exch pop gsave matrix invertmatrix concat 0.5 setgray 0 0 4 2 roll rectfill grestore { 0 1 _NXChannels 1 sub{ _NXProcArray exch get exec length _NXTotalBytes exch sub /_NXTotalBytes exch store}for _NXTotalBytes 0 le{exit}if }loop /_NXProcArray 5 array def }__NXbdef /colorimage{_NXDoImageOp}__NXbdef /alphaimage{1 add _NXDoImageOp}def %%EndDocument % EquationBuilder(TM) Expression % +.\Jr(=:/*V[StRFQAYmdjEEA;1nuN+[co.OZ7L#cq3iK%3:</XJ@:d#6;^G;Td47dud<A3@ % 5%MGToUOR5&d`'TuEDU)B-tbR".R+:s'$,VT6!0M"`kV$VD;,X_XRS2ihA"VLe`-6WLgVDQ[ % K-Oh*sQ3\5X&jqDA/d.K[>m^Q5J[c<T!&,[Tin#Mqf\EPQp`n<3FPaBu@nDhna*n=K8V8]^g % &Ph0W+B`H8/K#LT#`(=`_@:1Gn;L'ZfuJ,^7@LuTXn]hB>4eNjCohcYj01BLS-[+S+U8eYX. % Y&,XCQk:]W2:%%r2]!"c'0#;P9u6'4'E"r8;'4S`;^cO3:iNf(Mjm\7i;MLGJLF9([67tb<l % &:+?'XlkQeTh'3],cnY6Rh2EeQ<IaUOltA4o=al?.-fb]8:/pi^`>k;'dSQ(Dobd\CD/$B!, % XLH(8#*ER-.h&lYWS-!2XY4a^[&+35?qq]I4(>11#34Ib)("+X%8i`.g`$\M;rN6rmD3J/KO % K;%K<&+p7u:;-B]C20'O2%/h'S?8A2OP9u^m,t#`RC"+TI;VZ][>&lh3q:IH4!dLN#KVZI*. % XqY[@@?O\&>irf=ui@NN"QcHS!;oVN?lb>&re0e-qI3Q$'f\RM0hjpVM8[m0ZG&Q/=F9fE.f % r8)5sF2.\,qOn>ITHdPkljXCc[UT=PBAWg3\D_-JUk.SF9VKXAUe/^].o^amJuif9a9&djPW % O+$==O@%Bjkb&n:43V4"mYYSPZZQ$)YOP5S.?_&G9'du,\iK,>!7Z&PEdB&_)nM(:Y\p9L$# % sXri)Y_-QY==kXi'`/9B`2SYNLE]"A",p,Rc*"9t",Lh!,lUEVBpc+gVN5YMBCu+g3!fWcQA % MAs\G%%oXF0$iV1b6CWA]2FJR+bY_iP;K^29?^ZFrJ6I:.eHZ0_(nfaf<2@or8hET-=;Yr># % =7$Ucd@=UJWZ&t*FZ17`YeP$!&rVRZU%'I!<=N=iB[GEb2E=GIL-#eN=c:@!-$qY+TMKBx % EndExpression N 801 321 E aa S 15cba R bbf4e8c8 gsave /__NXbasematrix matrix currentmatrix def grestore %%EndProlog %%BeginSetup /EqBFont 18 dict dup begin /FontType 3 def /PaintType 0 def /FontMatrix [0.001 0 0 0.001 0 0] def /FontBBox [15 -2950 1360 1045] def /Encoding [/.notdef /rad1 /rad2 /rad3 /rad4 /rad5 /radext /int /pathint /sum /prod /cdot /vdot /ddot /hbar /bigcap /bigcup /parallel] def /Metrics 18 dict dup begin /.notdef 0 def /rad1 1000 def /rad2 1000 def /rad3 1000 def /rad4 1000 def /rad5 1048 def /radext 1048 def /int 500 def /pathint 500 def /sum 700 def /prod 700 def /cdot 1000 def /vdot 550 def /ddot 1000 def /hbar 500 def /bigcap 700 def /bigcup 700 def /parallel 500 def end def /BBox 18 dict dup begin /.notdef [0 0 0 0] def /rad1 [110 -1150 1020 0] def /rad2 [110 -1750 1020 0] def /rad3 [110 -2350 1020 0] def /rad4 [110 -2950 1020 0] def /rad5 [128 -2890 740 0] def /radext [700 -600 740 0] def /int [20 0 425 1045] def /pathint [20 0 425 1045] def /sum [15 5 640 715] def /prod [15 0 670 715] def /cdot [110 215 890 325] def /vdot [165 -5 280 770] def /ddot [110 -5 890 770] def /hbar [20 -10 480 685] def /bigcap [0 0 680 720] def /bigcup [0 0 680 720] def /parallel [65 -175 400 675] def end def /CharacterDefs 18 dict dup begin /.notdef { } def /rad1 { 980 0 moveto 464 -1048 lineto 248 -560 lineto 110 -656 lineto 125 -675 lineto 200 -624 lineto 420 -1150 lineto 460 -1150 lineto 1020 0 lineto 980 0 lineto closepath } def /rad2 { 980 0 moveto 460 -1600 lineto 245 -864 lineto 110 -1000 lineto 128 -1023 lineto 200 -950 lineto 424 -1750 lineto 456 -1750 lineto 1020 0 lineto 980 0 lineto closepath } def /rad3 { 980 0 moveto 464 -2144 lineto 248 -1160 lineto 110 -1368 lineto 130 -1390 lineto 192 -1288 lineto 424 -2350 lineto 464 -2350 lineto 1020 0 lineto 980 0 lineto closepath } def /rad4 { 980 0 moveto 880 -488 lineto 672 -1600 lineto 464 -2696 lineto 248 -1464 lineto 110 -1728 lineto 130 -1750 lineto 192 -1616 lineto 424 -2950 lineto 464 -2950 lineto 768 -1312 lineto 1020 0 lineto 980 0 lineto closepath } def /rad5 { 700 0 moveto 700 -2628 lineto 248 -1284 lineto 128 -1564 lineto 150 -1580 lineto 215 -1425 lineto 688 -2890 lineto 740 -2890 lineto 740 0 lineto 700 0 lineto closepath } def /radext { 700 0 moveto 700 -600 lineto 740 -600 lineto 740 0 lineto 700 0 lineto closepath } def /int { 245 390 moveto 225 235 195 145 180 100 curveto 165 60 150 50 140 40 curveto 135 30 85 0 50 20 curveto 35 25 25 45 25 55 curveto 20 80 30 100 55 110 curveto 80 125 93 110 110 90 curveto 135 60 155 135 160 195 curveto 165 270 170 340 190 510 curveto 205 650 lineto 220 810 230 890 265 960 curveto 275 980 295 1015 320 1030 curveto 350 1045 380 1030 395 1020 curveto 410 1010 425 980 400 955 curveto 385 940 347 925 330 950 curveto 310 970 290 955 290 940 curveto 280 900 285 745 270 610 curveto 245 390 lineto closepath } def /pathint { 245 390 moveto 225 235 195 145 180 100 curveto 165 60 150 50 140 40 curveto 135 30 85 0 50 20 curveto 35 25 25 45 25 55 curveto 20 80 30 100 55 110 curveto 80 125 93 110 110 90 curveto 135 60 155 135 160 195 curveto 165 270 170 340 190 510 curveto 205 650 lineto 220 810 230 890 265 960 curveto 275 980 295 1015 320 1030 curveto 350 1045 380 1030 395 1020 curveto 410 1010 425 980 400 955 curveto 385 940 347 925 330 950 curveto 310 970 290 955 290 940 curveto 280 900 285 745 270 610 curveto 245 390 lineto fill 20 setlinewidth 220 400 moveto 290 400 340 455 340 500 curveto 340 585 290 640 220 640 curveto 155 640 100 585 100 500 curveto 100 455 155 400 220 400 curveto stroke } def /sum { 330 335 moveto 20 715 lineto 575 715 lineto 600 545 lineto 575 535 lineto 550 625 525 685 395 685 curveto 160 685 lineto 411 370 lineto 160 90 lineto 470 90 lineto 565 90 590 140 610 210 curveto 640 200 lineto 585 5 lineto 15 5 lineto 330 335 lineto closepath } def /prod { 590 90 moveto 590 30 620 20 670 20 curveto 670 0 lineto 410 0 lineto 410 20 lineto 460 20 495 30 495 90 curveto 495 670 lineto 190 670 lineto 190 90 lineto 190 30 225 20 275 20 curveto 275 0 lineto 15 0 lineto 15 20 lineto 65 20 100 30 100 90 curveto 100 615 lineto 100 690 40 700 20 700 curveto 20 715 lineto 670 715 lineto 670 700 lineto 645 700 590 690 590 615 curveto 590 90 lineto closepath } def /cdot { 110 270 moveto 110 300 135 325 165 325 curveto 195 325 220 300 220 270 curveto 220 235 185 215 165 215 curveto 145 215 110 235 110 270 curveto fill 445 270 moveto 445 300 470 325 500 325 curveto 530 325 555 300 555 270 curveto 555 235 520 215 500 215 curveto 480 215 445 235 445 270 curveto fill 775 270 moveto 775 300 800 325 830 325 curveto 860 325 890 300 890 270 curveto 890 235 850 215 830 215 curveto 810 215 775 235 775 270 curveto fill } def /vdot { 220 770 moveto 250 770 280 750 280 720 curveto 280 680 250 660 220 660 curveto 185 660 165 695 165 720 curveto 165 740 185 770 220 770 curveto fill 220 440 moveto 250 440 280 415 280 385 curveto 280 355 250 330 220 330 curveto 185 330 165 365 165 385 curveto 165 405 185 440 220 440 curveto fill 220 105 moveto 250 105 280 80 280 50 curveto 280 20 250 -5 220 -5 curveto 185 -5 165 30 165 50 curveto 165 70 185 105 220 105 curveto fill } def /ddot { 165 720 moveto 205 720 220 700 220 665 curveto 220 630 195 610 165 610 curveto 130 620 110 645 110 665 curveto 110 690 130 720 165 720 curveto fill 500 440 moveto 530 440 555 415 555 385 curveto 555 355 530 330 500 330 curveto 465 330 445 365 445 385 curveto 445 405 465 440 500 440 curveto fill 830 155 moveto 860 155 890 130 890 100 curveto 890 70 860 45 830 45 curveto 795 45 775 80 775 100 curveto 775 120 795 155 830 155 curveto fill } def /hbar { 480 105 moveto 440 55 400 -10 335 -10 curveto 300 -10 290 10 290 35 curveto 290 65 370 350 370 365 curveto 370 380 355 390 340 390 curveto 290 390 205 275 185 250 curveto 140 180 125 125 95 0 curveto 20 0 lineto 150 500 lineto 160 545 180 600 180 615 curveto 180 625 170 640 135 640 curveto 110 640 lineto 100 655 lineto 165 660 215 670 270 685 curveto 275 680 lineto 155 240 lineto 155 235 lineto 190 290 290 440 380 440 curveto 400 440 450 435 450 370 curveto 450 335 365 80 365 55 curveto 365 45 375 40 385 40 curveto 405 40 425 70 465 115 curveto 480 105 lineto fill 55 545 moveto 55 520 lineto 310 560 lineto 310 585 lineto 55 545 lineto fill } def /bigcap { 615 400 moveto 615 575 500 655 340 655 curveto 180 655 70 575 70 400 curveto 70 0 lineto 5 0 lineto 5 400 lineto 5 610 140 720 340 720 curveto 545 720 680 610 680 400 curveto 680 0 lineto 615 0 lineto 615 400 lineto closepath } def /bigcup { 65 320 moveto 65 145 180 65 340 65 curveto 500 65 615 145 615 320 curveto 615 720 lineto 680 720 lineto 680 320 lineto 680 110 540 0 340 0 curveto 140 0 0 110 0 320 curveto 0 720 lineto 65 720 lineto 65 320 lineto closepath } def /parallel { 135 675 moveto 135 -175 lineto 65 -175 lineto 65 675 lineto 135 675 lineto fill 335 675 moveto 335 -175 lineto 265 -175 lineto 265 675 lineto 335 675 lineto fill 400 675 moveto } def end def /BuildChar { 0 begin /char exch def /fontdict exch def /charname fontdict /Encoding get char get def fontdict begin Metrics charname get 0 BBox charname get aload pop setcachedevice CharacterDefs charname get exec eofill end end } def /BuildChar load 0 3 dict put end definefont pop %%EndSetup 0 0 164.996643 35.640968 rectclip 0 0 164.996643 35.640968 rectclip 0 setgray 1 setalpha /EqBFont 24 selectfont 26.437834 10.244992 moveto <09> [0 0] xyshow 0 setgray 1 setalpha /EqBFont 24 selectfont 100.580162 10.244992 moveto <09> [0 0] xyshow 0 setgray 1 setalpha /Times-BoldItalic findfont 12 scalefont 49 exch defineuserobject 49 execuserobject setfont 160.996658 15.884993 moveto (.xnBxnAA) [-6 0 -6.671997 0 -30.211998 0 -35.266335 0 -6.671997 0 -32.203999 0 -42.970333 0 0 0 0 0] xyshow /Symbol findfont 9 scalefont 60 exch defineuserobject 60 execuserobject setfont 106.221664 2.281 moveto (=\245=\245) [-0.990005 27.523968 -73.152328 -27.523968 -0.990002 27.523968 0 0 0 0] xyshow /Times-Roman findfont 9 scalefont 51 exch defineuserobject 51 execuserobject setfont 111.612663 2.281 moveto (110) [-74.142334 0 -28.466333 11.804017 0 0 0 0] xyshow /Symbol findfont 12 scalefont 48 exch defineuserobject 48 execuserobject setfont 91.513 15.884993 moveto (++) [-74.142334 0 0 0 -9.921333 0] xyshow /Times-BoldItalic findfont 9 scalefont 67 exch defineuserobject 67 execuserobject setfont 126.116661 14.085017 moveto (nnnn) [-25.348999 -11.804017 -48.793327 11.804017 -25.349001 -11.804017 0 0 4.867859e-38 3.798623e+09] xyshow /Times-Roman findfont 12 scalefont 50 exch defineuserobject 50 execuserobject setfont 132.320663 15.884993 moveto (sincos) [4.667999 0 3.335999 0 -82.146332 0 5.327999 0 6 0 0 0 0 0] xyshow %%Trailer %%DocumentFonts: Times-BoldItalic %%+ Times-Roman %%+ Symbol %End of Graphics MathPictureEnd end :[font = text; inactive; preserveAspect; ] First, the command :[font = input; preserveAspect; startGroup; ] Integrate[x Cos[n x],{x,-Pi,Pi}] :[font = output; output; inactive; preserveAspect; endGroup; ] 0 ;[o] 0 :[font = text; inactive; preserveAspect; ] tells us that each An is zero. To find Bn we try ;[s] 7:0,0;20,1;21,2;22,3;42,4;43,5;44,6;53,-1; 7:1,11,8,Times,0,12,0,0,0;1,10,8,Times,2,12,0,0,0;1,10,8,Times,66,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Times,2,12,0,0,0;1,10,8,Times,66,12,0,0,0;1,11,8,Times,0,12,0,0,0; :[font = input; preserveAspect; startGroup; ] Integrate[x Sin[n x],{x,-Pi,Pi}] :[font = output; output; inactive; preserveAspect; endGroup; ] (-2*Pi*Cos[n*Pi])/n + (2*Sin[n*Pi])/n^2 ;[o] -2 Pi Cos[n Pi] 2 Sin[n Pi] --------------- + ----------- n 2 n :[font = text; inactive; preserveAspect; ] This is not exactly what we want because Mathematica does not know that n is an integer. Here is a better version: ;[s] 5:0,0;41,1;52,2;73,3;74,4;118,-1; 5:1,11,8,Times,0,12,0,0,0;1,10,8,Times,2,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Courier,0,12,0,0,0;1,11,8,Times,0,12,0,0,0; :[font = input; preserveAspect; startGroup; ] Integrate[x Sin[n x],{x,-Pi,Pi}] /. {Sin[n Pi]->0,Cos[n Pi]->(-1)^n} :[font = output; output; inactive; preserveAspect; endGroup; ] (-2*(-1)^n*Pi)/n ;[o] n -2 (-1) Pi ----------- n :[font = text; inactive; pageBreakBelow; preserveAspect; ] Here we have used the command /. which means "goes to". Note also that -> consists of the two characters - and >. Without special programming Mathematica has to be told that sin(p x)=0; this is done via the command Sin[n Pi]->0. Similar remarks apply to the other substitutions. ;[s] 15:0,0;30,1;33,2;74,3;76,4;109,5;111,6;117,7;118,8;148,9;159,10;185,11;186,12;222,13;234,14;286,-1; 15:1,11,8,Times,0,12,0,0,0;1,10,8,Courier,0,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Courier,0,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Courier,0,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Courier,0,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Times,2,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,0,0,Symbol,0,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Courier,0,12,0,0,0;1,11,8,Times,0,12,0,0,0; :[font = text; inactive; preserveAspect; ] Mathematica can be used in many cases to find Fourier coefficients, but sometimes it does not work. For example, although the absolute value function is represented in Mathematica by Abs, Mathematica does not know how to integrate it. So to find the Fourier coefficient of ;[s] 8:0,0;11,1;170,2;181,3;185,4;188,5;190,6;201,7;276,-1; 8:1,10,8,Times,2,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Times,2,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Courier,0,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Times,2,12,0,0,0;1,11,8,Times,0,12,0,0,0; :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 0; pictureWidth = 39; pictureHeight = 13; ] %! %%Creator: Mathematica %%AspectRatio: 0.333333 200 dict begin MathPictureStart /Courier findfont 10 scalefont setfont % Scaling calculations 0 1 0 1 [ [ 0 0 0 0 ] [ 39 13 0 0 ] ] MathScale % Start of Graphics %!PS-Adobe-2.0 EPSF-2.0 %%Title: %%Creator: EquationBuilder %%CreationDate: Sun Aug 2 09:22:00 1992 %%For: gray %%DocumentFonts: (atend) %%Pages: 0 1 %%BoundingBox: 0 0 39 13 %%Translation: 1 2.1 37.990669 0 %%EndComments %%BeginDocument: /usr/lib/NextStep/printPackage.ps %!PS-Adobe-2.0 %%Title: Printing Package %%EndComments %Version: 2.0 %Copyright: 1988, NeXT, Inc. /__NXdef{1 index where{pop pop pop}{def}ifelse}bind def /__NXbdef{1 index where{pop pop pop}{bind def}ifelse}bind def /UserObjects 10 array __NXdef /defineuserobject{ exch dup 1 add dup UserObjects length gt{ array dup 0 UserObjects putinterval /UserObjects exch def }{pop}ifelse UserObjects exch 3 -1 roll put }__NXbdef /undefineuserobject{UserObjects exch null put}__NXbdef /execuserobject{UserObjects exch get exec}__NXbdef /__NXRectPath{4 2 roll moveto 1 index 0 rlineto 0 exch rlineto neg 0 rlineto closepath}__NXbdef /__NXProcessRectArgs{ 1 index type /arraytype eq{ exch 0 4 2 index length 1 sub{ dup 3 add 1 exch{1 index exch get exch}for 5 1 roll 5 index exec }for pop pop }{exec}ifelse }__NXbdef /rectfill{gsave newpath {__NXRectPath fill} __NXProcessRectArgs grestore}__NXbdef /rectclip{newpath {__NXRectPath} __NXProcessRectArgs clip newpath}__NXbdef /rectstroke{ gsave newpath dup type /arraytype eq{dup length 6 eq}{false}ifelse{ {gsave __NXRectPath null concat stroke grestore} dup length array cvx copy dup 2 4 -1 roll put __NXProcessRectArgs }{{__NXRectPath stroke} __NXProcessRectArgs}ifelse grestore }__NXbdef /xyshow{ 0 1 3 index length 1 sub{ currentpoint 4 index 3 index 1 getinterval show 3 index 3 index 2 mul 1 add get add exch 3 index 3 index 2 mul get add exch moveto pop }for pop pop }__NXbdef /xshow{ 0 1 3 index length 1 sub{ currentpoint 4 index 3 index 1 getinterval show exch 3 index 3 index get add exch moveto pop }for pop pop }__NXbdef /yshow{ 0 1 3 index length 1 sub{ currentpoint 4 index 3 index 1 getinterval show 3 index 3 index get add moveto pop }for pop pop }__NXbdef /arct{arcto pop pop pop pop}__NXbdef /setbbox{pop pop pop pop}__NXbdef /ucache{}__NXbdef /ucachestatus{mark 0 0 0 0 0}__NXbdef /setucacheparams{cleartomark}__NXbdef /uappend{systemdict begin cvx exec end}__NXbdef /ueofill{gsave newpath uappend eofill grestore}__NXbdef /ufill{gsave newpath uappend fill grestore}__NXbdef /ustroke{ gsave newpath dup length 6 eq {exch uappend concat}{uappend}ifelse stroke grestore }__NXbdef /__NXustrokepathMatrix dup where {pop pop}{matrix def}ifelse /ustrokepath{ newpath dup length 6 eq{ exch uappend __NXustrokepathMatrix currentmatrix exch concat strokepath setmatrix }{uappend strokepath}ifelse } __NXbdef /upath{ [exch {/ucache cvx}if pathbbox /setbbox cvx {/moveto cvx}{/lineto cvx}{/curveto cvx}{/closepath cvx}pathforall]cvx } __NXbdef /setstrokeadjust{pop}__NXbdef /currentstrokeadjust{false}__NXbdef /selectfont{exch findfont exch dup type /arraytype eq {makefont}{scalefont}ifelse setfont}__NXbdef /_NXCombineArrays{ counttomark dup 2 add index dup length 3 -1 roll { 2 index length sub dup 4 1 roll 1 index exch 4 -1 roll putinterval exch }repeat pop pop pop }__NXbdef /setcmykcolor{ 1.0 exch sub dup dup 6 -1 roll sub dup 0 lt{pop 0}if 5 1 roll 4 -1 roll sub dup 0 lt{pop 0}if 3 1 roll exch sub dup 0 lt{pop 0}if setrgbcolor }__NXbdef /currentcmykcolor{currentrgbcolor 3{1.0 exch sub 3 1 roll}repeat 0}__NXbdef /flushgraphics{}def /setwindowtype{pop pop}def /currentwindowtype{pop 0}def /setalpha{pop}def /currentalpha{1.0}def /hidecursor{}def /obscurecursor{}def /revealcursor{}def /setcursor{4 {pop}repeat}bind def /showcursor{}def /NextStepEncoding where not{ /NextStepEncoding StandardEncoding 256 array copy def 0 [129/Agrave/Aacute/Acircumflex/Atilde/Adieresis/Aring/Ccedilla/Egrave /Eacute/Ecircumflex/Edieresis/Igrave/Iacute/Icircumflex/Idieresis /Eth/Ntilde/Ograve/Oacute/Ocircumflex/Otilde/Odieresis/Ugrave/Uacute /Ucircumflex/Udieresis/Yacute/Thorn/mu/multiply/divide/copyright 176/registered 181/brokenbar 190/logicalnot 192/onesuperior 201/twosuperior 204/threesuperior 209/plusminus/onequarter/onehalf/threequarters/agrave /aacute/acircumflex/atilde/adieresis/aring/ccedilla/egrave/eacute /ecircumflex/edieresis/igrave 226/iacute 228/icircumflex/idieresis/eth /ntilde 236/ograve/oacute/ocircumflex/otilde/odieresis 242/ugrave/uacute /ucircumflex 246/udieresis/yacute 252/thorn/ydieresis] {dup type /nametype eq {NextStepEncoding 2 index 2 index put pop 1 add}{exch pop}ifelse }forall pop /NextStepEncoding NextStepEncoding readonly def /_NXfstr 128 string dup 0 (_NX) putinterval def /findfont{ % Because we can never let NextStepEncoding get into % SharedFontDirectory, we cannot reencode a font to NextStepEncoding % if we are in shared mode. So if currentshared is true, % we call the normal findfont and return that /currentshared where {pop currentshared} {false} ifelse {//findfont exec} {dup _NXfstr 3 125 getinterval cvs length 3 add _NXfstr 0 3 -1 roll getinterval cvn exch FontDirectory 2 index known {pop FontDirectory exch get} {//findfont exec dup /Encoding get StandardEncoding eq { dup length dict exch {1 index /FID ne {2 index 3 1 roll put}{pop pop}ifelse}forall dup /Encoding NextStepEncoding put definefont }{exch pop} ifelse }ifelse }ifelse }bind def }{pop}ifelse /_NXProcArray 5 array __NXdef /_NXChannels 0 __NXdef /_NXTotalBytes 0 __NXdef /_NXDoImageOp{ 1 index{dup}{1}ifelse /_NXChannels exch store _NXChannels 2 add 2 roll _NXProcArray 0 _NXChannels getinterval astore pop 5 index 4 index mul 2 index{1 sub 8 idiv 1 add mul}{mul 1 sub 8 idiv 1 add}ifelse 4 index mul /_NXTotalBytes exch store pop exch pop gsave matrix invertmatrix concat 0.5 setgray 0 0 4 2 roll rectfill grestore { 0 1 _NXChannels 1 sub{ _NXProcArray exch get exec length _NXTotalBytes exch sub /_NXTotalBytes exch store}for _NXTotalBytes 0 le{exit}if }loop /_NXProcArray 5 array def }__NXbdef /colorimage{_NXDoImageOp}__NXbdef /alphaimage{1 add _NXDoImageOp}def %%EndDocument % EquationBuilder(TM) Expression % +.\Jr(=:/*V[StRFQAYmdjEEA;1nuN+[co.OZ7L#cq3iK%3:</XJ@:d#6;^G;Td47dud<A3@ % 5%MGToUOR5&d`'TuEDU)B-tbR".Ri'[Z@!!!E7_BJ.d+Ae6lj@P.-O?%B4'FH)c:eX1]TM0' % o;,/#l!%\^3YU,#41L183Q(&Hf"9NTgc2h<D&k0$!hQXOnBCNsA60Psq6cY@ca,g65@$1_&! % 4;jU%&gaK#*:He'hl`u7Z?9DM[q$LJgQEre;Q=PcK)GJ.)(?;#YbXU^a50F<)<u'U==-ELAi % r!a[RLF2mDOsRU,g!+]>r\r&/H8GXPW++sa)=J4LP9C"fAVo.moDa0Z0+%_t*EB0r45VlJ#Y % 4+Q2B&^h[CV\jFpDnZ6K$?u`)d\%]`7/.jW!;lg&-koj@+TNsecio.>-=rj"!%J925CJ5D6l % g<ln]#@%`<P!jnl\IlK&$$a#q-(\[Du*l0T^echCUu]H7H<F`F<8ci>i%k[ZR?sJru/u*FYU % "L3Etm3G"JK";8M'@g2r:L1LF+#ltWt&HjH]E8lp%x % EndExpression N 436 1b4 E 6 S ac62 R c2e399d5 gsave /__NXbasematrix matrix currentmatrix def grestore %%EndProlog %%BeginSetup %%EndSetup 0 0 38.990669 12.200001 rectclip 0 0 38.990669 12.200001 rectclip 0 setgray 1 setalpha 1 1 scale /Symbol findfont 12 scalefont 45 exch defineuserobject 45 execuserobject setfont 25.930666 3.124 moveto (|) [0 0] xyshow 1 1 scale 1 1 scale 45 execuserobject setfont 35.170666 3.124 moveto (|) [0 0] xyshow 1 1 scale 0 setgray 1 setalpha /Times-BoldItalic findfont 12 scalefont 44 exch defineuserobject 44 execuserobject setfont 28.750666 3.1 moveto (xx) [-27.750666 0 0 0 0 0] xyshow 45 execuserobject setfont 10.333334 3.1 moveto <ae> [0 0 0 0] xyshow %%Trailer %%DocumentFonts: Times-BoldItalic %%+ Symbol %End of Graphics MathPictureEnd end :[font = text; inactive; preserveAspect; ] we first observe that is an even function so that all the Bn's are zero. Furthermore, to find the An's we can modify the usual formulas by integrating from 0 to ¥ and multiplying by 2. Here is the Mathematica command that we want: ;[s] 11:0,0;58,1;59,2;60,3;100,4;101,5;102,6;165,7;166,8;202,9;213,10;236,-1; 11:1,11,8,Times,0,12,0,0,0;1,10,8,Times,2,12,0,0,0;1,11,8,Times,64,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Times,2,12,0,0,0;1,10,8,Times,66,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,0,0,Symbol,0,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Times,2,12,0,0,0;1,11,8,Times,0,12,0,0,0; :[font = input; preserveAspect; startGroup; ] Simplify[Integrate[x Cos[n x],{x,0,Pi}] /. {Sin[n Pi]->0,Cos[n Pi]->(-1)^n}] :[font = output; output; inactive; preserveAspect; endGroup; ] (-1 + (-1)^n)/n^2 ;[o] n -1 + (-1) ---------- 2 n :[font = text; inactive; preserveAspect; ] (The Mathematica command Simplify can be used to convert one expression into another, usually simpler, expression. However, Mathematica's idea of simplicity does not always coincide with that of a mathematician.) ;[s] 7:0,0;5,1;16,2;26,3;34,4;139,5;140,6;216,-1; 7:1,11,8,Times,0,12,0,0,0;1,10,8,Times,2,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Courier,0,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Times,2,12,0,0,0;1,11,8,Times,0,12,0,0,0; :[font = text; inactive; preserveAspect; ] Next let us automate the process of finding Fourier coefficients. We use the following Mathematica definitions: ;[s] 5:0,0;25,1;32,2;87,3;98,4;112,-1; 5:1,11,8,Times,0,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Times,2,12,0,0,0;1,11,8,Times,0,12,0,0,0; :[font = input; pageBreakBelow; dontNoPageBreakBelow; preserveAspect; ] A[n_,L_,f_]:= (1/L) Integrate[f[x] Cos[n Pi x /L], {x,-L,L}] A0[L_,f_]:= (1/(2L)) Integrate[f[x],{x,-L,L}] B[n_,L_,f_]:= (1/L) Integrate[f[x] Sin[n Pi x /L], {x,-L,L}] A[n_,f_]:=(1/Pi) Integrate[f[x] Cos[n x], {x,-Pi,Pi}] A0[f_]:=(1/(2Pi)) Integrate[f[x],{x,-Pi,Pi}] B[n_,f_]:=(1/Pi) Integrate[f[x] Sin[n x], {x,-Pi,Pi}] :[font = text; inactive; preserveAspect; ] Also, when we need to inform Mathematica that n is an integer we shall need: ;[s] 5:0,0;29,1;40,2;48,3;49,4;79,-1; 5:1,11,8,Times,0,12,0,0,0;1,10,8,Times,2,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Courier,0,12,0,0,0;1,11,8,Times,0,12,0,0,0; :[font = input; preserveAspect; ] TrigCoefRules[n_]:= {Sin[n Pi]->0,Sin[-n Pi]->0, Cos[n Pi]->(-1)^n, Cos[-n Pi]->(-1)^n} :[font = text; inactive; preserveAspect; ] Before using these formulas we must define a function f via the command f[x]:= ;[s] 4:0,0;55,1;57,2;74,3;81,-1; 4:1,11,8,Times,0,12,0,0,0;1,10,8,Courier,0,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Courier,0,12,0,0,0; :[font = text; inactive; preserveAspect; ] On the right hand side of this equation you should type the formula that defines f(x) using proper Mathematica syntax. Then if L and n are specific numbers, the command A[n,L,f] will yield the quantity An. (Mathematica is able to distinguish between A[n,L,f] and A[n,f] because the first is a function of 3 variables, and the second is a function of 2 variables.) ;[s] 22:0,0;82,1;83,2;84,3;85,4;101,5;112,6;132,7;133,8;139,9;140,10;176,11;185,12;209,13;210,14;211,15;214,16;225,17;257,18;265,19;270,20;276,21;371,-1; 22:1,11,8,Times,0,12,0,0,0;1,10,8,Times,2,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Times,2,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Times,2,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Times,2,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Times,2,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Courier,0,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Times,2,12,0,0,0;1,10,8,Times,66,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Times,2,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Courier,0,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Courier,0,12,0,0,0;1,11,8,Times,0,12,0,0,0; :[font = text; inactive; preserveAspect; ] We use these formulas to compute the Fourier coefficients of the function f which has period 2L and is given by ;[s] 5:0,0;75,1;76,2;96,3;98,4;114,-1; 5:1,11,8,Times,0,12,0,0,0;1,10,8,Times,2,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Times,2,12,0,0,0;1,11,8,Times,0,12,0,0,0; :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 0; pictureWidth = 135; pictureHeight = 14; ] %! %%Creator: Mathematica %%AspectRatio: 0.103704 200 dict begin MathPictureStart /Courier findfont 10 scalefont setfont % Scaling calculations 0 1 0 1 [ [ 0 0 0 0 ] [ 135 14 0 0 ] ] MathScale % Start of Graphics %!PS-Adobe-2.0 EPSF-2.0 %%Title: %%Creator: EquationBuilder %%CreationDate: Mon Aug 3 20:02:04 1992 %%For: gray %%DocumentFonts: (atend) %%Pages: 0 1 %%BoundingBox: 0 0 135 14 %%Translation: 1 2.46 133.101334 0 %%EndComments %%BeginDocument: /usr/lib/NextStep/printPackage.ps %!PS-Adobe-2.0 %%Title: Printing Package %%EndComments %Version: 2.0 %Copyright: 1988, NeXT, Inc. /__NXdef{1 index where{pop pop pop}{def}ifelse}bind def /__NXbdef{1 index where{pop pop pop}{bind def}ifelse}bind def /UserObjects 10 array __NXdef /defineuserobject{ exch dup 1 add dup UserObjects length gt{ array dup 0 UserObjects putinterval /UserObjects exch def }{pop}ifelse UserObjects exch 3 -1 roll put }__NXbdef /undefineuserobject{UserObjects exch null put}__NXbdef /execuserobject{UserObjects exch get exec}__NXbdef /__NXRectPath{4 2 roll moveto 1 index 0 rlineto 0 exch rlineto neg 0 rlineto closepath}__NXbdef /__NXProcessRectArgs{ 1 index type /arraytype eq{ exch 0 4 2 index length 1 sub{ dup 3 add 1 exch{1 index exch get exch}for 5 1 roll 5 index exec }for pop pop }{exec}ifelse }__NXbdef /rectfill{gsave newpath {__NXRectPath fill} __NXProcessRectArgs grestore}__NXbdef /rectclip{newpath {__NXRectPath} __NXProcessRectArgs clip newpath}__NXbdef /rectstroke{ gsave newpath dup type /arraytype eq{dup length 6 eq}{false}ifelse{ {gsave __NXRectPath null concat stroke grestore} dup length array cvx copy dup 2 4 -1 roll put __NXProcessRectArgs }{{__NXRectPath stroke} __NXProcessRectArgs}ifelse grestore }__NXbdef /xyshow{ 0 1 3 index length 1 sub{ currentpoint 4 index 3 index 1 getinterval show 3 index 3 index 2 mul 1 add get add exch 3 index 3 index 2 mul get add exch moveto pop }for pop pop }__NXbdef /xshow{ 0 1 3 index length 1 sub{ currentpoint 4 index 3 index 1 getinterval show exch 3 index 3 index get add exch moveto pop }for pop pop }__NXbdef /yshow{ 0 1 3 index length 1 sub{ currentpoint 4 index 3 index 1 getinterval show 3 index 3 index get add moveto pop }for pop pop }__NXbdef /arct{arcto pop pop pop pop}__NXbdef /setbbox{pop pop pop pop}__NXbdef /ucache{}__NXbdef /ucachestatus{mark 0 0 0 0 0}__NXbdef /setucacheparams{cleartomark}__NXbdef /uappend{systemdict begin cvx exec end}__NXbdef /ueofill{gsave newpath uappend eofill grestore}__NXbdef /ufill{gsave newpath uappend fill grestore}__NXbdef /ustroke{ gsave newpath dup length 6 eq {exch uappend concat}{uappend}ifelse stroke grestore }__NXbdef /__NXustrokepathMatrix dup where {pop pop}{matrix def}ifelse /ustrokepath{ newpath dup length 6 eq{ exch uappend __NXustrokepathMatrix currentmatrix exch concat strokepath setmatrix }{uappend strokepath}ifelse } __NXbdef /upath{ [exch {/ucache cvx}if pathbbox /setbbox cvx {/moveto cvx}{/lineto cvx}{/curveto cvx}{/closepath cvx}pathforall]cvx } __NXbdef /setstrokeadjust{pop}__NXbdef /currentstrokeadjust{false}__NXbdef /selectfont{exch findfont exch dup type /arraytype eq {makefont}{scalefont}ifelse setfont}__NXbdef /_NXCombineArrays{ counttomark dup 2 add index dup length 3 -1 roll { 2 index length sub dup 4 1 roll 1 index exch 4 -1 roll putinterval exch }repeat pop pop pop }__NXbdef /setcmykcolor{ 1.0 exch sub dup dup 6 -1 roll sub dup 0 lt{pop 0}if 5 1 roll 4 -1 roll sub dup 0 lt{pop 0}if 3 1 roll exch sub dup 0 lt{pop 0}if setrgbcolor }__NXbdef /currentcmykcolor{currentrgbcolor 3{1.0 exch sub 3 1 roll}repeat 0}__NXbdef /flushgraphics{}def /setwindowtype{pop pop}def /currentwindowtype{pop 0}def /setalpha{pop}def /currentalpha{1.0}def /hidecursor{}def /obscurecursor{}def /revealcursor{}def /setcursor{4 {pop}repeat}bind def /showcursor{}def /NextStepEncoding where not{ /NextStepEncoding StandardEncoding 256 array copy def 0 [129/Agrave/Aacute/Acircumflex/Atilde/Adieresis/Aring/Ccedilla/Egrave /Eacute/Ecircumflex/Edieresis/Igrave/Iacute/Icircumflex/Idieresis /Eth/Ntilde/Ograve/Oacute/Ocircumflex/Otilde/Odieresis/Ugrave/Uacute /Ucircumflex/Udieresis/Yacute/Thorn/mu/multiply/divide/copyright 176/registered 181/brokenbar 190/logicalnot 192/onesuperior 201/twosuperior 204/threesuperior 209/plusminus/onequarter/onehalf/threequarters/agrave /aacute/acircumflex/atilde/adieresis/aring/ccedilla/egrave/eacute /ecircumflex/edieresis/igrave 226/iacute 228/icircumflex/idieresis/eth /ntilde 236/ograve/oacute/ocircumflex/otilde/odieresis 242/ugrave/uacute /ucircumflex 246/udieresis/yacute 252/thorn/ydieresis] {dup type /nametype eq {NextStepEncoding 2 index 2 index put pop 1 add}{exch pop}ifelse }forall pop /NextStepEncoding NextStepEncoding readonly def /_NXfstr 128 string dup 0 (_NX) putinterval def /findfont{ % Because we can never let NextStepEncoding get into % SharedFontDirectory, we cannot reencode a font to NextStepEncoding % if we are in shared mode. So if currentshared is true, % we call the normal findfont and return that /currentshared where {pop currentshared} {false} ifelse {//findfont exec} {dup _NXfstr 3 125 getinterval cvs length 3 add _NXfstr 0 3 -1 roll getinterval cvn exch FontDirectory 2 index known {pop FontDirectory exch get} {//findfont exec dup /Encoding get StandardEncoding eq { dup length dict exch {1 index /FID ne {2 index 3 1 roll put}{pop pop}ifelse}forall dup /Encoding NextStepEncoding put definefont }{exch pop} ifelse }ifelse }ifelse }bind def }{pop}ifelse /_NXProcArray 5 array __NXdef /_NXChannels 0 __NXdef /_NXTotalBytes 0 __NXdef /_NXDoImageOp{ 1 index{dup}{1}ifelse /_NXChannels exch store _NXChannels 2 add 2 roll _NXProcArray 0 _NXChannels getinterval astore pop 5 index 4 index mul 2 index{1 sub 8 idiv 1 add mul}{mul 1 sub 8 idiv 1 add}ifelse 4 index mul /_NXTotalBytes exch store pop exch pop gsave matrix invertmatrix concat 0.5 setgray 0 0 4 2 roll rectfill grestore { 0 1 _NXChannels 1 sub{ _NXProcArray exch get exec length _NXTotalBytes exch sub /_NXTotalBytes exch store}for _NXTotalBytes 0 le{exit}if }loop /_NXProcArray 5 array def }__NXbdef /colorimage{_NXDoImageOp}__NXbdef /alphaimage{1 add _NXDoImageOp}def %%EndDocument % EquationBuilder(TM) Expression % +.\Jr(=:/*V[StRFQAYmdjEEA;1nuN+[co.OZ7L#cq3iK%3:</XJ@:d#6;^G;Td47dud<A3@ % 5%MGToUOR5&d`'TuEDU)B-tbR".R+:s&j%PRn`0M'99V1X;K=iupU+TapkbQ1TiO8qO##pFg % SV27Z%C%),,7&[ne(c'<W+g@4!;?.76d#C!iJ0@Ju`AW"Zk%:1$+0PmBo;2s;99LJh%SVCE) % MA.D#"W6tm1pQpJ92+!?qR"K@L+W)S/B'R1h#IHG.^Ce-GZ0;q'@:OkYb4X/t!-@Jmn]kV$L % ?^'Y"1+12PCG8OM\&Cc\sKh^4RL^t#og$%7t1$Fan?Uj`OROPEh+Lj0?%="h3c.cC(*"kX=c % P^WTs!9.>c!"(+[!%F_jKms+U`>h;TUr8!W*4^&0.(5Q,=*jf[:`gkK):ucOJMHYI*+Yl(7V % m?TQ^oDVjYjnu1P@\nA?]`,?[qCLpKdi20R'9;3HB<bLVS<T@O_m$P!BUr^bO2iOqjD>L;/# % T*f03S[ptTfq'*p3P2%8'/gR"k73-:(7&>&f,2"3-`j*m$!`QlH`Y`9sR%]XEnu*f^1GEIcT % 3$J%V^[9;MWg2pD96IbKH(=HTSa4OWMM;g)/5G,4rPi:TK7C5U2k!`C8[Gr.VYZCMONe1Tq2 % UjTX6VVJ.O39pa3&?AuXjP/I3f+X$.HB"EUXa/$,O2nc0"$`nEK=U<,,]8@-lrWF`BDJ:^.r % ^tq[5S[FOuUUf"qkCS*&rWC;-IuuWW*HKBoL2;gr-J;.rIL>(AR=KlS>C-e/#6P8LnD44G8> % f>n;D]/D)t-'(E>oQ!5o:J2ck/`hA%\_T`Z9O;8g"6H#XAqd\.%sT"@P(9&=`a9M*H^px % EndExpression N 688 2b0 E 74 S 120f2 R 27f3395 gsave /__NXbasematrix matrix currentmatrix def grestore %%EndProlog %%BeginSetup /EqBFont 18 dict dup begin /FontType 3 def /PaintType 0 def /FontMatrix [0.001 0 0 0.001 0 0] def /FontBBox [15 -2950 1360 1045] def /Encoding [/.notdef /rad1 /rad2 /rad3 /rad4 /rad5 /radext /int /pathint /sum /prod /cdot /vdot /ddot /hbar /bigcap /bigcup /parallel] def /Metrics 18 dict dup begin /.notdef 0 def /rad1 1000 def /rad2 1000 def /rad3 1000 def /rad4 1000 def /rad5 1048 def /radext 1048 def /int 500 def /pathint 500 def /sum 700 def /prod 700 def /cdot 1000 def /vdot 550 def /ddot 1000 def /hbar 500 def /bigcap 700 def /bigcup 700 def /parallel 500 def end def /BBox 18 dict dup begin /.notdef [0 0 0 0] def /rad1 [110 -1150 1020 0] def /rad2 [110 -1750 1020 0] def /rad3 [110 -2350 1020 0] def /rad4 [110 -2950 1020 0] def /rad5 [128 -2890 740 0] def /radext [700 -600 740 0] def /int [20 0 425 1045] def /pathint [20 0 425 1045] def /sum [15 5 640 715] def /prod [15 0 670 715] def /cdot [110 215 890 325] def /vdot [165 -5 280 770] def /ddot [110 -5 890 770] def /hbar [20 -10 480 685] def /bigcap [0 0 680 720] def /bigcup [0 0 680 720] def /parallel [65 -175 400 675] def end def /CharacterDefs 18 dict dup begin /.notdef { } def /rad1 { 980 0 moveto 464 -1048 lineto 248 -560 lineto 110 -656 lineto 125 -675 lineto 200 -624 lineto 420 -1150 lineto 460 -1150 lineto 1020 0 lineto 980 0 lineto closepath } def /rad2 { 980 0 moveto 460 -1600 lineto 245 -864 lineto 110 -1000 lineto 128 -1023 lineto 200 -950 lineto 424 -1750 lineto 456 -1750 lineto 1020 0 lineto 980 0 lineto closepath } def /rad3 { 980 0 moveto 464 -2144 lineto 248 -1160 lineto 110 -1368 lineto 130 -1390 lineto 192 -1288 lineto 424 -2350 lineto 464 -2350 lineto 1020 0 lineto 980 0 lineto closepath } def /rad4 { 980 0 moveto 880 -488 lineto 672 -1600 lineto 464 -2696 lineto 248 -1464 lineto 110 -1728 lineto 130 -1750 lineto 192 -1616 lineto 424 -2950 lineto 464 -2950 lineto 768 -1312 lineto 1020 0 lineto 980 0 lineto closepath } def /rad5 { 700 0 moveto 700 -2628 lineto 248 -1284 lineto 128 -1564 lineto 150 -1580 lineto 215 -1425 lineto 688 -2890 lineto 740 -2890 lineto 740 0 lineto 700 0 lineto closepath } def /radext { 700 0 moveto 700 -600 lineto 740 -600 lineto 740 0 lineto 700 0 lineto closepath } def /int { 245 390 moveto 225 235 195 145 180 100 curveto 165 60 150 50 140 40 curveto 135 30 85 0 50 20 curveto 35 25 25 45 25 55 curveto 20 80 30 100 55 110 curveto 80 125 93 110 110 90 curveto 135 60 155 135 160 195 curveto 165 270 170 340 190 510 curveto 205 650 lineto 220 810 230 890 265 960 curveto 275 980 295 1015 320 1030 curveto 350 1045 380 1030 395 1020 curveto 410 1010 425 980 400 955 curveto 385 940 347 925 330 950 curveto 310 970 290 955 290 940 curveto 280 900 285 745 270 610 curveto 245 390 lineto closepath } def /pathint { 245 390 moveto 225 235 195 145 180 100 curveto 165 60 150 50 140 40 curveto 135 30 85 0 50 20 curveto 35 25 25 45 25 55 curveto 20 80 30 100 55 110 curveto 80 125 93 110 110 90 curveto 135 60 155 135 160 195 curveto 165 270 170 340 190 510 curveto 205 650 lineto 220 810 230 890 265 960 curveto 275 980 295 1015 320 1030 curveto 350 1045 380 1030 395 1020 curveto 410 1010 425 980 400 955 curveto 385 940 347 925 330 950 curveto 310 970 290 955 290 940 curveto 280 900 285 745 270 610 curveto 245 390 lineto fill 20 setlinewidth 220 400 moveto 290 400 340 455 340 500 curveto 340 585 290 640 220 640 curveto 155 640 100 585 100 500 curveto 100 455 155 400 220 400 curveto stroke } def /sum { 330 335 moveto 20 715 lineto 575 715 lineto 600 545 lineto 575 535 lineto 550 625 525 685 395 685 curveto 160 685 lineto 411 370 lineto 160 90 lineto 470 90 lineto 565 90 590 140 610 210 curveto 640 200 lineto 585 5 lineto 15 5 lineto 330 335 lineto closepath } def /prod { 590 90 moveto 590 30 620 20 670 20 curveto 670 0 lineto 410 0 lineto 410 20 lineto 460 20 495 30 495 90 curveto 495 670 lineto 190 670 lineto 190 90 lineto 190 30 225 20 275 20 curveto 275 0 lineto 15 0 lineto 15 20 lineto 65 20 100 30 100 90 curveto 100 615 lineto 100 690 40 700 20 700 curveto 20 715 lineto 670 715 lineto 670 700 lineto 645 700 590 690 590 615 curveto 590 90 lineto closepath } def /cdot { 110 270 moveto 110 300 135 325 165 325 curveto 195 325 220 300 220 270 curveto 220 235 185 215 165 215 curveto 145 215 110 235 110 270 curveto fill 445 270 moveto 445 300 470 325 500 325 curveto 530 325 555 300 555 270 curveto 555 235 520 215 500 215 curveto 480 215 445 235 445 270 curveto fill 775 270 moveto 775 300 800 325 830 325 curveto 860 325 890 300 890 270 curveto 890 235 850 215 830 215 curveto 810 215 775 235 775 270 curveto fill } def /vdot { 220 770 moveto 250 770 280 750 280 720 curveto 280 680 250 660 220 660 curveto 185 660 165 695 165 720 curveto 165 740 185 770 220 770 curveto fill 220 440 moveto 250 440 280 415 280 385 curveto 280 355 250 330 220 330 curveto 185 330 165 365 165 385 curveto 165 405 185 440 220 440 curveto fill 220 105 moveto 250 105 280 80 280 50 curveto 280 20 250 -5 220 -5 curveto 185 -5 165 30 165 50 curveto 165 70 185 105 220 105 curveto fill } def /ddot { 165 720 moveto 205 720 220 700 220 665 curveto 220 630 195 610 165 610 curveto 130 620 110 645 110 665 curveto 110 690 130 720 165 720 curveto fill 500 440 moveto 530 440 555 415 555 385 curveto 555 355 530 330 500 330 curveto 465 330 445 365 445 385 curveto 445 405 465 440 500 440 curveto fill 830 155 moveto 860 155 890 130 890 100 curveto 890 70 860 45 830 45 curveto 795 45 775 80 775 100 curveto 775 120 795 155 830 155 curveto fill } def /hbar { 480 105 moveto 440 55 400 -10 335 -10 curveto 300 -10 290 10 290 35 curveto 290 65 370 350 370 365 curveto 370 380 355 390 340 390 curveto 290 390 205 275 185 250 curveto 140 180 125 125 95 0 curveto 20 0 lineto 150 500 lineto 160 545 180 600 180 615 curveto 180 625 170 640 135 640 curveto 110 640 lineto 100 655 lineto 165 660 215 670 270 685 curveto 275 680 lineto 155 240 lineto 155 235 lineto 190 290 290 440 380 440 curveto 400 440 450 435 450 370 curveto 450 335 365 80 365 55 curveto 365 45 375 40 385 40 curveto 405 40 425 70 465 115 curveto 480 105 lineto fill 55 545 moveto 55 520 lineto 310 560 lineto 310 585 lineto 55 545 lineto fill } def /bigcap { 615 400 moveto 615 575 500 655 340 655 curveto 180 655 70 575 70 400 curveto 70 0 lineto 5 0 lineto 5 400 lineto 5 610 140 720 340 720 curveto 545 720 680 610 680 400 curveto 680 0 lineto 615 0 lineto 615 400 lineto closepath } def /bigcup { 65 320 moveto 65 145 180 65 340 65 curveto 500 65 615 145 615 320 curveto 615 720 lineto 680 720 lineto 680 320 lineto 680 110 540 0 340 0 curveto 140 0 0 110 0 320 curveto 0 720 lineto 65 720 lineto 65 320 lineto closepath } def /parallel { 135 675 moveto 135 -175 lineto 65 -175 lineto 65 675 lineto 135 675 lineto fill 335 675 moveto 335 -175 lineto 265 -175 lineto 265 675 lineto 335 675 lineto fill 400 675 moveto } def end def /BuildChar { 0 begin /char exch def /fontdict exch def /charname fontdict /Encoding get char get def fontdict begin Metrics charname get 0 BBox charname get aload pop setcachedevice CharacterDefs charname get exec eofill end end } def /BuildChar load 0 3 dict put end definefont pop %%EndSetup 0 0 134.101334 13.572704 rectclip 0 0 134.101334 13.572704 rectclip 0 setgray 1 setalpha 1 1 scale /Symbol findfont 12 scalefont 48 exch defineuserobject 48 execuserobject setfont 5.56 3.568 moveto (\() [0 0] xyshow 1 1 scale 1 1 scale 48 execuserobject setfont 17.200001 3.562 moveto (\)) [0 0] xyshow 1 1 scale 0 setgray 1 setalpha /Times-BoldItalic findfont 12 scalefont 49 exch defineuserobject 49 execuserobject setfont 130.101334 3.46 moveto (.LxLexf) [-7.332001 0 -19.254662 0 -20.58667 0 -47.949333 0 -24.618668 0 -9.360001 0 0 0 0 0] xyshow 48 execuserobject setfont 112.848 3.46 moveto (<<-=) [-19.254662 0 -19.920006 0 -48.615997 0 0 0 -43.587997 0] xyshow /Times-BoldItalic findfont 9 scalefont 67 exch defineuserobject 67 execuserobject setfont 40.306667 8.414704 moveto (x) [0 0 -48.793327 11.804017] xyshow /Symbol findfont 9 scalefont 60 exch defineuserobject 60 execuserobject setfont 31.089333 29.804968 moveto () [-0.990005 27.523968] xyshow /Times-Roman findfont 12 scalefont 50 exch defineuserobject 50 execuserobject setfont 46.006668 3.46 moveto (,) [0 0 3.335999 0] xyshow %%Trailer %%DocumentFonts: Times-BoldItalic %%+ Times-Roman %%+ Symbol %End of Graphics MathPictureEnd end :[font = text; inactive; preserveAspect; ] First we define the function f via the command ;[s] 3:0,0;30,1;31,2;49,-1; 3:1,11,8,Times,0,12,0,0,0;1,10,8,Times,2,12,0,0,0;1,11,8,Times,0,12,0,0,0; :[font = input; preserveAspect; ] f[x_]:=E^x :[font = text; inactive; preserveAspect; ] Next we type :[font = input; preserveAspect; startGroup; ] Simplify[A[n,L,f] /. TrigCoefRules[n]] :[font = output; output; inactive; preserveAspect; endGroup; ] ((-1)^n*(-1 + E^(2*L))*L)/ (E^L*(L^2 + n^2*Pi^2)) ;[o] n 2 L (-1) (-1 + E ) L ------------------- L 2 2 2 E (L + n Pi ) :[font = text; inactive; preserveAspect; ] Similarly :[font = input; preserveAspect; startGroup; ] Simplify[A0[L,f] /. TrigCoefRules[n]] :[font = output; output; inactive; preserveAspect; endGroup; ] (-E^(-L) + E^L)/(2*L) ;[o] -L L -E + E --------- 2 L :[font = input; preserveAspect; startGroup; ] Simplify[B[n,L,f] /. TrigCoefRules[n]] :[font = output; output; inactive; pageBreakBelow; dontNoPageBreakBelow; preserveAspect; endGroup; ] ((-1)^n*(1 - E^(2*L))*n*Pi)/ (E^L*(L^2 + n^2*Pi^2)) ;[o] n 2 L (-1) (1 - E ) n Pi --------------------- L 2 2 2 E (L + n Pi ) :[font = text; inactive; preserveAspect; ] Therefore :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 0; pictureWidth = 235; pictureHeight = 73; ] %! %%Creator: Mathematica %%AspectRatio: 0.310638 200 dict begin MathPictureStart /Courier findfont 10 scalefont setfont % Scaling calculations 0 1 0 1 [ [ 0 0 0 0 ] [ 235 73 0 0 ] ] MathScale % Start of Graphics %!PS-Adobe-2.0 EPSF-2.0 %%Title: %%Creator: EquationBuilder %%CreationDate: Sun Aug 2 12:23:37 1992 %%For: gray %%DocumentFonts: (atend) %%Pages: 0 1 %%BoundingBox: 0 0 235 73 %%Translation: 1 32.385971 233.59433 0 %%EndComments %%BeginDocument: /usr/lib/NextStep/printPackage.ps %!PS-Adobe-2.0 %%Title: Printing Package %%EndComments %Version: 2.0 %Copyright: 1988, NeXT, Inc. /__NXdef{1 index where{pop pop pop}{def}ifelse}bind def /__NXbdef{1 index where{pop pop pop}{bind def}ifelse}bind def /UserObjects 10 array __NXdef /defineuserobject{ exch dup 1 add dup UserObjects length gt{ array dup 0 UserObjects putinterval /UserObjects exch def }{pop}ifelse UserObjects exch 3 -1 roll put }__NXbdef /undefineuserobject{UserObjects exch null put}__NXbdef /execuserobject{UserObjects exch get exec}__NXbdef /__NXRectPath{4 2 roll moveto 1 index 0 rlineto 0 exch rlineto neg 0 rlineto closepath}__NXbdef /__NXProcessRectArgs{ 1 index type /arraytype eq{ exch 0 4 2 index length 1 sub{ dup 3 add 1 exch{1 index exch get exch}for 5 1 roll 5 index exec }for pop pop }{exec}ifelse }__NXbdef /rectfill{gsave newpath {__NXRectPath fill} __NXProcessRectArgs grestore}__NXbdef /rectclip{newpath {__NXRectPath} __NXProcessRectArgs clip newpath}__NXbdef /rectstroke{ gsave newpath dup type /arraytype eq{dup length 6 eq}{false}ifelse{ {gsave __NXRectPath null concat stroke grestore} dup length array cvx copy dup 2 4 -1 roll put __NXProcessRectArgs }{{__NXRectPath stroke} __NXProcessRectArgs}ifelse grestore }__NXbdef /xyshow{ 0 1 3 index length 1 sub{ currentpoint 4 index 3 index 1 getinterval show 3 index 3 index 2 mul 1 add get add exch 3 index 3 index 2 mul get add exch moveto pop }for pop pop }__NXbdef /xshow{ 0 1 3 index length 1 sub{ currentpoint 4 index 3 index 1 getinterval show exch 3 index 3 index get add exch moveto pop }for pop pop }__NXbdef /yshow{ 0 1 3 index length 1 sub{ currentpoint 4 index 3 index 1 getinterval show 3 index 3 index get add moveto pop }for pop pop }__NXbdef /arct{arcto pop pop pop pop}__NXbdef /setbbox{pop pop pop pop}__NXbdef /ucache{}__NXbdef /ucachestatus{mark 0 0 0 0 0}__NXbdef /setucacheparams{cleartomark}__NXbdef /uappend{systemdict begin cvx exec end}__NXbdef /ueofill{gsave newpath uappend eofill grestore}__NXbdef /ufill{gsave newpath uappend fill grestore}__NXbdef /ustroke{ gsave newpath dup length 6 eq {exch uappend concat}{uappend}ifelse stroke grestore }__NXbdef /__NXustrokepathMatrix dup where {pop pop}{matrix def}ifelse /ustrokepath{ newpath dup length 6 eq{ exch uappend __NXustrokepathMatrix currentmatrix exch concat strokepath setmatrix }{uappend strokepath}ifelse } __NXbdef /upath{ [exch {/ucache cvx}if pathbbox /setbbox cvx {/moveto cvx}{/lineto cvx}{/curveto cvx}{/closepath cvx}pathforall]cvx } __NXbdef /setstrokeadjust{pop}__NXbdef /currentstrokeadjust{false}__NXbdef /selectfont{exch findfont exch dup type /arraytype eq {makefont}{scalefont}ifelse setfont}__NXbdef /_NXCombineArrays{ counttomark dup 2 add index dup length 3 -1 roll { 2 index length sub dup 4 1 roll 1 index exch 4 -1 roll putinterval exch }repeat pop pop pop }__NXbdef /setcmykcolor{ 1.0 exch sub dup dup 6 -1 roll sub dup 0 lt{pop 0}if 5 1 roll 4 -1 roll sub dup 0 lt{pop 0}if 3 1 roll exch sub dup 0 lt{pop 0}if setrgbcolor }__NXbdef /currentcmykcolor{currentrgbcolor 3{1.0 exch sub 3 1 roll}repeat 0}__NXbdef /flushgraphics{}def /setwindowtype{pop pop}def /currentwindowtype{pop 0}def /setalpha{pop}def /currentalpha{1.0}def /hidecursor{}def /obscurecursor{}def /revealcursor{}def /setcursor{4 {pop}repeat}bind def /showcursor{}def /NextStepEncoding where not{ /NextStepEncoding StandardEncoding 256 array copy def 0 [129/Agrave/Aacute/Acircumflex/Atilde/Adieresis/Aring/Ccedilla/Egrave /Eacute/Ecircumflex/Edieresis/Igrave/Iacute/Icircumflex/Idieresis /Eth/Ntilde/Ograve/Oacute/Ocircumflex/Otilde/Odieresis/Ugrave/Uacute /Ucircumflex/Udieresis/Yacute/Thorn/mu/multiply/divide/copyright 176/registered 181/brokenbar 190/logicalnot 192/onesuperior 201/twosuperior 204/threesuperior 209/plusminus/onequarter/onehalf/threequarters/agrave /aacute/acircumflex/atilde/adieresis/aring/ccedilla/egrave/eacute /ecircumflex/edieresis/igrave 226/iacute 228/icircumflex/idieresis/eth /ntilde 236/ograve/oacute/ocircumflex/otilde/odieresis 242/ugrave/uacute /ucircumflex 246/udieresis/yacute 252/thorn/ydieresis] {dup type /nametype eq {NextStepEncoding 2 index 2 index put pop 1 add}{exch pop}ifelse }forall pop /NextStepEncoding NextStepEncoding readonly def /_NXfstr 128 string dup 0 (_NX) putinterval def /findfont{ % Because we can never let NextStepEncoding get into % SharedFontDirectory, we cannot reencode a font to NextStepEncoding % if we are in shared mode. So if currentshared is true, % we call the normal findfont and return that /currentshared where {pop currentshared} {false} ifelse {//findfont exec} {dup _NXfstr 3 125 getinterval cvs length 3 add _NXfstr 0 3 -1 roll getinterval cvn exch FontDirectory 2 index known {pop FontDirectory exch get} {//findfont exec dup /Encoding get StandardEncoding eq { dup length dict exch {1 index /FID ne {2 index 3 1 roll put}{pop pop}ifelse}forall dup /Encoding NextStepEncoding put definefont }{exch pop} ifelse }ifelse }ifelse }bind def }{pop}ifelse /_NXProcArray 5 array __NXdef /_NXChannels 0 __NXdef /_NXTotalBytes 0 __NXdef /_NXDoImageOp{ 1 index{dup}{1}ifelse /_NXChannels exch store _NXChannels 2 add 2 roll _NXProcArray 0 _NXChannels getinterval astore pop 5 index 4 index mul 2 index{1 sub 8 idiv 1 add mul}{mul 1 sub 8 idiv 1 add}ifelse 4 index mul /_NXTotalBytes exch store pop exch pop gsave matrix invertmatrix concat 0.5 setgray 0 0 4 2 roll rectfill grestore { 0 1 _NXChannels 1 sub{ _NXProcArray exch get exec length _NXTotalBytes exch sub /_NXTotalBytes exch store}for _NXTotalBytes 0 le{exit}if }loop /_NXProcArray 5 array def }__NXbdef /colorimage{_NXDoImageOp}__NXbdef /alphaimage{1 add _NXDoImageOp}def %%EndDocument % EquationBuilder(TM) Expression % +.\Jr(=:/*V[StRFQAYmdjEEA;1nuN+[co.OZ7L#cq3iK%3:</XJ@:d#6;^G;Td47dud<A3@ % 5%MGToUOR5&d`'TuEDU)B-tbR".R+:s'G8MC/F0M'9RV1X@j>0<$V+TapkbQ1TiO8qO##pFg % SV27Z%C%),,7&[ne(c'<W+g@4!;?.76d#C!iJ0@Ju#pfko-I!De60PsqoC6ST`H3Y&RQNSa! % @DLl#%/AmNI\'6A,tj,(dX?ECi7fq9kP.$9Ro3U63%qEkeIUua<?be9c,m9*]%t&fi)]`"$k % 2B9i8`k4Di$0(>]nE&2_/$':0Eu\6hQ(KJ!ECi0)1^.OYK$O\1$>6\J'/mS6<]?W?(/")+m< % I7-3eSQ%urE"#.*.J%gt#^uc(;Lj&jnHXqY$g%\&/<Z)=M(4[u11>]d0TZnd)KCFt@Z2I4S: % V/1#U(AiWRo%C\/?G[(c[q^*G*7@#DrMT&Fp-=\/[Fm.>3LDfe,`<.Gn[LY\<!E0XeFtHr!? % IV2t@F)52@M_"S1I>M<.eK+CitUFI@W8KNQ,8FhZa_aneVNQhdWgMap%8;E(c0Zg/\Bmpps$ % h,T*R@pi+)'<G:8D0Yb?X>0,UrNL-#W%]J,H6T9[PnS[SUO1FQ%[(bLmsWT"3Ni;@QJ1\Kqc % 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matrix currentmatrix def grestore %%EndProlog %%BeginSetup /EqBFont 18 dict dup begin /FontType 3 def /PaintType 0 def /FontMatrix [0.001 0 0 0.001 0 0] def /FontBBox [15 -2950 1360 1045] def /Encoding [/.notdef /rad1 /rad2 /rad3 /rad4 /rad5 /radext /int /pathint /sum /prod /cdot /vdot /ddot /hbar /bigcap /bigcup /parallel] def /Metrics 18 dict dup begin /.notdef 0 def /rad1 1000 def /rad2 1000 def /rad3 1000 def /rad4 1000 def /rad5 1048 def /radext 1048 def /int 500 def /pathint 500 def /sum 700 def /prod 700 def /cdot 1000 def /vdot 550 def /ddot 1000 def /hbar 500 def /bigcap 700 def /bigcup 700 def /parallel 500 def end def /BBox 18 dict dup begin /.notdef [0 0 0 0] def /rad1 [110 -1150 1020 0] def /rad2 [110 -1750 1020 0] def /rad3 [110 -2350 1020 0] def /rad4 [110 -2950 1020 0] def /rad5 [128 -2890 740 0] def /radext [700 -600 740 0] def /int [20 0 425 1045] def /pathint [20 0 425 1045] def /sum [15 5 640 715] def /prod [15 0 670 715] def /cdot [110 215 890 325] def /vdot [165 -5 280 770] def /ddot [110 -5 890 770] def /hbar [20 -10 480 685] def /bigcap [0 0 680 720] def /bigcup [0 0 680 720] def /parallel [65 -175 400 675] def end def /CharacterDefs 18 dict dup begin /.notdef { } def /rad1 { 980 0 moveto 464 -1048 lineto 248 -560 lineto 110 -656 lineto 125 -675 lineto 200 -624 lineto 420 -1150 lineto 460 -1150 lineto 1020 0 lineto 980 0 lineto closepath } def /rad2 { 980 0 moveto 460 -1600 lineto 245 -864 lineto 110 -1000 lineto 128 -1023 lineto 200 -950 lineto 424 -1750 lineto 456 -1750 lineto 1020 0 lineto 980 0 lineto closepath } def /rad3 { 980 0 moveto 464 -2144 lineto 248 -1160 lineto 110 -1368 lineto 130 -1390 lineto 192 -1288 lineto 424 -2350 lineto 464 -2350 lineto 1020 0 lineto 980 0 lineto closepath } def /rad4 { 980 0 moveto 880 -488 lineto 672 -1600 lineto 464 -2696 lineto 248 -1464 lineto 110 -1728 lineto 130 -1750 lineto 192 -1616 lineto 424 -2950 lineto 464 -2950 lineto 768 -1312 lineto 1020 0 lineto 980 0 lineto closepath } def /rad5 { 700 0 moveto 700 -2628 lineto 248 -1284 lineto 128 -1564 lineto 150 -1580 lineto 215 -1425 lineto 688 -2890 lineto 740 -2890 lineto 740 0 lineto 700 0 lineto closepath } def /radext { 700 0 moveto 700 -600 lineto 740 -600 lineto 740 0 lineto 700 0 lineto closepath } def /int { 245 390 moveto 225 235 195 145 180 100 curveto 165 60 150 50 140 40 curveto 135 30 85 0 50 20 curveto 35 25 25 45 25 55 curveto 20 80 30 100 55 110 curveto 80 125 93 110 110 90 curveto 135 60 155 135 160 195 curveto 165 270 170 340 190 510 curveto 205 650 lineto 220 810 230 890 265 960 curveto 275 980 295 1015 320 1030 curveto 350 1045 380 1030 395 1020 curveto 410 1010 425 980 400 955 curveto 385 940 347 925 330 950 curveto 310 970 290 955 290 940 curveto 280 900 285 745 270 610 curveto 245 390 lineto closepath } def /pathint { 245 390 moveto 225 235 195 145 180 100 curveto 165 60 150 50 140 40 curveto 135 30 85 0 50 20 curveto 35 25 25 45 25 55 curveto 20 80 30 100 55 110 curveto 80 125 93 110 110 90 curveto 135 60 155 135 160 195 curveto 165 270 170 340 190 510 curveto 205 650 lineto 220 810 230 890 265 960 curveto 275 980 295 1015 320 1030 curveto 350 1045 380 1030 395 1020 curveto 410 1010 425 980 400 955 curveto 385 940 347 925 330 950 curveto 310 970 290 955 290 940 curveto 280 900 285 745 270 610 curveto 245 390 lineto fill 20 setlinewidth 220 400 moveto 290 400 340 455 340 500 curveto 340 585 290 640 220 640 curveto 155 640 100 585 100 500 curveto 100 455 155 400 220 400 curveto stroke } def /sum { 330 335 moveto 20 715 lineto 575 715 lineto 600 545 lineto 575 535 lineto 550 625 525 685 395 685 curveto 160 685 lineto 411 370 lineto 160 90 lineto 470 90 lineto 565 90 590 140 610 210 curveto 640 200 lineto 585 5 lineto 15 5 lineto 330 335 lineto closepath } def /prod { 590 90 moveto 590 30 620 20 670 20 curveto 670 0 lineto 410 0 lineto 410 20 lineto 460 20 495 30 495 90 curveto 495 670 lineto 190 670 lineto 190 90 lineto 190 30 225 20 275 20 curveto 275 0 lineto 15 0 lineto 15 20 lineto 65 20 100 30 100 90 curveto 100 615 lineto 100 690 40 700 20 700 curveto 20 715 lineto 670 715 lineto 670 700 lineto 645 700 590 690 590 615 curveto 590 90 lineto closepath } def /cdot { 110 270 moveto 110 300 135 325 165 325 curveto 195 325 220 300 220 270 curveto 220 235 185 215 165 215 curveto 145 215 110 235 110 270 curveto fill 445 270 moveto 445 300 470 325 500 325 curveto 530 325 555 300 555 270 curveto 555 235 520 215 500 215 curveto 480 215 445 235 445 270 curveto fill 775 270 moveto 775 300 800 325 830 325 curveto 860 325 890 300 890 270 curveto 890 235 850 215 830 215 curveto 810 215 775 235 775 270 curveto fill } def /vdot { 220 770 moveto 250 770 280 750 280 720 curveto 280 680 250 660 220 660 curveto 185 660 165 695 165 720 curveto 165 740 185 770 220 770 curveto fill 220 440 moveto 250 440 280 415 280 385 curveto 280 355 250 330 220 330 curveto 185 330 165 365 165 385 curveto 165 405 185 440 220 440 curveto fill 220 105 moveto 250 105 280 80 280 50 curveto 280 20 250 -5 220 -5 curveto 185 -5 165 30 165 50 curveto 165 70 185 105 220 105 curveto fill } def /ddot { 165 720 moveto 205 720 220 700 220 665 curveto 220 630 195 610 165 610 curveto 130 620 110 645 110 665 curveto 110 690 130 720 165 720 curveto fill 500 440 moveto 530 440 555 415 555 385 curveto 555 355 530 330 500 330 curveto 465 330 445 365 445 385 curveto 445 405 465 440 500 440 curveto fill 830 155 moveto 860 155 890 130 890 100 curveto 890 70 860 45 830 45 curveto 795 45 775 80 775 100 curveto 775 120 795 155 830 155 curveto fill } def /hbar { 480 105 moveto 440 55 400 -10 335 -10 curveto 300 -10 290 10 290 35 curveto 290 65 370 350 370 365 curveto 370 380 355 390 340 390 curveto 290 390 205 275 185 250 curveto 140 180 125 125 95 0 curveto 20 0 lineto 150 500 lineto 160 545 180 600 180 615 curveto 180 625 170 640 135 640 curveto 110 640 lineto 100 655 lineto 165 660 215 670 270 685 curveto 275 680 lineto 155 240 lineto 155 235 lineto 190 290 290 440 380 440 curveto 400 440 450 435 450 370 curveto 450 335 365 80 365 55 curveto 365 45 375 40 385 40 curveto 405 40 425 70 465 115 curveto 480 105 lineto fill 55 545 moveto 55 520 lineto 310 560 lineto 310 585 lineto 55 545 lineto fill } def /bigcap { 615 400 moveto 615 575 500 655 340 655 curveto 180 655 70 575 70 400 curveto 70 0 lineto 5 0 lineto 5 400 lineto 5 610 140 720 340 720 curveto 545 720 680 610 680 400 curveto 680 0 lineto 615 0 lineto 615 400 lineto closepath } def /bigcup { 65 320 moveto 65 145 180 65 340 65 curveto 500 65 615 145 615 320 curveto 615 720 lineto 680 720 lineto 680 320 lineto 680 110 540 0 340 0 curveto 140 0 0 110 0 320 curveto 0 720 lineto 65 720 lineto 65 320 lineto closepath } def /parallel { 135 675 moveto 135 -175 lineto 65 -175 lineto 65 675 lineto 135 675 lineto fill 335 675 moveto 335 -175 lineto 265 -175 lineto 265 675 lineto 335 675 lineto fill 400 675 moveto } def end def /BuildChar { 0 begin /char exch def /fontdict exch def /charname fontdict /Encoding get char get def fontdict begin Metrics charname get 0 BBox charname get aload pop setcachedevice CharacterDefs charname get exec eofill end end } def /BuildChar load 0 3 dict put end definefont pop %%EndSetup 0 0 234.59433 72.771942 rectclip 0 0 234.59433 72.771942 rectclip 0 setgray 1 setalpha 0.48 setlinewidth 25.282667 55.950962 moveto 67.649002 55.950962 lineto stroke 0 setgray 1 setalpha /EqBFont 24 selectfont 79.382835 47.310963 moveto <09> [0 0] xyshow 0 setgray 1 setalpha 1 1 scale /Symbol findfont 12 scalefont 48 exch defineuserobject 48 execuserobject setfont 97.97934 61.441238 moveto (\() [0 0] xyshow 1 1 scale 1 1 scale 48 execuserobject setfont 118.874001 61.435238 moveto (\)) [0 0] xyshow 1 1 scale 0 setgray 1 setalpha 1 1.292876 scale 48 execuserobject setfont 130.166 46.867771 moveto (\() [0 0] xyshow 1 0.773469 scale 1 1.291382 scale 48 execuserobject setfont 170.254333 46.919365 moveto (\)) [0 0] xyshow 1 0.774364 scale 0 setgray 1 setalpha 1 1.292876 scale 48 execuserobject setfont 116.169678 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7:1,11,8,Times,0,12,0,0,0;1,10,8,Times,2,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Times,2,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Times,2,12,0,0,0;1,11,8,Times,0,12,0,0,0; :[font = subsection; inactive; preserveAspect; startGroup; ] Exercises :[font = text; inactive; preserveAspect; ] Find the Fourier coefficients of the following functions, and write out the complete Fourier series: :[font = subsubsection; inactive; preserveAspect; startGroup; ] #1 :[font = text; inactive; preserveAspect; ] x2 (-p < x < p) ;[s] 9:0,0;1,1;2,2;40,3;41,4;44,5;46,6;48,7;49,8;50,-1; 9:1,11,8,Times,0,12,0,0,0;1,9,7,Times,32,10,0,0,0;1,11,8,Times,0,12,0,0,0;1,0,0,Symbol,0,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Times,3,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,0,0,Symbol,0,12,0,0,0;1,11,8,Times,0,12,0,0,0; :[font = input; preserveAspect; ] f1[x_]:=x^2 :[font = input; preserveAspect; startGroup; ] A0[f1] :[font = output; output; inactive; preserveAspect; endGroup; ] Pi^2/3 ;[o] 2 Pi --- 3 :[font = input; preserveAspect; startGroup; ] Simplify[A[n,f1] /. TrigCoefRules[n]] :[font = output; output; inactive; preserveAspect; endGroup; ] (4*(-1)^n)/n^2 ;[o] n 4 (-1) ------- 2 n :[font = text; inactive; preserveAspect; ] Hence :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 0; pictureWidth = 144; pictureHeight = 36; endGroup; ] %! %%Creator: Mathematica %%AspectRatio: 0.25 200 dict begin MathPictureStart /Courier findfont 10 scalefont setfont % Scaling calculations 0 1 0 1 [ [ 0 0 0 0 ] [ 144 36 0 0 ] ] MathScale % Start of Graphics %!PS-Adobe-2.0 EPSF-2.0 %%Title: %%Creator: EquationBuilder %%CreationDate: Mon Aug 3 07:33:52 1992 %%For: gray %%DocumentFonts: (atend) %%Pages: 0 1 %%BoundingBox: 0 0 144 36 %%Translation: 1 14.884993 142.363678 0 %%EndComments %%BeginDocument: /usr/lib/NextStep/printPackage.ps %!PS-Adobe-2.0 %%Title: Printing Package %%EndComments %Version: 2.0 %Copyright: 1988, NeXT, Inc. /__NXdef{1 index where{pop pop pop}{def}ifelse}bind def /__NXbdef{1 index where{pop pop pop}{bind def}ifelse}bind def /UserObjects 10 array __NXdef /defineuserobject{ exch dup 1 add dup UserObjects length gt{ array dup 0 UserObjects putinterval /UserObjects exch def }{pop}ifelse UserObjects exch 3 -1 roll put }__NXbdef /undefineuserobject{UserObjects exch null put}__NXbdef /execuserobject{UserObjects exch get exec}__NXbdef /__NXRectPath{4 2 roll moveto 1 index 0 rlineto 0 exch rlineto neg 0 rlineto closepath}__NXbdef /__NXProcessRectArgs{ 1 index type /arraytype eq{ exch 0 4 2 index length 1 sub{ dup 3 add 1 exch{1 index exch get exch}for 5 1 roll 5 index exec }for pop pop }{exec}ifelse }__NXbdef /rectfill{gsave newpath {__NXRectPath fill} __NXProcessRectArgs grestore}__NXbdef /rectclip{newpath {__NXRectPath} __NXProcessRectArgs clip newpath}__NXbdef /rectstroke{ gsave newpath dup type /arraytype eq{dup length 6 eq}{false}ifelse{ {gsave __NXRectPath null concat stroke grestore} dup length array cvx copy dup 2 4 -1 roll put __NXProcessRectArgs }{{__NXRectPath stroke} __NXProcessRectArgs}ifelse grestore }__NXbdef /xyshow{ 0 1 3 index length 1 sub{ currentpoint 4 index 3 index 1 getinterval show 3 index 3 index 2 mul 1 add get add exch 3 index 3 index 2 mul get add exch moveto pop }for pop pop }__NXbdef /xshow{ 0 1 3 index length 1 sub{ currentpoint 4 index 3 index 1 getinterval show exch 3 index 3 index get add exch moveto pop }for pop pop }__NXbdef /yshow{ 0 1 3 index length 1 sub{ currentpoint 4 index 3 index 1 getinterval show 3 index 3 index get add moveto pop }for pop pop }__NXbdef /arct{arcto pop pop pop pop}__NXbdef /setbbox{pop pop pop pop}__NXbdef /ucache{}__NXbdef /ucachestatus{mark 0 0 0 0 0}__NXbdef /setucacheparams{cleartomark}__NXbdef /uappend{systemdict begin cvx exec end}__NXbdef /ueofill{gsave newpath uappend eofill grestore}__NXbdef /ufill{gsave newpath uappend fill grestore}__NXbdef /ustroke{ gsave newpath dup length 6 eq {exch uappend concat}{uappend}ifelse stroke grestore }__NXbdef /__NXustrokepathMatrix dup where {pop 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/NextStepEncoding NextStepEncoding readonly def /_NXfstr 128 string dup 0 (_NX) putinterval def /findfont{ % Because we can never let NextStepEncoding get into % SharedFontDirectory, we cannot reencode a font to NextStepEncoding % if we are in shared mode. 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exch get exec length _NXTotalBytes exch sub /_NXTotalBytes exch store}for _NXTotalBytes 0 le{exit}if }loop /_NXProcArray 5 array def }__NXbdef /colorimage{_NXDoImageOp}__NXbdef /alphaimage{1 add _NXDoImageOp}def %%EndDocument % EquationBuilder(TM) Expression % +.\Jr(=:/*V[StRFQAYmdjEEA;1nuN+[co.OZ7L#cq3iK%3:</XJ@:d#6;^G;Td47dud<A3@ % 5%MGToUOR5&d`'TuEDU)B-tbR".R+:nND,VT6!5U$5)SqI@LR55(=Tg/YZ`?^*:b*44kjeT] % p^_3=B>@PZ-dtPS8[n?`e+Q,0;)ut^=;<6Hl%@`i0rdKeEK[-&FF=^!e]Qc(1->="q0H2.ZK % !T0$!9peU_%Sh>d^TT.-Vr^=F>bl!Uph#9!E$>MfXt;%PWWg$pcXVHiXdrBH)G9@WAQ*:KEM % Qd$7`h;P6"WZ$io;:F?G9`M'?gWY:8VZ!E_j'4edo%-]>?_1JmipmY"hlB3sNlq!k![/L@d> % OJ2+C1hWF2P(Y%$.?^cQ*.!8lO:iKNETDk$X*+.YIJ+7>hK1o2&r8sb115bT@GiE)E2kM0>' % utK&=X*8IDT)<`XOVZG)YBh+b668-_?(S"TV&9gF]9IMt7)hhL1'UOrZ&5mYKh_)ksXrKhC& % hN$Pu*ieo)!/;W=q1a7dQJ5c_5K7?!9M+ht!.NjCXUmYg3@dg9EAUls"):8dZ<bQ*7)/^9'% % 3[o4&;c.`JYJg?$1Q*)P"`2pO+dd@8p6XWAgrWr;QA!<9*.,E)qi3IfVjEta^cop^_#^=XNG % 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500 def /pathint 500 def /sum 700 def /prod 700 def /cdot 1000 def /vdot 550 def /ddot 1000 def /hbar 500 def /bigcap 700 def /bigcup 700 def /parallel 500 def end def /BBox 18 dict dup begin /.notdef [0 0 0 0] def /rad1 [110 -1150 1020 0] def /rad2 [110 -1750 1020 0] def /rad3 [110 -2350 1020 0] def /rad4 [110 -2950 1020 0] def /rad5 [128 -2890 740 0] def /radext [700 -600 740 0] def /int [20 0 425 1045] def /pathint [20 0 425 1045] def /sum [15 5 640 715] def /prod [15 0 670 715] def /cdot [110 215 890 325] def /vdot [165 -5 280 770] def /ddot [110 -5 890 770] def /hbar [20 -10 480 685] def /bigcap [0 0 680 720] def /bigcup [0 0 680 720] def /parallel [65 -175 400 675] def end def /CharacterDefs 18 dict dup begin /.notdef { } def /rad1 { 980 0 moveto 464 -1048 lineto 248 -560 lineto 110 -656 lineto 125 -675 lineto 200 -624 lineto 420 -1150 lineto 460 -1150 lineto 1020 0 lineto 980 0 lineto closepath } def /rad2 { 980 0 moveto 460 -1600 lineto 245 -864 lineto 110 -1000 lineto 128 -1023 lineto 200 -950 lineto 424 -1750 lineto 456 -1750 lineto 1020 0 lineto 980 0 lineto closepath } def /rad3 { 980 0 moveto 464 -2144 lineto 248 -1160 lineto 110 -1368 lineto 130 -1390 lineto 192 -1288 lineto 424 -2350 lineto 464 -2350 lineto 1020 0 lineto 980 0 lineto closepath } def /rad4 { 980 0 moveto 880 -488 lineto 672 -1600 lineto 464 -2696 lineto 248 -1464 lineto 110 -1728 lineto 130 -1750 lineto 192 -1616 lineto 424 -2950 lineto 464 -2950 lineto 768 -1312 lineto 1020 0 lineto 980 0 lineto closepath } def /rad5 { 700 0 moveto 700 -2628 lineto 248 -1284 lineto 128 -1564 lineto 150 -1580 lineto 215 -1425 lineto 688 -2890 lineto 740 -2890 lineto 740 0 lineto 700 0 lineto closepath } def /radext { 700 0 moveto 700 -600 lineto 740 -600 lineto 740 0 lineto 700 0 lineto closepath } def /int { 245 390 moveto 225 235 195 145 180 100 curveto 165 60 150 50 140 40 curveto 135 30 85 0 50 20 curveto 35 25 25 45 25 55 curveto 20 80 30 100 55 110 curveto 80 125 93 110 110 90 curveto 135 60 155 135 160 195 curveto 165 270 170 340 190 510 curveto 205 650 lineto 220 810 230 890 265 960 curveto 275 980 295 1015 320 1030 curveto 350 1045 380 1030 395 1020 curveto 410 1010 425 980 400 955 curveto 385 940 347 925 330 950 curveto 310 970 290 955 290 940 curveto 280 900 285 745 270 610 curveto 245 390 lineto closepath } def /pathint { 245 390 moveto 225 235 195 145 180 100 curveto 165 60 150 50 140 40 curveto 135 30 85 0 50 20 curveto 35 25 25 45 25 55 curveto 20 80 30 100 55 110 curveto 80 125 93 110 110 90 curveto 135 60 155 135 160 195 curveto 165 270 170 340 190 510 curveto 205 650 lineto 220 810 230 890 265 960 curveto 275 980 295 1015 320 1030 curveto 350 1045 380 1030 395 1020 curveto 410 1010 425 980 400 955 curveto 385 940 347 925 330 950 curveto 310 970 290 955 290 940 curveto 280 900 285 745 270 610 curveto 245 390 lineto fill 20 setlinewidth 220 400 moveto 290 400 340 455 340 500 curveto 340 585 290 640 220 640 curveto 155 640 100 585 100 500 curveto 100 455 155 400 220 400 curveto stroke } def /sum { 330 335 moveto 20 715 lineto 575 715 lineto 600 545 lineto 575 535 lineto 550 625 525 685 395 685 curveto 160 685 lineto 411 370 lineto 160 90 lineto 470 90 lineto 565 90 590 140 610 210 curveto 640 200 lineto 585 5 lineto 15 5 lineto 330 335 lineto closepath } def /prod { 590 90 moveto 590 30 620 20 670 20 curveto 670 0 lineto 410 0 lineto 410 20 lineto 460 20 495 30 495 90 curveto 495 670 lineto 190 670 lineto 190 90 lineto 190 30 225 20 275 20 curveto 275 0 lineto 15 0 lineto 15 20 lineto 65 20 100 30 100 90 curveto 100 615 lineto 100 690 40 700 20 700 curveto 20 715 lineto 670 715 lineto 670 700 lineto 645 700 590 690 590 615 curveto 590 90 lineto closepath } def /cdot { 110 270 moveto 110 300 135 325 165 325 curveto 195 325 220 300 220 270 curveto 220 235 185 215 165 215 curveto 145 215 110 235 110 270 curveto fill 445 270 moveto 445 300 470 325 500 325 curveto 530 325 555 300 555 270 curveto 555 235 520 215 500 215 curveto 480 215 445 235 445 270 curveto fill 775 270 moveto 775 300 800 325 830 325 curveto 860 325 890 300 890 270 curveto 890 235 850 215 830 215 curveto 810 215 775 235 775 270 curveto fill } def /vdot { 220 770 moveto 250 770 280 750 280 720 curveto 280 680 250 660 220 660 curveto 185 660 165 695 165 720 curveto 165 740 185 770 220 770 curveto fill 220 440 moveto 250 440 280 415 280 385 curveto 280 355 250 330 220 330 curveto 185 330 165 365 165 385 curveto 165 405 185 440 220 440 curveto fill 220 105 moveto 250 105 280 80 280 50 curveto 280 20 250 -5 220 -5 curveto 185 -5 165 30 165 50 curveto 165 70 185 105 220 105 curveto fill } def /ddot { 165 720 moveto 205 720 220 700 220 665 curveto 220 630 195 610 165 610 curveto 130 620 110 645 110 665 curveto 110 690 130 720 165 720 curveto fill 500 440 moveto 530 440 555 415 555 385 curveto 555 355 530 330 500 330 curveto 465 330 445 365 445 385 curveto 445 405 465 440 500 440 curveto fill 830 155 moveto 860 155 890 130 890 100 curveto 890 70 860 45 830 45 curveto 795 45 775 80 775 100 curveto 775 120 795 155 830 155 curveto fill } def /hbar { 480 105 moveto 440 55 400 -10 335 -10 curveto 300 -10 290 10 290 35 curveto 290 65 370 350 370 365 curveto 370 380 355 390 340 390 curveto 290 390 205 275 185 250 curveto 140 180 125 125 95 0 curveto 20 0 lineto 150 500 lineto 160 545 180 600 180 615 curveto 180 625 170 640 135 640 curveto 110 640 lineto 100 655 lineto 165 660 215 670 270 685 curveto 275 680 lineto 155 240 lineto 155 235 lineto 190 290 290 440 380 440 curveto 400 440 450 435 450 370 curveto 450 335 365 80 365 55 curveto 365 45 375 40 385 40 curveto 405 40 425 70 465 115 curveto 480 105 lineto fill 55 545 moveto 55 520 lineto 310 560 lineto 310 585 lineto 55 545 lineto fill } def /bigcap { 615 400 moveto 615 575 500 655 340 655 curveto 180 655 70 575 70 400 curveto 70 0 lineto 5 0 lineto 5 400 lineto 5 610 140 720 340 720 curveto 545 720 680 610 680 400 curveto 680 0 lineto 615 0 lineto 615 400 lineto closepath } def /bigcup { 65 320 moveto 65 145 180 65 340 65 curveto 500 65 615 145 615 320 curveto 615 720 lineto 680 720 lineto 680 320 lineto 680 110 540 0 340 0 curveto 140 0 0 110 0 320 curveto 0 720 lineto 65 720 lineto 65 320 lineto closepath } def /parallel { 135 675 moveto 135 -175 lineto 65 -175 lineto 65 675 lineto 135 675 lineto fill 335 675 moveto 335 -175 lineto 265 -175 lineto 265 675 lineto 335 675 lineto fill 400 675 moveto } def end def /BuildChar { 0 begin /char exch def /fontdict exch def /charname fontdict /Encoding get char get def fontdict begin Metrics charname get 0 BBox charname get aload pop setcachedevice CharacterDefs charname get exec eofill end end } def /BuildChar load 0 3 dict put end definefont pop %%EndSetup 0 0 143.363678 35.640968 rectclip 0 0 143.363678 35.640968 rectclip 0 setgray 1 setalpha 0.48 setlinewidth 25.954666 18.884993 moveto 39.242668 18.884993 lineto stroke 0 setgray 1 setalpha /EqBFont 24 selectfont 58.976501 10.244992 moveto <09> [0 0] xyshow 0 setgray 1 setalpha 1 1 scale /Symbol findfont 12 scalefont 47 exch defineuserobject 47 execuserobject setfont 77.573006 23.792992 moveto (\() [0 0] xyshow 1 1 scale 1 1 scale 47 execuserobject setfont 98.467667 23.786991 moveto (\)) [0 0] xyshow 1 1 scale 0 setgray 1 setalpha 0.48 setlinewidth 76.509003 18.884993 moveto 109.695671 18.884993 lineto stroke 0 setgray 1 setalpha /Times-BoldItalic findfont 12 scalefont 48 exch defineuserobject 48 execuserobject setfont 136.363663 15.884993 moveto (xnnx) [-6.671997 0 -42.77533 -9.118704 -85.916336 9.118704 0 0 0 0] xyshow /Times-Roman findfont 9 scalefont 49 exch defineuserobject 49 execuserobject setfont 93.588333 11.120993 moveto (2122) [-23.57933 -8.839993 -36.966335 25.758696 -26.042667 -7.199999 0 0 0 0] xyshow /Symbol findfont 9 scalefont 50 exch defineuserobject 50 execuserobject setfont 64.618004 2.281 moveto (=\245) [-0.990002 27.523968 0 0 0 0] xyshow 47 execuserobject setfont 82.373001 23.684992 moveto (-+p=) [-40.463669 -7.799999 -15.454666 7.799999 -10.421333 -7.799999 0 0 0 0] xyshow /Times-Roman findfont 12 scalefont 51 exch defineuserobject 51 execuserobject setfont 111.695671 15.884993 moveto (cos143) [5.328003 0 6 0 -31.396004 7.799999 -40.463669 -7.799999 -21.565334 -8.231412 0 0 0 0] xyshow /Times-BoldItalic findfont 9 scalefont 52 exch defineuserobject 52 execuserobject setfont 102.991669 28.400019 moveto (nn) [-43.827667 -26.119019 0 0 0 0] xyshow %%Trailer %%DocumentFonts: Times-BoldItalic %%+ Times-Roman %%+ Symbol %End of Graphics MathPictureEnd end :[font = subsubsection; inactive; preserveAspect; startGroup; ] #2 :[font = text; inactive; preserveAspect; ] x3 (-p < x < p) ;[s] 9:0,0;1,1;2,2;42,3;43,4;46,5;48,6;50,7;51,8;52,-1; 9:1,11,8,Times,0,12,0,0,0;1,9,7,Times,32,10,0,0,0;1,11,8,Times,0,12,0,0,0;1,0,0,Symbol,0,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Times,3,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,0,0,Symbol,0,12,0,0,0;1,11,8,Times,0,12,0,0,0; :[font = input; preserveAspect; ] f2[x_]:=x^3 :[font = input; preserveAspect; startGroup; ] Simplify[B[n,f2] /. TrigCoefRules[n]] :[font = output; output; inactive; pageBreakBelow; dontNoPageBreakBelow; preserveAspect; endGroup; ] (2*(-1)^n*(6 - n^2*Pi^2))/n^3 ;[o] n 2 2 2 (-1) (6 - n Pi ) -------------------- 3 n :[font = text; inactive; preserveAspect; ] Hence :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 0; pictureWidth = 164; pictureHeight = 36; endGroup; ] %! %%Creator: Mathematica %%AspectRatio: 0.219512 200 dict begin MathPictureStart /Courier findfont 10 scalefont setfont % Scaling calculations 0 1 0 1 [ [ 0 0 0 0 ] [ 164 36 0 0 ] ] MathScale % Start of Graphics %!PS-Adobe-2.0 EPSF-2.0 %%Title: %%Creator: EquationBuilder %%CreationDate: Mon Aug 3 07:36:44 1992 %%For: gray %%DocumentFonts: (atend) %%Pages: 0 1 %%BoundingBox: 0 0 164 36 %%Translation: 1 14.884993 162.771667 0 %%EndComments %%BeginDocument: /usr/lib/NextStep/printPackage.ps %!PS-Adobe-2.0 %%Title: Printing Package %%EndComments %Version: 2.0 %Copyright: 1988, NeXT, Inc. /__NXdef{1 index where{pop pop pop}{def}ifelse}bind def /__NXbdef{1 index where{pop pop pop}{bind def}ifelse}bind def /UserObjects 10 array __NXdef /defineuserobject{ exch dup 1 add dup UserObjects length gt{ array dup 0 UserObjects putinterval /UserObjects exch def }{pop}ifelse UserObjects exch 3 -1 roll put }__NXbdef /undefineuserobject{UserObjects exch null put}__NXbdef /execuserobject{UserObjects exch get exec}__NXbdef /__NXRectPath{4 2 roll moveto 1 index 0 rlineto 0 exch rlineto neg 0 rlineto closepath}__NXbdef /__NXProcessRectArgs{ 1 index type /arraytype eq{ exch 0 4 2 index length 1 sub{ dup 3 add 1 exch{1 index exch get exch}for 5 1 roll 5 index exec }for pop pop }{exec}ifelse }__NXbdef /rectfill{gsave newpath {__NXRectPath fill} __NXProcessRectArgs grestore}__NXbdef /rectclip{newpath {__NXRectPath} __NXProcessRectArgs clip newpath}__NXbdef /rectstroke{ gsave newpath dup type /arraytype eq{dup length 6 eq}{false}ifelse{ {gsave __NXRectPath null concat stroke grestore} dup length array cvx copy dup 2 4 -1 roll put __NXProcessRectArgs }{{__NXRectPath stroke} __NXProcessRectArgs}ifelse grestore }__NXbdef /xyshow{ 0 1 3 index length 1 sub{ currentpoint 4 index 3 index 1 getinterval show 3 index 3 index 2 mul 1 add get add exch 3 index 3 index 2 mul get add exch moveto pop }for pop pop }__NXbdef /xshow{ 0 1 3 index length 1 sub{ currentpoint 4 index 3 index 1 getinterval show exch 3 index 3 index get add exch moveto pop }for pop pop }__NXbdef /yshow{ 0 1 3 index length 1 sub{ currentpoint 4 index 3 index 1 getinterval show 3 index 3 index get add moveto pop }for pop pop }__NXbdef /arct{arcto pop pop pop pop}__NXbdef /setbbox{pop pop pop pop}__NXbdef /ucache{}__NXbdef /ucachestatus{mark 0 0 0 0 0}__NXbdef /setucacheparams{cleartomark}__NXbdef /uappend{systemdict begin cvx exec end}__NXbdef /ueofill{gsave newpath uappend eofill grestore}__NXbdef /ufill{gsave newpath uappend fill grestore}__NXbdef /ustroke{ gsave newpath dup length 6 eq {exch uappend concat}{uappend}ifelse stroke grestore }__NXbdef /__NXustrokepathMatrix dup where {pop pop}{matrix def}ifelse /ustrokepath{ newpath dup length 6 eq{ exch uappend __NXustrokepathMatrix currentmatrix exch concat strokepath setmatrix }{uappend strokepath}ifelse } __NXbdef /upath{ [exch {/ucache cvx}if pathbbox /setbbox cvx {/moveto cvx}{/lineto cvx}{/curveto cvx}{/closepath cvx}pathforall]cvx } __NXbdef /setstrokeadjust{pop}__NXbdef /currentstrokeadjust{false}__NXbdef /selectfont{exch findfont exch dup type /arraytype eq {makefont}{scalefont}ifelse setfont}__NXbdef /_NXCombineArrays{ counttomark dup 2 add index dup length 3 -1 roll { 2 index length sub dup 4 1 roll 1 index exch 4 -1 roll putinterval exch }repeat pop pop pop }__NXbdef /setcmykcolor{ 1.0 exch sub dup dup 6 -1 roll sub dup 0 lt{pop 0}if 5 1 roll 4 -1 roll sub dup 0 lt{pop 0}if 3 1 roll exch sub dup 0 lt{pop 0}if setrgbcolor }__NXbdef /currentcmykcolor{currentrgbcolor 3{1.0 exch sub 3 1 roll}repeat 0}__NXbdef /flushgraphics{}def /setwindowtype{pop pop}def /currentwindowtype{pop 0}def /setalpha{pop}def /currentalpha{1.0}def /hidecursor{}def /obscurecursor{}def /revealcursor{}def /setcursor{4 {pop}repeat}bind def /showcursor{}def /NextStepEncoding where not{ /NextStepEncoding StandardEncoding 256 array copy def 0 [129/Agrave/Aacute/Acircumflex/Atilde/Adieresis/Aring/Ccedilla/Egrave /Eacute/Ecircumflex/Edieresis/Igrave/Iacute/Icircumflex/Idieresis /Eth/Ntilde/Ograve/Oacute/Ocircumflex/Otilde/Odieresis/Ugrave/Uacute /Ucircumflex/Udieresis/Yacute/Thorn/mu/multiply/divide/copyright 176/registered 181/brokenbar 190/logicalnot 192/onesuperior 201/twosuperior 204/threesuperior 209/plusminus/onequarter/onehalf/threequarters/agrave /aacute/acircumflex/atilde/adieresis/aring/ccedilla/egrave/eacute /ecircumflex/edieresis/igrave 226/iacute 228/icircumflex/idieresis/eth /ntilde 236/ograve/oacute/ocircumflex/otilde/odieresis 242/ugrave/uacute /ucircumflex 246/udieresis/yacute 252/thorn/ydieresis] {dup type /nametype eq {NextStepEncoding 2 index 2 index put pop 1 add}{exch pop}ifelse }forall pop /NextStepEncoding NextStepEncoding readonly def /_NXfstr 128 string dup 0 (_NX) putinterval def /findfont{ % Because we can never let NextStepEncoding get into % SharedFontDirectory, we cannot reencode a font to NextStepEncoding % if we are in shared mode. So if currentshared is true, % we call the normal findfont and return that /currentshared where {pop currentshared} {false} ifelse {//findfont exec} {dup _NXfstr 3 125 getinterval cvs length 3 add _NXfstr 0 3 -1 roll getinterval cvn exch FontDirectory 2 index known {pop FontDirectory exch get} {//findfont exec dup /Encoding get StandardEncoding eq { dup length dict exch {1 index /FID ne {2 index 3 1 roll put}{pop pop}ifelse}forall dup /Encoding NextStepEncoding put definefont }{exch pop} ifelse }ifelse }ifelse }bind def }{pop}ifelse /_NXProcArray 5 array __NXdef /_NXChannels 0 __NXdef /_NXTotalBytes 0 __NXdef /_NXDoImageOp{ 1 index{dup}{1}ifelse /_NXChannels exch store _NXChannels 2 add 2 roll _NXProcArray 0 _NXChannels getinterval astore pop 5 index 4 index mul 2 index{1 sub 8 idiv 1 add mul}{mul 1 sub 8 idiv 1 add}ifelse 4 index mul /_NXTotalBytes exch store pop exch pop gsave matrix invertmatrix concat 0.5 setgray 0 0 4 2 roll rectfill grestore { 0 1 _NXChannels 1 sub{ _NXProcArray exch get exec length _NXTotalBytes exch sub /_NXTotalBytes exch store}for _NXTotalBytes 0 le{exit}if }loop /_NXProcArray 5 array def }__NXbdef /colorimage{_NXDoImageOp}__NXbdef /alphaimage{1 add _NXDoImageOp}def %%EndDocument % EquationBuilder(TM) Expression % +.\Jr(=:/*V[StRFQAYmdjEEA;1nuN+[co.OZ7L#cq3iK%3:</XJ@:d#6;^G;Td47dud<A3@ % 5%MGToUOR5&d`'TuEDU)B-tbR".R+:nNN,VT6!O<O^$SqI@LR55(=Tg/YZ`?^*:b*44kjeT] % p^_3=B>@PZ-dtPS8[n?`e+Q,0;)ut^=;<6Hl%@`i0rdKeEK[-&FF=^!e]Qc(1->="q0H2-/J % uDmo!9peU_%Sh>d^TT.-Vr^=F>bl!Uph#9!LRM9hQ($VSi1'USjlt)#TD.';`"-s)AHLE!e` % 00(#Y7s!%KB$!-(Ul*^bTK%53aneFM`%WhZR8E<c-#7G>Np(jshV6E[pR0Ap*JGYj!X&mA\a % OEU]Ke`Vf8Th'3],c_c+*[(rf`.g`$\^B8Kh"`m9I>!P,>n<S2@M*e;/]J]V$N:iLGSuY\J. % )g3<fe:>]cfmbWejiJ!1:k^>DSI';PAs>Tr;BfV-p2-@m6&8^aRkj#0`.c@5qctDI=9=P>D; % tUj[p'!X]>Y@S*;A!.r7&6o"=HU7'5UnrK5:L;s]G"/KA'!)'_5"H5</80`g6$31GW7kWrJN % Iu\s=>[uK\Q8l$Pefg-D59J<97JQP9lU2A(;jsB8l?+mZj@loBr2c&R!Xa"P9hV]VC5^X@CH % $4RBk*>B:8FHj6!KlT&OMcbi'ARYlG+D!!#4`@P"k2k*JCA?/X=NSkWsGX-D/VJfQA#D*"PJ % >&p++K&d-*Ni8,",KVqQ&n9+[SDHR\A*Xh!!+n(SaB1OY$E9a/fPU6KX0HE,$[.C[#-a7anp % !sG>^DmA,SQ6NVR`jWb?2dSV46t&Gbo3<!0ZCf8O&@LHQ8(WLCBuh&]"LLR0WY)Ed]q]9<>G % ?5230cd5".m[AEAdccnTP8k?j_*")5T:2(\\`O/<9[EM^qWf"iTD-c.T!hW]f`o0o?aT@Vf+ % NXcMU4#W;bUlq$6DC$5W?`^/c"l!Sc6:nn#tt@i(h8f8GMcp1I@kd$1JVPdZ<GJIj?&^!$Id % t%n>fk0.54.1Em,]ehE+m-G_UKT!X]!Am4hQ^$m]3\GmdG7WCea5EEF&qO9^@)D=B02?XtEJ % 7i7GM6(S.r)ioB%b<?jJ&Cpr'ah@e7&HjH]E8lp%x % EndExpression N 896 380 E f3 S 18249 R 94e0dad2 gsave /__NXbasematrix matrix currentmatrix def grestore %%EndProlog %%BeginSetup /EqBFont 18 dict dup begin /FontType 3 def /PaintType 0 def /FontMatrix [0.001 0 0 0.001 0 0] def /FontBBox [15 -2950 1360 1045] def /Encoding [/.notdef /rad1 /rad2 /rad3 /rad4 /rad5 /radext /int /pathint /sum /prod /cdot /vdot /ddot /hbar /bigcap /bigcup /parallel] def /Metrics 18 dict dup begin /.notdef 0 def /rad1 1000 def /rad2 1000 def /rad3 1000 def /rad4 1000 def /rad5 1048 def /radext 1048 def /int 500 def /pathint 500 def /sum 700 def /prod 700 def /cdot 1000 def /vdot 550 def /ddot 1000 def /hbar 500 def /bigcap 700 def /bigcup 700 def /parallel 500 def end def /BBox 18 dict dup begin /.notdef [0 0 0 0] def /rad1 [110 -1150 1020 0] def /rad2 [110 -1750 1020 0] def /rad3 [110 -2350 1020 0] def /rad4 [110 -2950 1020 0] def /rad5 [128 -2890 740 0] def /radext [700 -600 740 0] def /int [20 0 425 1045] def /pathint [20 0 425 1045] def /sum [15 5 640 715] def /prod [15 0 670 715] def /cdot [110 215 890 325] def /vdot [165 -5 280 770] def /ddot [110 -5 890 770] def /hbar [20 -10 480 685] def /bigcap [0 0 680 720] def /bigcup [0 0 680 720] def /parallel [65 -175 400 675] def end def /CharacterDefs 18 dict dup begin /.notdef { } def /rad1 { 980 0 moveto 464 -1048 lineto 248 -560 lineto 110 -656 lineto 125 -675 lineto 200 -624 lineto 420 -1150 lineto 460 -1150 lineto 1020 0 lineto 980 0 lineto closepath } def /rad2 { 980 0 moveto 460 -1600 lineto 245 -864 lineto 110 -1000 lineto 128 -1023 lineto 200 -950 lineto 424 -1750 lineto 456 -1750 lineto 1020 0 lineto 980 0 lineto closepath } def /rad3 { 980 0 moveto 464 -2144 lineto 248 -1160 lineto 110 -1368 lineto 130 -1390 lineto 192 -1288 lineto 424 -2350 lineto 464 -2350 lineto 1020 0 lineto 980 0 lineto closepath } def /rad4 { 980 0 moveto 880 -488 lineto 672 -1600 lineto 464 -2696 lineto 248 -1464 lineto 110 -1728 lineto 130 -1750 lineto 192 -1616 lineto 424 -2950 lineto 464 -2950 lineto 768 -1312 lineto 1020 0 lineto 980 0 lineto closepath } def /rad5 { 700 0 moveto 700 -2628 lineto 248 -1284 lineto 128 -1564 lineto 150 -1580 lineto 215 -1425 lineto 688 -2890 lineto 740 -2890 lineto 740 0 lineto 700 0 lineto closepath } def /radext { 700 0 moveto 700 -600 lineto 740 -600 lineto 740 0 lineto 700 0 lineto closepath } def /int { 245 390 moveto 225 235 195 145 180 100 curveto 165 60 150 50 140 40 curveto 135 30 85 0 50 20 curveto 35 25 25 45 25 55 curveto 20 80 30 100 55 110 curveto 80 125 93 110 110 90 curveto 135 60 155 135 160 195 curveto 165 270 170 340 190 510 curveto 205 650 lineto 220 810 230 890 265 960 curveto 275 980 295 1015 320 1030 curveto 350 1045 380 1030 395 1020 curveto 410 1010 425 980 400 955 curveto 385 940 347 925 330 950 curveto 310 970 290 955 290 940 curveto 280 900 285 745 270 610 curveto 245 390 lineto closepath } def /pathint { 245 390 moveto 225 235 195 145 180 100 curveto 165 60 150 50 140 40 curveto 135 30 85 0 50 20 curveto 35 25 25 45 25 55 curveto 20 80 30 100 55 110 curveto 80 125 93 110 110 90 curveto 135 60 155 135 160 195 curveto 165 270 170 340 190 510 curveto 205 650 lineto 220 810 230 890 265 960 curveto 275 980 295 1015 320 1030 curveto 350 1045 380 1030 395 1020 curveto 410 1010 425 980 400 955 curveto 385 940 347 925 330 950 curveto 310 970 290 955 290 940 curveto 280 900 285 745 270 610 curveto 245 390 lineto fill 20 setlinewidth 220 400 moveto 290 400 340 455 340 500 curveto 340 585 290 640 220 640 curveto 155 640 100 585 100 500 curveto 100 455 155 400 220 400 curveto stroke } def /sum { 330 335 moveto 20 715 lineto 575 715 lineto 600 545 lineto 575 535 lineto 550 625 525 685 395 685 curveto 160 685 lineto 411 370 lineto 160 90 lineto 470 90 lineto 565 90 590 140 610 210 curveto 640 200 lineto 585 5 lineto 15 5 lineto 330 335 lineto closepath } def /prod { 590 90 moveto 590 30 620 20 670 20 curveto 670 0 lineto 410 0 lineto 410 20 lineto 460 20 495 30 495 90 curveto 495 670 lineto 190 670 lineto 190 90 lineto 190 30 225 20 275 20 curveto 275 0 lineto 15 0 lineto 15 20 lineto 65 20 100 30 100 90 curveto 100 615 lineto 100 690 40 700 20 700 curveto 20 715 lineto 670 715 lineto 670 700 lineto 645 700 590 690 590 615 curveto 590 90 lineto closepath } def /cdot { 110 270 moveto 110 300 135 325 165 325 curveto 195 325 220 300 220 270 curveto 220 235 185 215 165 215 curveto 145 215 110 235 110 270 curveto fill 445 270 moveto 445 300 470 325 500 325 curveto 530 325 555 300 555 270 curveto 555 235 520 215 500 215 curveto 480 215 445 235 445 270 curveto fill 775 270 moveto 775 300 800 325 830 325 curveto 860 325 890 300 890 270 curveto 890 235 850 215 830 215 curveto 810 215 775 235 775 270 curveto fill } def /vdot { 220 770 moveto 250 770 280 750 280 720 curveto 280 680 250 660 220 660 curveto 185 660 165 695 165 720 curveto 165 740 185 770 220 770 curveto fill 220 440 moveto 250 440 280 415 280 385 curveto 280 355 250 330 220 330 curveto 185 330 165 365 165 385 curveto 165 405 185 440 220 440 curveto fill 220 105 moveto 250 105 280 80 280 50 curveto 280 20 250 -5 220 -5 curveto 185 -5 165 30 165 50 curveto 165 70 185 105 220 105 curveto fill } def /ddot { 165 720 moveto 205 720 220 700 220 665 curveto 220 630 195 610 165 610 curveto 130 620 110 645 110 665 curveto 110 690 130 720 165 720 curveto fill 500 440 moveto 530 440 555 415 555 385 curveto 555 355 530 330 500 330 curveto 465 330 445 365 445 385 curveto 445 405 465 440 500 440 curveto fill 830 155 moveto 860 155 890 130 890 100 curveto 890 70 860 45 830 45 curveto 795 45 775 80 775 100 curveto 775 120 795 155 830 155 curveto fill } def /hbar { 480 105 moveto 440 55 400 -10 335 -10 curveto 300 -10 290 10 290 35 curveto 290 65 370 350 370 365 curveto 370 380 355 390 340 390 curveto 290 390 205 275 185 250 curveto 140 180 125 125 95 0 curveto 20 0 lineto 150 500 lineto 160 545 180 600 180 615 curveto 180 625 170 640 135 640 curveto 110 640 lineto 100 655 lineto 165 660 215 670 270 685 curveto 275 680 lineto 155 240 lineto 155 235 lineto 190 290 290 440 380 440 curveto 400 440 450 435 450 370 curveto 450 335 365 80 365 55 curveto 365 45 375 40 385 40 curveto 405 40 425 70 465 115 curveto 480 105 lineto fill 55 545 moveto 55 520 lineto 310 560 lineto 310 585 lineto 55 545 lineto fill } def /bigcap { 615 400 moveto 615 575 500 655 340 655 curveto 180 655 70 575 70 400 curveto 70 0 lineto 5 0 lineto 5 400 lineto 5 610 140 720 340 720 curveto 545 720 680 610 680 400 curveto 680 0 lineto 615 0 lineto 615 400 lineto closepath } def /bigcup { 65 320 moveto 65 145 180 65 340 65 curveto 500 65 615 145 615 320 curveto 615 720 lineto 680 720 lineto 680 320 lineto 680 110 540 0 340 0 curveto 140 0 0 110 0 320 curveto 0 720 lineto 65 720 lineto 65 320 lineto closepath } def /parallel { 135 675 moveto 135 -175 lineto 65 -175 lineto 65 675 lineto 135 675 lineto fill 335 675 moveto 335 -175 lineto 265 -175 lineto 265 675 lineto 335 675 lineto fill 400 675 moveto } def end def /BuildChar { 0 begin /char exch def /fontdict exch def /charname fontdict /Encoding get char get def fontdict begin Metrics charname get 0 BBox charname get aload pop setcachedevice CharacterDefs charname get exec eofill end end } def /BuildChar load 0 3 dict put end definefont pop %%EndSetup 0 0 163.771667 35.705971 rectclip 0 0 163.771667 35.705971 rectclip 0 setgray 1 setalpha /EqBFont 24 selectfont 33.767166 10.244992 moveto <09> [0 0] xyshow 0 setgray 1 setalpha 1 1 scale /Symbol findfont 12 scalefont 47 exch defineuserobject 47 execuserobject setfont 52.363667 24.375265 moveto (\() [0 0] xyshow 1 1 scale 1 1 scale 47 execuserobject setfont 73.258331 24.369265 moveto (\)) [0 0] xyshow 1 1 scale 0 setgray 1 setalpha 1 1.292876 scale 47 execuserobject setfont 84.550339 18.198387 moveto (\() [0 0] xyshow 1 0.773469 scale 1 1.291382 scale 47 execuserobject setfont 127.071671 18.216797 moveto (\)) [0 0] xyshow 1 0.774364 scale 0 setgray 1 setalpha 0.48 setlinewidth 51.299667 18.884993 moveto 132.095673 18.884993 lineto stroke 0 setgray 1 setalpha /Times-BoldItalic findfont 12 scalefont 48 exch defineuserobject 48 execuserobject setfont 156.771667 15.884993 moveto (xnnnx) [-6.671997 0 -64.587997 -9.118704 21.759995 17.500977 -106.271667 -8.382273 0 0 0 0] xyshow /Times-Roman findfont 9 scalefont 49 exch defineuserobject 49 execuserobject setfont 92.18367 11.120994 moveto (3213) [21.759995 17.500977 -69.143997 -26.340969 -37.799667 18.558697 0 0 0 0] xyshow /Symbol findfont 9 scalefont 50 exch defineuserobject 50 execuserobject setfont 39.408665 2.281 moveto (=\245) [-0.989998 27.523968 0 0 0 0] xyshow 47 execuserobject setfont 119.643669 24.267265 moveto (p--=) [-21.626671 0 -40.853333 0 -41.130333 -8.382273 0 0 0 0] xyshow /Times-Roman findfont 12 scalefont 51 exch defineuserobject 51 execuserobject setfont 134.095673 15.884993 moveto (sin612) [4.667999 0 3.335999 0 -52.749336 8.382273 -22.931999 0 -40.463669 -8.382273 0 0 0 0] xyshow /Times-BoldItalic findfont 9 scalefont 52 exch defineuserobject 52 execuserobject setfont 77.782333 28.982294 moveto (nn) [-43.827667 -26.701294 0 0 0 0] xyshow %%Trailer %%DocumentFonts: Times-BoldItalic %%+ Times-Roman %%+ Symbol %End of Graphics MathPictureEnd end :[font = subsubsection; inactive; preserveAspect; startGroup; ] #3 :[font = text; inactive; preserveAspect; ] x4 (-p < x < p) ;[s] 9:0,0;1,1;2,2;43,3;44,4;47,5;49,6;51,7;52,8;53,-1; 9:1,11,8,Times,0,12,0,0,0;1,9,7,Times,32,10,0,0,0;1,11,8,Times,0,12,0,0,0;1,0,0,Symbol,0,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Times,3,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,0,0,Symbol,0,12,0,0,0;1,11,8,Times,0,12,0,0,0; :[font = input; preserveAspect; ] f3[x_]:=x^4 :[font = input; preserveAspect; startGroup; ] A0[f3] :[font = output; output; inactive; preserveAspect; endGroup; ] Pi^4/5 ;[o] 4 Pi --- 5 :[font = input; preserveAspect; startGroup; ] Simplify[A[n,f3] /. TrigCoefRules[n]] :[font = output; output; inactive; preserveAspect; endGroup; ] (8*(-1)^n*(-6 + n^2*Pi^2))/n^4 ;[o] n 2 2 8 (-1) (-6 + n Pi ) --------------------- 4 n :[font = text; inactive; preserveAspect; ] Hence :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 0; pictureWidth = 201; pictureHeight = 36; endGroup; ] %! %%Creator: Mathematica %%AspectRatio: 0.179104 200 dict begin MathPictureStart /Courier findfont 10 scalefont setfont % Scaling calculations 0 1 0 1 [ [ 0 0 0 0 ] [ 201 36 0 0 ] ] MathScale % Start of Graphics %!PS-Adobe-2.0 EPSF-2.0 %%Title: %%Creator: EquationBuilder %%CreationDate: Mon Aug 3 07:35:20 1992 %%For: gray %%DocumentFonts: (atend) %%Pages: 0 1 %%BoundingBox: 0 0 201 36 %%Translation: 1 14.884993 199.227661 0 %%EndComments %%BeginDocument: /usr/lib/NextStep/printPackage.ps %!PS-Adobe-2.0 %%Title: Printing Package %%EndComments %Version: 2.0 %Copyright: 1988, NeXT, Inc. /__NXdef{1 index where{pop pop pop}{def}ifelse}bind def /__NXbdef{1 index where{pop pop pop}{bind def}ifelse}bind def /UserObjects 10 array __NXdef /defineuserobject{ exch dup 1 add dup UserObjects length gt{ array dup 0 UserObjects putinterval /UserObjects exch def }{pop}ifelse UserObjects exch 3 -1 roll put }__NXbdef /undefineuserobject{UserObjects exch null put}__NXbdef /execuserobject{UserObjects exch get exec}__NXbdef /__NXRectPath{4 2 roll moveto 1 index 0 rlineto 0 exch rlineto neg 0 rlineto closepath}__NXbdef /__NXProcessRectArgs{ 1 index type /arraytype eq{ exch 0 4 2 index length 1 sub{ dup 3 add 1 exch{1 index exch get exch}for 5 1 roll 5 index exec }for pop pop }{exec}ifelse }__NXbdef /rectfill{gsave newpath {__NXRectPath fill} __NXProcessRectArgs grestore}__NXbdef /rectclip{newpath {__NXRectPath} __NXProcessRectArgs clip newpath}__NXbdef /rectstroke{ gsave newpath dup type /arraytype eq{dup length 6 eq}{false}ifelse{ {gsave __NXRectPath null concat stroke grestore} dup length array cvx copy dup 2 4 -1 roll put __NXProcessRectArgs }{{__NXRectPath stroke} __NXProcessRectArgs}ifelse grestore }__NXbdef /xyshow{ 0 1 3 index length 1 sub{ currentpoint 4 index 3 index 1 getinterval show 3 index 3 index 2 mul 1 add get add exch 3 index 3 index 2 mul get add exch moveto pop }for pop pop }__NXbdef /xshow{ 0 1 3 index length 1 sub{ currentpoint 4 index 3 index 1 getinterval show exch 3 index 3 index get add exch moveto pop }for pop pop }__NXbdef /yshow{ 0 1 3 index length 1 sub{ currentpoint 4 index 3 index 1 getinterval show 3 index 3 index get add moveto pop }for pop pop }__NXbdef /arct{arcto pop pop pop pop}__NXbdef /setbbox{pop pop pop pop}__NXbdef /ucache{}__NXbdef /ucachestatus{mark 0 0 0 0 0}__NXbdef /setucacheparams{cleartomark}__NXbdef /uappend{systemdict begin cvx exec end}__NXbdef /ueofill{gsave newpath uappend eofill grestore}__NXbdef /ufill{gsave newpath uappend fill grestore}__NXbdef /ustroke{ gsave newpath dup length 6 eq {exch uappend concat}{uappend}ifelse stroke grestore }__NXbdef /__NXustrokepathMatrix dup where {pop pop}{matrix def}ifelse /ustrokepath{ newpath dup length 6 eq{ exch uappend __NXustrokepathMatrix currentmatrix exch concat strokepath setmatrix }{uappend strokepath}ifelse } __NXbdef /upath{ [exch {/ucache cvx}if pathbbox /setbbox cvx {/moveto cvx}{/lineto cvx}{/curveto cvx}{/closepath cvx}pathforall]cvx } __NXbdef /setstrokeadjust{pop}__NXbdef /currentstrokeadjust{false}__NXbdef /selectfont{exch findfont exch dup type /arraytype eq {makefont}{scalefont}ifelse setfont}__NXbdef /_NXCombineArrays{ counttomark dup 2 add index dup length 3 -1 roll { 2 index length sub dup 4 1 roll 1 index exch 4 -1 roll putinterval exch }repeat pop pop pop }__NXbdef /setcmykcolor{ 1.0 exch sub dup dup 6 -1 roll sub dup 0 lt{pop 0}if 5 1 roll 4 -1 roll sub dup 0 lt{pop 0}if 3 1 roll exch sub dup 0 lt{pop 0}if setrgbcolor }__NXbdef /currentcmykcolor{currentrgbcolor 3{1.0 exch sub 3 1 roll}repeat 0}__NXbdef /flushgraphics{}def /setwindowtype{pop pop}def /currentwindowtype{pop 0}def /setalpha{pop}def /currentalpha{1.0}def /hidecursor{}def /obscurecursor{}def /revealcursor{}def /setcursor{4 {pop}repeat}bind def /showcursor{}def /NextStepEncoding where not{ /NextStepEncoding StandardEncoding 256 array copy def 0 [129/Agrave/Aacute/Acircumflex/Atilde/Adieresis/Aring/Ccedilla/Egrave /Eacute/Ecircumflex/Edieresis/Igrave/Iacute/Icircumflex/Idieresis /Eth/Ntilde/Ograve/Oacute/Ocircumflex/Otilde/Odieresis/Ugrave/Uacute /Ucircumflex/Udieresis/Yacute/Thorn/mu/multiply/divide/copyright 176/registered 181/brokenbar 190/logicalnot 192/onesuperior 201/twosuperior 204/threesuperior 209/plusminus/onequarter/onehalf/threequarters/agrave /aacute/acircumflex/atilde/adieresis/aring/ccedilla/egrave/eacute /ecircumflex/edieresis/igrave 226/iacute 228/icircumflex/idieresis/eth /ntilde 236/ograve/oacute/ocircumflex/otilde/odieresis 242/ugrave/uacute /ucircumflex 246/udieresis/yacute 252/thorn/ydieresis] {dup type /nametype eq {NextStepEncoding 2 index 2 index put pop 1 add}{exch pop}ifelse }forall pop /NextStepEncoding NextStepEncoding readonly def /_NXfstr 128 string dup 0 (_NX) putinterval def /findfont{ % Because we can never let NextStepEncoding get into % SharedFontDirectory, we cannot reencode a font to NextStepEncoding % if we are in shared mode. So if currentshared is true, % we call the normal findfont and return that /currentshared where {pop currentshared} {false} ifelse {//findfont exec} {dup _NXfstr 3 125 getinterval cvs length 3 add _NXfstr 0 3 -1 roll getinterval cvn exch FontDirectory 2 index known {pop FontDirectory exch get} {//findfont exec dup /Encoding get StandardEncoding eq { dup length dict exch {1 index /FID ne {2 index 3 1 roll put}{pop pop}ifelse}forall dup /Encoding NextStepEncoding put definefont }{exch pop} ifelse }ifelse }ifelse }bind def }{pop}ifelse /_NXProcArray 5 array __NXdef /_NXChannels 0 __NXdef /_NXTotalBytes 0 __NXdef /_NXDoImageOp{ 1 index{dup}{1}ifelse /_NXChannels exch store _NXChannels 2 add 2 roll _NXProcArray 0 _NXChannels getinterval astore pop 5 index 4 index mul 2 index{1 sub 8 idiv 1 add mul}{mul 1 sub 8 idiv 1 add}ifelse 4 index mul /_NXTotalBytes exch store pop exch pop gsave matrix invertmatrix concat 0.5 setgray 0 0 4 2 roll rectfill grestore { 0 1 _NXChannels 1 sub{ _NXProcArray exch get exec length _NXTotalBytes exch sub /_NXTotalBytes exch store}for _NXTotalBytes 0 le{exit}if }loop /_NXProcArray 5 array def }__NXbdef /colorimage{_NXDoImageOp}__NXbdef /alphaimage{1 add _NXDoImageOp}def %%EndDocument % EquationBuilder(TM) Expression % +.\Jr(=:/*V[StRFQAYmdjEEA;1nuN+[co.OZ7L#cq3iK%3:</XJ@:d#6;^G;Td47dud<A3@ % 5%MGToUOR5&d`'TuEDU)B-tbR".R+:s'6,VT6!0M"`tV$VD;,X_XRS2ihA"VLe`-6WLgVDQ[ % K-Oh*sQ3\5X&jqDA/d.K[>m^Q5J[c<T!&,[Tin#Mqf\EPQp`n<3FPaBu@nDhna*n=K8V8]^g % &Ph&W+?nM8/K#LT#`(=`_@:1Gn;L'ZfuJ,^7@LuTXn]hB>4eNjCohcYj01BLS-[+S+U8eYX. % Y&,XCQk:]W2:%%r2]!"c'0j"WqmeFDUH9g+OI+?QUi&SZ2K=elLsJ]1---PSZ?o)d@bRg16Q % .A6p"(a:"$d,jGJ,b[bLMBo5N[iQ\$Ur8!W*4ZZRCZ#MD?[dZuB3c%@"Cfc2$lC-X(cYMt>\ % fA57NX#[cmHYIT0-.IEW_9Kh.*s*3$9XEQ.I1*!PlOBGE2@kU4!0-V]R(^E5r<H5&MdELT!2 % 2@?^L!,De%9p`bPK%drQ5ND".li0?^]&E&UE+p6YR$4'M_BrNDeR!^Jd8RRWU7^U$p(o@TeL % E8p<",TefFCJ^"XB9.P!k`!>,h)Mp,bd\m_OW$7AbokrQYN:pP=&O:#'[=7Au_P*51p8k0G* % O%X7`HKC8(aInE+ukR01Sr^N*JK3J7g&`j,Xs`LSTE_ItBsNhMYAV1d@^Qo[4l85TIWPefg- % D59J<99M?U:3$B^f%F]c9&e.(Q7hc24IQ>?C.'Z+.LEhn)Z(d4`NFJY5/S0&,f0RIa'SfX'. % ?BO!;li<_\HP6?OA6(_4S]l?Fk$i#,cd*P`Lfol]:mOWTBi8"X'uoqag9RRI<8Xm>D#36:e` % k,Zat_2;DfI@g1r:.3ZA!Rl&7!&nqGl7n`FFb-@g)$[:<.h<h$PS@\@7@#N:,FY]WV=P+n.] % \d&,k4aEP'hFL=,TW,+Ut9H:%?&+!NKA,7jU'd__\5P!Z,MW:$in<h$h?Y6I-:ZWi<a/oR'i % ^S\9A6k5=tS3Tr/eT+]CE+HJ`@J#rg)+`$!/.P,oeJ$SMTd?4:o72FTs/SWU'#/n-F%^&s>a % `Z,L90EiFEQ8*dW%`7'G-o:Wk4d.1=3-qse7nlN(Q_,Hi_L5:W^^tY00uk5ql=lbu4$eiS?^ % ZFrJ6I:.eHZ0_0E?D75Qu2-U]jXm9GlQ8*XEIa%0mS6@*"E43eUpH@mH(V*Z2^"L59:U$N^l % +#_Yh^+<k;$![du`$%Ecnx % EndExpression N 996 3e4 E f6 S 1ad00 R 38fdb6b8 gsave /__NXbasematrix matrix currentmatrix def grestore %%EndProlog %%BeginSetup /EqBFont 18 dict dup begin /FontType 3 def /PaintType 0 def /FontMatrix [0.001 0 0 0.001 0 0] def /FontBBox [15 -2950 1360 1045] def /Encoding [/.notdef /rad1 /rad2 /rad3 /rad4 /rad5 /radext /int /pathint /sum /prod /cdot /vdot /ddot /hbar /bigcap /bigcup /parallel] def /Metrics 18 dict dup begin /.notdef 0 def /rad1 1000 def /rad2 1000 def /rad3 1000 def /rad4 1000 def /rad5 1048 def /radext 1048 def /int 500 def /pathint 500 def /sum 700 def /prod 700 def /cdot 1000 def /vdot 550 def /ddot 1000 def /hbar 500 def /bigcap 700 def /bigcup 700 def /parallel 500 def end def /BBox 18 dict dup begin /.notdef [0 0 0 0] def /rad1 [110 -1150 1020 0] def /rad2 [110 -1750 1020 0] def /rad3 [110 -2350 1020 0] def /rad4 [110 -2950 1020 0] def /rad5 [128 -2890 740 0] def /radext [700 -600 740 0] def /int [20 0 425 1045] def /pathint [20 0 425 1045] def /sum [15 5 640 715] def /prod [15 0 670 715] def /cdot [110 215 890 325] def /vdot [165 -5 280 770] def /ddot [110 -5 890 770] def /hbar [20 -10 480 685] def /bigcap [0 0 680 720] def /bigcup [0 0 680 720] def /parallel [65 -175 400 675] def end def /CharacterDefs 18 dict dup begin /.notdef { } def /rad1 { 980 0 moveto 464 -1048 lineto 248 -560 lineto 110 -656 lineto 125 -675 lineto 200 -624 lineto 420 -1150 lineto 460 -1150 lineto 1020 0 lineto 980 0 lineto closepath } def /rad2 { 980 0 moveto 460 -1600 lineto 245 -864 lineto 110 -1000 lineto 128 -1023 lineto 200 -950 lineto 424 -1750 lineto 456 -1750 lineto 1020 0 lineto 980 0 lineto closepath } def /rad3 { 980 0 moveto 464 -2144 lineto 248 -1160 lineto 110 -1368 lineto 130 -1390 lineto 192 -1288 lineto 424 -2350 lineto 464 -2350 lineto 1020 0 lineto 980 0 lineto closepath } def /rad4 { 980 0 moveto 880 -488 lineto 672 -1600 lineto 464 -2696 lineto 248 -1464 lineto 110 -1728 lineto 130 -1750 lineto 192 -1616 lineto 424 -2950 lineto 464 -2950 lineto 768 -1312 lineto 1020 0 lineto 980 0 lineto closepath } def /rad5 { 700 0 moveto 700 -2628 lineto 248 -1284 lineto 128 -1564 lineto 150 -1580 lineto 215 -1425 lineto 688 -2890 lineto 740 -2890 lineto 740 0 lineto 700 0 lineto closepath } def /radext { 700 0 moveto 700 -600 lineto 740 -600 lineto 740 0 lineto 700 0 lineto closepath } def /int { 245 390 moveto 225 235 195 145 180 100 curveto 165 60 150 50 140 40 curveto 135 30 85 0 50 20 curveto 35 25 25 45 25 55 curveto 20 80 30 100 55 110 curveto 80 125 93 110 110 90 curveto 135 60 155 135 160 195 curveto 165 270 170 340 190 510 curveto 205 650 lineto 220 810 230 890 265 960 curveto 275 980 295 1015 320 1030 curveto 350 1045 380 1030 395 1020 curveto 410 1010 425 980 400 955 curveto 385 940 347 925 330 950 curveto 310 970 290 955 290 940 curveto 280 900 285 745 270 610 curveto 245 390 lineto closepath } def /pathint { 245 390 moveto 225 235 195 145 180 100 curveto 165 60 150 50 140 40 curveto 135 30 85 0 50 20 curveto 35 25 25 45 25 55 curveto 20 80 30 100 55 110 curveto 80 125 93 110 110 90 curveto 135 60 155 135 160 195 curveto 165 270 170 340 190 510 curveto 205 650 lineto 220 810 230 890 265 960 curveto 275 980 295 1015 320 1030 curveto 350 1045 380 1030 395 1020 curveto 410 1010 425 980 400 955 curveto 385 940 347 925 330 950 curveto 310 970 290 955 290 940 curveto 280 900 285 745 270 610 curveto 245 390 lineto fill 20 setlinewidth 220 400 moveto 290 400 340 455 340 500 curveto 340 585 290 640 220 640 curveto 155 640 100 585 100 500 curveto 100 455 155 400 220 400 curveto stroke } def /sum { 330 335 moveto 20 715 lineto 575 715 lineto 600 545 lineto 575 535 lineto 550 625 525 685 395 685 curveto 160 685 lineto 411 370 lineto 160 90 lineto 470 90 lineto 565 90 590 140 610 210 curveto 640 200 lineto 585 5 lineto 15 5 lineto 330 335 lineto closepath } def /prod { 590 90 moveto 590 30 620 20 670 20 curveto 670 0 lineto 410 0 lineto 410 20 lineto 460 20 495 30 495 90 curveto 495 670 lineto 190 670 lineto 190 90 lineto 190 30 225 20 275 20 curveto 275 0 lineto 15 0 lineto 15 20 lineto 65 20 100 30 100 90 curveto 100 615 lineto 100 690 40 700 20 700 curveto 20 715 lineto 670 715 lineto 670 700 lineto 645 700 590 690 590 615 curveto 590 90 lineto closepath } def /cdot { 110 270 moveto 110 300 135 325 165 325 curveto 195 325 220 300 220 270 curveto 220 235 185 215 165 215 curveto 145 215 110 235 110 270 curveto fill 445 270 moveto 445 300 470 325 500 325 curveto 530 325 555 300 555 270 curveto 555 235 520 215 500 215 curveto 480 215 445 235 445 270 curveto fill 775 270 moveto 775 300 800 325 830 325 curveto 860 325 890 300 890 270 curveto 890 235 850 215 830 215 curveto 810 215 775 235 775 270 curveto fill } def /vdot { 220 770 moveto 250 770 280 750 280 720 curveto 280 680 250 660 220 660 curveto 185 660 165 695 165 720 curveto 165 740 185 770 220 770 curveto fill 220 440 moveto 250 440 280 415 280 385 curveto 280 355 250 330 220 330 curveto 185 330 165 365 165 385 curveto 165 405 185 440 220 440 curveto fill 220 105 moveto 250 105 280 80 280 50 curveto 280 20 250 -5 220 -5 curveto 185 -5 165 30 165 50 curveto 165 70 185 105 220 105 curveto fill } def /ddot { 165 720 moveto 205 720 220 700 220 665 curveto 220 630 195 610 165 610 curveto 130 620 110 645 110 665 curveto 110 690 130 720 165 720 curveto fill 500 440 moveto 530 440 555 415 555 385 curveto 555 355 530 330 500 330 curveto 465 330 445 365 445 385 curveto 445 405 465 440 500 440 curveto fill 830 155 moveto 860 155 890 130 890 100 curveto 890 70 860 45 830 45 curveto 795 45 775 80 775 100 curveto 775 120 795 155 830 155 curveto fill } def /hbar { 480 105 moveto 440 55 400 -10 335 -10 curveto 300 -10 290 10 290 35 curveto 290 65 370 350 370 365 curveto 370 380 355 390 340 390 curveto 290 390 205 275 185 250 curveto 140 180 125 125 95 0 curveto 20 0 lineto 150 500 lineto 160 545 180 600 180 615 curveto 180 625 170 640 135 640 curveto 110 640 lineto 100 655 lineto 165 660 215 670 270 685 curveto 275 680 lineto 155 240 lineto 155 235 lineto 190 290 290 440 380 440 curveto 400 440 450 435 450 370 curveto 450 335 365 80 365 55 curveto 365 45 375 40 385 40 curveto 405 40 425 70 465 115 curveto 480 105 lineto fill 55 545 moveto 55 520 lineto 310 560 lineto 310 585 lineto 55 545 lineto fill } def /bigcap { 615 400 moveto 615 575 500 655 340 655 curveto 180 655 70 575 70 400 curveto 70 0 lineto 5 0 lineto 5 400 lineto 5 610 140 720 340 720 curveto 545 720 680 610 680 400 curveto 680 0 lineto 615 0 lineto 615 400 lineto closepath } def /bigcup { 65 320 moveto 65 145 180 65 340 65 curveto 500 65 615 145 615 320 curveto 615 720 lineto 680 720 lineto 680 320 lineto 680 110 540 0 340 0 curveto 140 0 0 110 0 320 curveto 0 720 lineto 65 720 lineto 65 320 lineto closepath } def /parallel { 135 675 moveto 135 -175 lineto 65 -175 lineto 65 675 lineto 135 675 lineto fill 335 675 moveto 335 -175 lineto 265 -175 lineto 265 675 lineto 335 675 lineto fill 400 675 moveto } def end def /BuildChar { 0 begin /char exch def /fontdict exch def /charname fontdict /Encoding get char get def fontdict begin Metrics charname get 0 BBox charname get aload pop setcachedevice CharacterDefs charname get exec eofill end end } def /BuildChar load 0 3 dict put end definefont pop %%EndSetup 0 0 200.227661 35.705971 rectclip 0 0 200.227661 35.705971 rectclip 0 setgray 1 setalpha 0.48 setlinewidth 25.954666 18.884993 moveto 39.242668 18.884993 lineto stroke 0 setgray 1 setalpha /EqBFont 24 selectfont 58.976501 10.244992 moveto <09> [0 0] xyshow 0 setgray 1 setalpha 1 1 scale /Symbol findfont 12 scalefont 47 exch defineuserobject 47 execuserobject setfont 77.573006 24.375265 moveto (\() [0 0] xyshow 1 1 scale 1 1 scale 47 execuserobject setfont 98.467667 24.369265 moveto (\)) [0 0] xyshow 1 1 scale 0 setgray 1 setalpha 1 1.292876 scale 47 execuserobject setfont 109.759674 18.198387 moveto (\() [0 0] xyshow 1 0.773469 scale 1 1.291382 scale 47 execuserobject setfont 161.535675 18.216797 moveto (\)) [0 0] xyshow 1 0.774364 scale 0 setgray 1 setalpha 0.48 setlinewidth 76.509003 18.884993 moveto 166.559662 18.884993 lineto stroke 0 setgray 1 setalpha /Times-BoldItalic findfont 12 scalefont 48 exch defineuserobject 48 execuserobject setfont 193.227661 15.884993 moveto (xnnnx) [-6.671997 0 -71.207329 -9.118704 26.387337 17.500977 -140.735672 -8.382273 0 0 0 0] xyshow /Times-Roman findfont 9 scalefont 49 exch defineuserobject 49 execuserobject setfont 122.020332 11.120993 moveto (42144) [26.387337 17.500977 -78.398666 -26.340969 -36.966335 25.758696 -26.042667 -7.199999 0 0 0 0] xyshow /Symbol findfont 9 scalefont 50 exch defineuserobject 50 execuserobject setfont 64.618004 2.281 moveto (=\245) [-0.990002 27.523968 0 0 0 0] xyshow 47 execuserobject setfont 154.107666 24.267265 moveto (p+--+p=) [-21.626663 0 -17.921333 0 -32.186668 0 -40.463669 -8.382273 -15.454666 7.799999 -10.421333 -7.799999 0 0 0 0] xyshow /Times-Roman findfont 12 scalefont 51 exch defineuserobject 51 execuserobject setfont 168.559662 15.884993 moveto (cos6185) [5.328003 0 6 0 -56.073326 8.382273 -32.186668 0 -40.463669 -8.382273 -21.565334 -8.231412 0 0 0 0] xyshow /Times-BoldItalic findfont 9 scalefont 52 exch defineuserobject 52 execuserobject setfont 102.991669 28.982294 moveto (nn) [-43.827667 -26.701294 0 0 0 0] xyshow %%Trailer %%DocumentFonts: Times-BoldItalic %%+ Times-Roman %%+ Symbol %End of Graphics MathPictureEnd end :[font = subsubsection; inactive; preserveAspect; startGroup; ] #4 :[font = text; inactive; preserveAspect; ] sinh x (-L < x < L) ;[s] 7:0,0;43,1;44,2;47,3;49,4;51,5;52,6;53,-1; 7:1,11,8,Times,0,12,0,0,0;1,10,8,Times,2,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Times,3,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Times,2,12,0,0,0;1,11,8,Times,0,12,0,0,0; :[font = input; preserveAspect; startGroup; ] Simplify[B[n,L,Sinh] /. TrigCoefRules[n]] :[font = output; output; inactive; pageBreakBelow; dontNoPageBreakBelow; preserveAspect; endGroup; ] (-2*(-1)^n*n*Pi*Sinh[L])/(L^2 + n^2*Pi^2) ;[o] n -2 (-1) n Pi Sinh[L] --------------------- 2 2 2 L + n Pi :[font = text; inactive; preserveAspect; ] Hence :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 0; pictureWidth = 188; pictureHeight = 36; endGroup; ] %! %%Creator: Mathematica %%AspectRatio: 0.191489 200 dict begin MathPictureStart /Courier findfont 10 scalefont setfont % Scaling calculations 0 1 0 1 [ [ 0 0 0 0 ] [ 188 36 0 0 ] ] MathScale % Start of Graphics %!PS-Adobe-2.0 EPSF-2.0 %%Title: %%Creator: EquationBuilder %%CreationDate: Mon Aug 3 07:40:19 1992 %%For: gray %%DocumentFonts: (atend) %%Pages: 0 1 %%BoundingBox: 0 0 188 36 %%Translation: 1 14.884993 186.817001 0 %%EndComments %%BeginDocument: /usr/lib/NextStep/printPackage.ps %!PS-Adobe-2.0 %%Title: Printing Package %%EndComments %Version: 2.0 %Copyright: 1988, NeXT, Inc. /__NXdef{1 index where{pop pop pop}{def}ifelse}bind def /__NXbdef{1 index where{pop pop pop}{bind def}ifelse}bind def /UserObjects 10 array __NXdef /defineuserobject{ exch dup 1 add dup UserObjects length gt{ array dup 0 UserObjects putinterval /UserObjects exch def }{pop}ifelse UserObjects exch 3 -1 roll put }__NXbdef /undefineuserobject{UserObjects exch null put}__NXbdef /execuserobject{UserObjects exch get exec}__NXbdef /__NXRectPath{4 2 roll moveto 1 index 0 rlineto 0 exch rlineto neg 0 rlineto closepath}__NXbdef /__NXProcessRectArgs{ 1 index type /arraytype eq{ exch 0 4 2 index length 1 sub{ dup 3 add 1 exch{1 index exch get exch}for 5 1 roll 5 index exec }for pop pop }{exec}ifelse }__NXbdef /rectfill{gsave newpath {__NXRectPath fill} __NXProcessRectArgs grestore}__NXbdef /rectclip{newpath {__NXRectPath} __NXProcessRectArgs clip newpath}__NXbdef /rectstroke{ gsave newpath dup type /arraytype eq{dup length 6 eq}{false}ifelse{ {gsave __NXRectPath null concat stroke grestore} dup length array cvx copy dup 2 4 -1 roll put __NXProcessRectArgs }{{__NXRectPath stroke} __NXProcessRectArgs}ifelse grestore }__NXbdef /xyshow{ 0 1 3 index length 1 sub{ currentpoint 4 index 3 index 1 getinterval show 3 index 3 index 2 mul 1 add get add exch 3 index 3 index 2 mul get add exch moveto pop }for pop pop }__NXbdef /xshow{ 0 1 3 index length 1 sub{ currentpoint 4 index 3 index 1 getinterval show exch 3 index 3 index get add exch moveto pop }for pop pop }__NXbdef /yshow{ 0 1 3 index length 1 sub{ currentpoint 4 index 3 index 1 getinterval show 3 index 3 index get add moveto pop }for pop pop }__NXbdef /arct{arcto pop pop pop pop}__NXbdef /setbbox{pop pop pop pop}__NXbdef /ucache{}__NXbdef /ucachestatus{mark 0 0 0 0 0}__NXbdef /setucacheparams{cleartomark}__NXbdef /uappend{systemdict begin cvx exec end}__NXbdef /ueofill{gsave newpath uappend eofill grestore}__NXbdef /ufill{gsave newpath uappend fill grestore}__NXbdef /ustroke{ gsave newpath dup length 6 eq {exch uappend concat}{uappend}ifelse stroke grestore }__NXbdef /__NXustrokepathMatrix dup where {pop pop}{matrix def}ifelse /ustrokepath{ newpath dup length 6 eq{ exch uappend __NXustrokepathMatrix currentmatrix exch concat strokepath setmatrix }{uappend strokepath}ifelse } __NXbdef /upath{ [exch {/ucache cvx}if pathbbox /setbbox cvx {/moveto cvx}{/lineto cvx}{/curveto cvx}{/closepath cvx}pathforall]cvx } __NXbdef /setstrokeadjust{pop}__NXbdef /currentstrokeadjust{false}__NXbdef /selectfont{exch findfont exch dup type /arraytype eq {makefont}{scalefont}ifelse setfont}__NXbdef /_NXCombineArrays{ counttomark dup 2 add index dup length 3 -1 roll { 2 index length sub dup 4 1 roll 1 index exch 4 -1 roll putinterval exch }repeat pop pop pop }__NXbdef /setcmykcolor{ 1.0 exch sub dup dup 6 -1 roll sub dup 0 lt{pop 0}if 5 1 roll 4 -1 roll sub dup 0 lt{pop 0}if 3 1 roll exch sub dup 0 lt{pop 0}if setrgbcolor }__NXbdef /currentcmykcolor{currentrgbcolor 3{1.0 exch sub 3 1 roll}repeat 0}__NXbdef /flushgraphics{}def /setwindowtype{pop pop}def /currentwindowtype{pop 0}def /setalpha{pop}def /currentalpha{1.0}def /hidecursor{}def /obscurecursor{}def /revealcursor{}def /setcursor{4 {pop}repeat}bind def /showcursor{}def /NextStepEncoding where not{ /NextStepEncoding StandardEncoding 256 array copy def 0 [129/Agrave/Aacute/Acircumflex/Atilde/Adieresis/Aring/Ccedilla/Egrave /Eacute/Ecircumflex/Edieresis/Igrave/Iacute/Icircumflex/Idieresis /Eth/Ntilde/Ograve/Oacute/Ocircumflex/Otilde/Odieresis/Ugrave/Uacute /Ucircumflex/Udieresis/Yacute/Thorn/mu/multiply/divide/copyright 176/registered 181/brokenbar 190/logicalnot 192/onesuperior 201/twosuperior 204/threesuperior 209/plusminus/onequarter/onehalf/threequarters/agrave /aacute/acircumflex/atilde/adieresis/aring/ccedilla/egrave/eacute /ecircumflex/edieresis/igrave 226/iacute 228/icircumflex/idieresis/eth /ntilde 236/ograve/oacute/ocircumflex/otilde/odieresis 242/ugrave/uacute /ucircumflex 246/udieresis/yacute 252/thorn/ydieresis] {dup type /nametype eq {NextStepEncoding 2 index 2 index put pop 1 add}{exch pop}ifelse }forall pop /NextStepEncoding NextStepEncoding readonly def /_NXfstr 128 string dup 0 (_NX) putinterval def /findfont{ % Because we can never let NextStepEncoding get into % SharedFontDirectory, we cannot reencode a font to NextStepEncoding % if we are in shared mode. So if currentshared is true, % we call the normal findfont and return that /currentshared where {pop currentshared} {false} ifelse {//findfont exec} {dup _NXfstr 3 125 getinterval cvs length 3 add _NXfstr 0 3 -1 roll getinterval cvn exch FontDirectory 2 index known {pop FontDirectory exch get} {//findfont exec dup /Encoding get StandardEncoding eq { dup length dict exch {1 index /FID ne {2 index 3 1 roll put}{pop pop}ifelse}forall dup /Encoding NextStepEncoding put definefont }{exch pop} ifelse }ifelse }ifelse }bind def }{pop}ifelse /_NXProcArray 5 array __NXdef /_NXChannels 0 __NXdef /_NXTotalBytes 0 __NXdef /_NXDoImageOp{ 1 index{dup}{1}ifelse /_NXChannels exch store _NXChannels 2 add 2 roll _NXProcArray 0 _NXChannels getinterval astore pop 5 index 4 index mul 2 index{1 sub 8 idiv 1 add mul}{mul 1 sub 8 idiv 1 add}ifelse 4 index mul /_NXTotalBytes exch store pop exch pop gsave matrix invertmatrix concat 0.5 setgray 0 0 4 2 roll rectfill grestore { 0 1 _NXChannels 1 sub{ _NXProcArray exch get exec length _NXTotalBytes exch sub /_NXTotalBytes exch store}for _NXTotalBytes 0 le{exit}if }loop /_NXProcArray 5 array def }__NXbdef /colorimage{_NXDoImageOp}__NXbdef /alphaimage{1 add _NXDoImageOp}def %%EndDocument % EquationBuilder(TM) Expression % +.\Jr(=:/*V[StRFQAYmdjEEA;1nuN+[co.OZ7L#cq3iK%3:</XJ@:d#6;^G;Td47dud<A3@ % 5%MGToUOR5&d`'TuEDU)B-tbR".R+:nNZ,VT6!n0.m/SqI@LR55(=Tg/YZ`?^*:b*44kjeT] % p^_3=B>@PZ-dtPS8[n?`e+Q,0;)ut^=;<6Hl%@`i0rdKeEK[-&FF=^!e]Qc(1->="q0H2,DK % X5B0!9pdj_3BUegmfg@R*I%uRIkY(KpRl$Th@ZT\7@@F$AnjLO9.Ei1`A`.'23rjAkX)jLef % [!9kZE@;6aa9fNej6%,_g_k@l0\0ClpbL7ehtkIYb+baf])%R;phlFS;#_+oeP8i@SJ](qp) % [;+:Y`E'd$KTm*V!'PiFLdf+Qia6a,ZKL(d"Wf'ZK+Ft0iIDU\L`7>LOcj.+d1KL:FbG,tic % ?Dh\=4ojXr(PndSXWtL<KP9,)moja96o"4Ah1[)SK">@EFKW5ZCt/YSeSo/r#=F1P3IeI(5F % CS0j1h$/kkXcL>A/Ouu+.KJcedRiB/R_O-`&mV$0mKWkLb$ojPuArBb\'S@Q/f!l/;rr@r=H % eT)e5T5RgYS<0g&;cU"js_/U+^?YNc$oe"8gZNEo&q8@ZiKoT*+X^l?65@a#3qP]m*u(JA[% % :d)em[h;QGtJo3VJk`pdN9P,-"[V:c0KWZ8)a8On8gYnQDmUINa"T$kUr.%Ra_b%EkjH7LHo % i!(;C."VSpDL:ULn,jfs#2h,HJcLm!$=0>%m\?.)jt!N013V!NBmNl2+n#\!m?MEUZpL>dCO % >9Dh/5=1!9",`6lU(R`1OdPdod\t(6a>8Z'#N#!.Z@?O>4#06:_*?1d$$n@g`,m5ceKO5VSb % 4jQX"hI&bj,E!A!S["\X>$jm'q:.P]rOuOS@"Hlje+Mp*n4DHObh7Qk8!0cZ`c_DOt2VM7S/ % JDW,'AX_I*`a'_&Agh@UC*07+:/@;TWkiVP*!B2c2fT<K%F,9P3:hF95QN6N'Yb]geO%q0e# % (QidrF287D\ES)Y&taef:CS;K<r?>t8ooYr4$,.;s2j<s)X*f8pWWO@bTX/V!5mi<"C?J,cb % LE(@Pam2^)S![8(:S5&8:jk$ZJ-#0l5fKiT[33Fs@CQmk"Q-[#^(>aUOTZ\5fK)KL$DbFcq[ % 1Ib0*"9<Z%)e?QrsF/&-P`.J.qo9"98E%x % EndExpression N 945 3b1 E f7 S 191e7 R a64b1054 gsave /__NXbasematrix matrix currentmatrix def grestore %%EndProlog %%BeginSetup /EqBFont 18 dict dup begin /FontType 3 def /PaintType 0 def /FontMatrix [0.001 0 0 0.001 0 0] def /FontBBox [15 -2950 1360 1045] def /Encoding [/.notdef /rad1 /rad2 /rad3 /rad4 /rad5 /radext /int /pathint /sum /prod /cdot /vdot /ddot /hbar /bigcap /bigcup /parallel] def /Metrics 18 dict dup begin /.notdef 0 def /rad1 1000 def /rad2 1000 def /rad3 1000 def /rad4 1000 def /rad5 1048 def /radext 1048 def /int 500 def /pathint 500 def /sum 700 def /prod 700 def /cdot 1000 def /vdot 550 def /ddot 1000 def /hbar 500 def /bigcap 700 def /bigcup 700 def /parallel 500 def end def /BBox 18 dict dup begin /.notdef [0 0 0 0] def /rad1 [110 -1150 1020 0] def /rad2 [110 -1750 1020 0] def /rad3 [110 -2350 1020 0] def /rad4 [110 -2950 1020 0] def /rad5 [128 -2890 740 0] def /radext [700 -600 740 0] def /int [20 0 425 1045] def /pathint [20 0 425 1045] def /sum [15 5 640 715] def /prod [15 0 670 715] def /cdot [110 215 890 325] def /vdot [165 -5 280 770] def /ddot [110 -5 890 770] def /hbar [20 -10 480 685] def /bigcap [0 0 680 720] def /bigcup [0 0 680 720] def /parallel [65 -175 400 675] def end def /CharacterDefs 18 dict dup begin /.notdef { } def /rad1 { 980 0 moveto 464 -1048 lineto 248 -560 lineto 110 -656 lineto 125 -675 lineto 200 -624 lineto 420 -1150 lineto 460 -1150 lineto 1020 0 lineto 980 0 lineto closepath } def /rad2 { 980 0 moveto 460 -1600 lineto 245 -864 lineto 110 -1000 lineto 128 -1023 lineto 200 -950 lineto 424 -1750 lineto 456 -1750 lineto 1020 0 lineto 980 0 lineto closepath } def /rad3 { 980 0 moveto 464 -2144 lineto 248 -1160 lineto 110 -1368 lineto 130 -1390 lineto 192 -1288 lineto 424 -2350 lineto 464 -2350 lineto 1020 0 lineto 980 0 lineto closepath } def /rad4 { 980 0 moveto 880 -488 lineto 672 -1600 lineto 464 -2696 lineto 248 -1464 lineto 110 -1728 lineto 130 -1750 lineto 192 -1616 lineto 424 -2950 lineto 464 -2950 lineto 768 -1312 lineto 1020 0 lineto 980 0 lineto closepath } def /rad5 { 700 0 moveto 700 -2628 lineto 248 -1284 lineto 128 -1564 lineto 150 -1580 lineto 215 -1425 lineto 688 -2890 lineto 740 -2890 lineto 740 0 lineto 700 0 lineto closepath } def /radext { 700 0 moveto 700 -600 lineto 740 -600 lineto 740 0 lineto 700 0 lineto closepath } def /int { 245 390 moveto 225 235 195 145 180 100 curveto 165 60 150 50 140 40 curveto 135 30 85 0 50 20 curveto 35 25 25 45 25 55 curveto 20 80 30 100 55 110 curveto 80 125 93 110 110 90 curveto 135 60 155 135 160 195 curveto 165 270 170 340 190 510 curveto 205 650 lineto 220 810 230 890 265 960 curveto 275 980 295 1015 320 1030 curveto 350 1045 380 1030 395 1020 curveto 410 1010 425 980 400 955 curveto 385 940 347 925 330 950 curveto 310 970 290 955 290 940 curveto 280 900 285 745 270 610 curveto 245 390 lineto closepath } def /pathint { 245 390 moveto 225 235 195 145 180 100 curveto 165 60 150 50 140 40 curveto 135 30 85 0 50 20 curveto 35 25 25 45 25 55 curveto 20 80 30 100 55 110 curveto 80 125 93 110 110 90 curveto 135 60 155 135 160 195 curveto 165 270 170 340 190 510 curveto 205 650 lineto 220 810 230 890 265 960 curveto 275 980 295 1015 320 1030 curveto 350 1045 380 1030 395 1020 curveto 410 1010 425 980 400 955 curveto 385 940 347 925 330 950 curveto 310 970 290 955 290 940 curveto 280 900 285 745 270 610 curveto 245 390 lineto fill 20 setlinewidth 220 400 moveto 290 400 340 455 340 500 curveto 340 585 290 640 220 640 curveto 155 640 100 585 100 500 curveto 100 455 155 400 220 400 curveto stroke } def /sum { 330 335 moveto 20 715 lineto 575 715 lineto 600 545 lineto 575 535 lineto 550 625 525 685 395 685 curveto 160 685 lineto 411 370 lineto 160 90 lineto 470 90 lineto 565 90 590 140 610 210 curveto 640 200 lineto 585 5 lineto 15 5 lineto 330 335 lineto closepath } def /prod { 590 90 moveto 590 30 620 20 670 20 curveto 670 0 lineto 410 0 lineto 410 20 lineto 460 20 495 30 495 90 curveto 495 670 lineto 190 670 lineto 190 90 lineto 190 30 225 20 275 20 curveto 275 0 lineto 15 0 lineto 15 20 lineto 65 20 100 30 100 90 curveto 100 615 lineto 100 690 40 700 20 700 curveto 20 715 lineto 670 715 lineto 670 700 lineto 645 700 590 690 590 615 curveto 590 90 lineto closepath } def /cdot { 110 270 moveto 110 300 135 325 165 325 curveto 195 325 220 300 220 270 curveto 220 235 185 215 165 215 curveto 145 215 110 235 110 270 curveto fill 445 270 moveto 445 300 470 325 500 325 curveto 530 325 555 300 555 270 curveto 555 235 520 215 500 215 curveto 480 215 445 235 445 270 curveto fill 775 270 moveto 775 300 800 325 830 325 curveto 860 325 890 300 890 270 curveto 890 235 850 215 830 215 curveto 810 215 775 235 775 270 curveto fill } def /vdot { 220 770 moveto 250 770 280 750 280 720 curveto 280 680 250 660 220 660 curveto 185 660 165 695 165 720 curveto 165 740 185 770 220 770 curveto fill 220 440 moveto 250 440 280 415 280 385 curveto 280 355 250 330 220 330 curveto 185 330 165 365 165 385 curveto 165 405 185 440 220 440 curveto fill 220 105 moveto 250 105 280 80 280 50 curveto 280 20 250 -5 220 -5 curveto 185 -5 165 30 165 50 curveto 165 70 185 105 220 105 curveto fill } def /ddot { 165 720 moveto 205 720 220 700 220 665 curveto 220 630 195 610 165 610 curveto 130 620 110 645 110 665 curveto 110 690 130 720 165 720 curveto fill 500 440 moveto 530 440 555 415 555 385 curveto 555 355 530 330 500 330 curveto 465 330 445 365 445 385 curveto 445 405 465 440 500 440 curveto fill 830 155 moveto 860 155 890 130 890 100 curveto 890 70 860 45 830 45 curveto 795 45 775 80 775 100 curveto 775 120 795 155 830 155 curveto fill } def /hbar { 480 105 moveto 440 55 400 -10 335 -10 curveto 300 -10 290 10 290 35 curveto 290 65 370 350 370 365 curveto 370 380 355 390 340 390 curveto 290 390 205 275 185 250 curveto 140 180 125 125 95 0 curveto 20 0 lineto 150 500 lineto 160 545 180 600 180 615 curveto 180 625 170 640 135 640 curveto 110 640 lineto 100 655 lineto 165 660 215 670 270 685 curveto 275 680 lineto 155 240 lineto 155 235 lineto 190 290 290 440 380 440 curveto 400 440 450 435 450 370 curveto 450 335 365 80 365 55 curveto 365 45 375 40 385 40 curveto 405 40 425 70 465 115 curveto 480 105 lineto fill 55 545 moveto 55 520 lineto 310 560 lineto 310 585 lineto 55 545 lineto fill } def /bigcap { 615 400 moveto 615 575 500 655 340 655 curveto 180 655 70 575 70 400 curveto 70 0 lineto 5 0 lineto 5 400 lineto 5 610 140 720 340 720 curveto 545 720 680 610 680 400 curveto 680 0 lineto 615 0 lineto 615 400 lineto closepath } def /bigcup { 65 320 moveto 65 145 180 65 340 65 curveto 500 65 615 145 615 320 curveto 615 720 lineto 680 720 lineto 680 320 lineto 680 110 540 0 340 0 curveto 140 0 0 110 0 320 curveto 0 720 lineto 65 720 lineto 65 320 lineto closepath } def /parallel { 135 675 moveto 135 -175 lineto 65 -175 lineto 65 675 lineto 135 675 lineto fill 335 675 moveto 335 -175 lineto 265 -175 lineto 265 675 lineto 335 675 lineto fill 400 675 moveto } def end def /BuildChar { 0 begin /char exch def /fontdict exch def /charname fontdict /Encoding get char get def fontdict begin Metrics charname get 0 BBox charname get aload pop setcachedevice CharacterDefs charname get exec eofill end end } def /BuildChar load 0 3 dict put end definefont pop %%EndSetup 0 0 187.817001 35.640968 rectclip 0 0 187.817001 35.640968 rectclip 0 setgray 1 setalpha /EqBFont 24 selectfont 87.995163 10.244992 moveto <09> [0 0] xyshow 0 setgray 1 setalpha 1 1 scale /Symbol findfont 12 scalefont 47 exch defineuserobject 47 execuserobject setfont 113.470497 23.792992 moveto (\() [0 0] xyshow 1 1 scale 1 1 scale 47 execuserobject setfont 134.365158 23.786991 moveto (\)) [0 0] xyshow 1 1 scale 0 setgray 1 setalpha 0.48 setlinewidth 105.527664 18.884993 moveto 156.140991 18.884993 lineto stroke 0 setgray 1 setalpha /Times-Roman findfont 12 scalefont 51 exch defineuserobject 51 execuserobject setfont 158.141006 15.884993 moveto (sin1sinh2sinh) [4.667999 0 3.335999 0 -38.619843 7.799999 -70.678497 -7.799999 4.667999 0 3.336002 0 6 0 -28.592003 0 -41.258667 0 4.668 0 3.335999 0 6 0 0 0 0 0] xyshow /Times-BoldItalic findfont 12 scalefont 48 exch defineuserobject 48 execuserobject setfont 180.817001 15.884993 moveto (xnnLnLx) [-6.671997 0 -43.164001 -9.118704 -24.953339 0 0.206833 16.918703 -27.383827 -7.799999 -55.846672 0 0 0 0 0] xyshow 47 execuserobject setfont 143.352997 6.766289 moveto (p+-p=) [-21.626663 0 -3.455841 16.918703 -70.011826 -7.799999 -15.921333 0 0 0 0 0] xyshow /Symbol findfont 9 scalefont 50 exch defineuserobject 50 execuserobject setfont 144.343155 28.400019 moveto (+=\245) [-50.70649 -26.119019 -0.990005 27.523968 0 0 0 0] xyshow /Times-BoldItalic findfont 9 scalefont 52 exch defineuserobject 52 execuserobject setfont 138.88916 28.400019 moveto (nn) [-50.706497 -26.119019 0 0 0 0] xyshow /Times-Roman findfont 9 scalefont 49 exch defineuserobject 49 execuserobject setfont 149.940994 11.120993 moveto (22211) [-12.287994 0 -24.293335 0 36.374496 17.279026 -50.706497 -26.119019 0 0 0 0] xyshow %%Trailer %%DocumentFonts: Times-BoldItalic %%+ Times-Roman %%+ Symbol %End of Graphics MathPictureEnd end :[font = subsubsection; inactive; preserveAspect; startGroup; ] #5 :[font = text; inactive; preserveAspect; ] a x3 + b x2 + c x + d (-L < x < L) ;[s] 12:0,0;3,1;4,2;5,3;11,4;12,5;36,6;37,7;40,8;42,9;44,10;45,11;46,-1; 12:1,11,8,Times,0,12,0,0,0;1,9,7,Times,32,10,0,0,0;1,9,7,Times,0,10,0,0,0;1,11,8,Times,0,12,0,0,0;1,9,7,Times,32,10,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Times,2,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Times,3,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Times,2,12,0,0,0;1,11,8,Times,0,12,0,0,0; :[font = input; preserveAspect; ] f5[x_]:=a x^3 + b x^2 + c x + d :[font = input; preserveAspect; startGroup; ] Simplify[A0[f5]] :[font = output; output; inactive; preserveAspect; endGroup; ] d + (b*Pi^2)/3 ;[o] 2 b Pi d + ----- 3 :[font = input; preserveAspect; startGroup; ] Simplify[A[n,f5] /. TrigCoefRules[n]] :[font = output; output; inactive; preserveAspect; endGroup; ] (4*(-1)^n*b)/n^2 ;[o] n 4 (-1) b --------- 2 n :[font = input; preserveAspect; startGroup; ] Simplify[B[n,f5] /. TrigCoefRules[n]] :[font = output; output; inactive; dontNoPageBreakBelow; preserveAspect; endGroup; ] (2*(-1)^n*(6*a - c*n^2 - a*n^2*Pi^2))/n^3 ;[o] n 2 2 2 2 (-1) (6 a - c n - a n Pi ) ------------------------------- 3 n :[font = text; inactive; preserveAspect; ] Hence :[font = postscript; PostScript; formatAsPostScript; output; inactive; pageBreakBelow; dontNoPageBreakBelow; preserveAspect; pictureLeft = 0; pictureWidth = 288; pictureHeight = 73; endGroup; endGroup; endGroup; ] %! %%Creator: Mathematica %%AspectRatio: 0.253472 200 dict begin MathPictureStart /Courier findfont 10 scalefont setfont % Scaling calculations 0 1 0 1 [ [ 0 0 0 0 ] [ 288 73 0 0 ] ] MathScale % Start of Graphics %!PS-Adobe-2.0 EPSF-2.0 %%Title: %%Creator: EquationBuilder %%CreationDate: Mon Aug 3 07:50:57 1992 %%For: gray %%DocumentFonts: (atend) %%Pages: 0 1 %%BoundingBox: 0 0 288 73 %%Translation: 1 32.35347 286.218323 0 %%EndComments %%BeginDocument: /usr/lib/NextStep/printPackage.ps %!PS-Adobe-2.0 %%Title: Printing Package %%EndComments %Version: 2.0 %Copyright: 1988, NeXT, Inc. /__NXdef{1 index where{pop pop pop}{def}ifelse}bind def /__NXbdef{1 index where{pop pop pop}{bind def}ifelse}bind def /UserObjects 10 array __NXdef /defineuserobject{ exch dup 1 add dup UserObjects length gt{ array dup 0 UserObjects putinterval /UserObjects exch def }{pop}ifelse UserObjects exch 3 -1 roll put }__NXbdef /undefineuserobject{UserObjects exch null put}__NXbdef /execuserobject{UserObjects exch get exec}__NXbdef /__NXRectPath{4 2 roll moveto 1 index 0 rlineto 0 exch rlineto neg 0 rlineto closepath}__NXbdef /__NXProcessRectArgs{ 1 index type /arraytype eq{ exch 0 4 2 index length 1 sub{ dup 3 add 1 exch{1 index exch get exch}for 5 1 roll 5 index exec }for pop pop }{exec}ifelse }__NXbdef /rectfill{gsave newpath {__NXRectPath fill} __NXProcessRectArgs grestore}__NXbdef /rectclip{newpath {__NXRectPath} __NXProcessRectArgs clip newpath}__NXbdef /rectstroke{ gsave newpath dup type /arraytype eq{dup length 6 eq}{false}ifelse{ {gsave __NXRectPath null concat stroke grestore} dup length array cvx copy dup 2 4 -1 roll put __NXProcessRectArgs }{{__NXRectPath stroke} __NXProcessRectArgs}ifelse grestore }__NXbdef /xyshow{ 0 1 3 index length 1 sub{ currentpoint 4 index 3 index 1 getinterval show 3 index 3 index 2 mul 1 add get add exch 3 index 3 index 2 mul get add exch moveto pop }for pop pop }__NXbdef /xshow{ 0 1 3 index length 1 sub{ currentpoint 4 index 3 index 1 getinterval show exch 3 index 3 index get add exch moveto pop }for pop pop }__NXbdef /yshow{ 0 1 3 index length 1 sub{ currentpoint 4 index 3 index 1 getinterval show 3 index 3 index get add moveto pop }for pop pop }__NXbdef /arct{arcto pop pop pop pop}__NXbdef /setbbox{pop pop pop pop}__NXbdef /ucache{}__NXbdef /ucachestatus{mark 0 0 0 0 0}__NXbdef /setucacheparams{cleartomark}__NXbdef /uappend{systemdict begin cvx exec end}__NXbdef /ueofill{gsave newpath uappend eofill grestore}__NXbdef /ufill{gsave newpath uappend fill grestore}__NXbdef /ustroke{ gsave newpath dup length 6 eq {exch uappend concat}{uappend}ifelse stroke grestore }__NXbdef /__NXustrokepathMatrix dup where {pop pop}{matrix def}ifelse /ustrokepath{ newpath dup length 6 eq{ exch uappend __NXustrokepathMatrix currentmatrix exch concat strokepath setmatrix }{uappend strokepath}ifelse } __NXbdef /upath{ [exch {/ucache cvx}if pathbbox /setbbox cvx {/moveto cvx}{/lineto cvx}{/curveto cvx}{/closepath cvx}pathforall]cvx } __NXbdef /setstrokeadjust{pop}__NXbdef /currentstrokeadjust{false}__NXbdef /selectfont{exch findfont exch dup type /arraytype eq {makefont}{scalefont}ifelse setfont}__NXbdef /_NXCombineArrays{ counttomark dup 2 add index dup length 3 -1 roll { 2 index length sub dup 4 1 roll 1 index exch 4 -1 roll putinterval exch }repeat pop pop pop }__NXbdef /setcmykcolor{ 1.0 exch sub dup dup 6 -1 roll sub dup 0 lt{pop 0}if 5 1 roll 4 -1 roll sub dup 0 lt{pop 0}if 3 1 roll exch sub dup 0 lt{pop 0}if setrgbcolor }__NXbdef /currentcmykcolor{currentrgbcolor 3{1.0 exch sub 3 1 roll}repeat 0}__NXbdef /flushgraphics{}def /setwindowtype{pop pop}def /currentwindowtype{pop 0}def /setalpha{pop}def /currentalpha{1.0}def /hidecursor{}def /obscurecursor{}def /revealcursor{}def /setcursor{4 {pop}repeat}bind def /showcursor{}def /NextStepEncoding where not{ /NextStepEncoding StandardEncoding 256 array copy def 0 [129/Agrave/Aacute/Acircumflex/Atilde/Adieresis/Aring/Ccedilla/Egrave /Eacute/Ecircumflex/Edieresis/Igrave/Iacute/Icircumflex/Idieresis /Eth/Ntilde/Ograve/Oacute/Ocircumflex/Otilde/Odieresis/Ugrave/Uacute /Ucircumflex/Udieresis/Yacute/Thorn/mu/multiply/divide/copyright 176/registered 181/brokenbar 190/logicalnot 192/onesuperior 201/twosuperior 204/threesuperior 209/plusminus/onequarter/onehalf/threequarters/agrave /aacute/acircumflex/atilde/adieresis/aring/ccedilla/egrave/eacute /ecircumflex/edieresis/igrave 226/iacute 228/icircumflex/idieresis/eth /ntilde 236/ograve/oacute/ocircumflex/otilde/odieresis 242/ugrave/uacute /ucircumflex 246/udieresis/yacute 252/thorn/ydieresis] {dup type /nametype eq {NextStepEncoding 2 index 2 index put pop 1 add}{exch pop}ifelse }forall pop /NextStepEncoding NextStepEncoding readonly def /_NXfstr 128 string dup 0 (_NX) putinterval def /findfont{ % Because we can never let NextStepEncoding get into % SharedFontDirectory, we cannot reencode a font to NextStepEncoding % if we are in shared mode. So if currentshared is true, % we call the normal findfont and return that /currentshared where {pop currentshared} {false} ifelse {//findfont exec} {dup _NXfstr 3 125 getinterval cvs length 3 add _NXfstr 0 3 -1 roll getinterval cvn exch FontDirectory 2 index known {pop FontDirectory exch get} {//findfont exec dup /Encoding get StandardEncoding eq { dup length dict exch {1 index /FID ne {2 index 3 1 roll put}{pop pop}ifelse}forall dup /Encoding NextStepEncoding put definefont }{exch pop} ifelse }ifelse }ifelse }bind def }{pop}ifelse /_NXProcArray 5 array __NXdef /_NXChannels 0 __NXdef /_NXTotalBytes 0 __NXdef /_NXDoImageOp{ 1 index{dup}{1}ifelse /_NXChannels exch store _NXChannels 2 add 2 roll _NXProcArray 0 _NXChannels getinterval astore pop 5 index 4 index mul 2 index{1 sub 8 idiv 1 add mul}{mul 1 sub 8 idiv 1 add}ifelse 4 index mul /_NXTotalBytes exch store pop exch pop gsave matrix invertmatrix concat 0.5 setgray 0 0 4 2 roll rectfill grestore { 0 1 _NXChannels 1 sub{ _NXProcArray exch get exec length _NXTotalBytes exch sub /_NXTotalBytes exch store}for _NXTotalBytes 0 le{exit}if }loop /_NXProcArray 5 array def }__NXbdef /colorimage{_NXDoImageOp}__NXbdef /alphaimage{1 add _NXDoImageOp}def %%EndDocument % EquationBuilder(TM) Expression % +.\Jr(=:/*V[StRFQAYmdjEEA;1nuN+[co.OZ7L#cq3iK%3:</XJ@:d#6;^G;Td47dud<A3@ % 5%MGToUOR5&d`'TuEDU)B-tbR".R+A`%L8MC/FJ7MIf8P=@",X_XRS2ihA"VLe`-6WLgVDQ[ % K-Oh*sQ3\5X&jqDA/d.K[>m^Q5J[c<T!&,[T,,#-*fVCOf-4$/$D*CiTN0oVO8WY+m-,:c8_ % >n9PU_,k7M[U_6r$#hFL<[t(VWhoSS>gES:`%W-ScRQKJgl!!LagS.EQ')7_T:uD3jL-bT_9 % !2+ET'2!1-@`:dOX8WTmn#q.9%`%[f.kK]aMm)Y.l7Y6T0H*ElG';on^%EXj,*F]?s;imI08 % 148`9))<96cUE%bIK@Zch+:&B!QRIDNUgD'!]jD5JV;sJ@=T(*=PkR88>:teY"0a0J@5abSp % 1hc8)\uk>\IK!#d1t8MZCmi+,1\',_gK@i0`@NK_-Vu0JDiRUP>l*#T8dcG$*[c#g/gr/<N6 % oRZ7NnGi4='8bjMcOpE:h0p3r;+tc1d#c!_C&d!uTP+9"XQolQP#+]qKWT9C?'#LMk8TpN"S % 9u)5jd-RAYok1uUr8!W*#3X!@M0:ljdV#pe,q+M=Hr$HJ&Y'<G(\AF.%D4$\Xa=QKRm'ujL$ % 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iG!=;dp<;8*qc5[-*j0Hb7%E6R-\rk/Je.]cV1%^`/n*$[&DZ>V30'uZM2lX`Zcf%2d<VZjX % qLg&Qt+9FG`FeeK:]1JWUcnt5YqZ%%cG_E(0Wag:l*Fg/,Z7>+I?$r0hJ(:GTf:J\kiA8<<. % EObn0._LH3!J@Z;gt<6!"G9RpT$"k^^hqS7Ua(a)$NTL'RNTTU']!AS#%6[g.#<jL#E"[XUH % )3R7?0F28!q,'C]b8OG?O9&>Z"C)L+-WhKrJcN+_![du`$%Ecnx % EndExpression N 1480 5c8 E 2b S 28f77 R 66edadb2 gsave /__NXbasematrix matrix currentmatrix def grestore %%EndProlog %%BeginSetup /EqBFont 18 dict dup begin /FontType 3 def /PaintType 0 def /FontMatrix [0.001 0 0 0.001 0 0] def /FontBBox [15 -2950 1360 1045] def /Encoding [/.notdef /rad1 /rad2 /rad3 /rad4 /rad5 /radext /int /pathint /sum /prod /cdot /vdot /ddot /hbar /bigcap /bigcup /parallel] def /Metrics 18 dict dup begin /.notdef 0 def /rad1 1000 def /rad2 1000 def /rad3 1000 def /rad4 1000 def /rad5 1048 def /radext 1048 def /int 500 def /pathint 500 def /sum 700 def /prod 700 def /cdot 1000 def /vdot 550 def /ddot 1000 def /hbar 500 def /bigcap 700 def /bigcup 700 def /parallel 500 def end def /BBox 18 dict dup begin /.notdef [0 0 0 0] def /rad1 [110 -1150 1020 0] def /rad2 [110 -1750 1020 0] def /rad3 [110 -2350 1020 0] def /rad4 [110 -2950 1020 0] def /rad5 [128 -2890 740 0] def /radext [700 -600 740 0] def /int [20 0 425 1045] def /pathint [20 0 425 1045] def /sum [15 5 640 715] def /prod [15 0 670 715] def /cdot [110 215 890 325] def /vdot [165 -5 280 770] def /ddot [110 -5 890 770] def /hbar [20 -10 480 685] def /bigcap [0 0 680 720] def /bigcup [0 0 680 720] def /parallel [65 -175 400 675] def end def /CharacterDefs 18 dict dup begin /.notdef { } def /rad1 { 980 0 moveto 464 -1048 lineto 248 -560 lineto 110 -656 lineto 125 -675 lineto 200 -624 lineto 420 -1150 lineto 460 -1150 lineto 1020 0 lineto 980 0 lineto closepath } def /rad2 { 980 0 moveto 460 -1600 lineto 245 -864 lineto 110 -1000 lineto 128 -1023 lineto 200 -950 lineto 424 -1750 lineto 456 -1750 lineto 1020 0 lineto 980 0 lineto closepath } def /rad3 { 980 0 moveto 464 -2144 lineto 248 -1160 lineto 110 -1368 lineto 130 -1390 lineto 192 -1288 lineto 424 -2350 lineto 464 -2350 lineto 1020 0 lineto 980 0 lineto closepath } def /rad4 { 980 0 moveto 880 -488 lineto 672 -1600 lineto 464 -2696 lineto 248 -1464 lineto 110 -1728 lineto 130 -1750 lineto 192 -1616 lineto 424 -2950 lineto 464 -2950 lineto 768 -1312 lineto 1020 0 lineto 980 0 lineto closepath } def /rad5 { 700 0 moveto 700 -2628 lineto 248 -1284 lineto 128 -1564 lineto 150 -1580 lineto 215 -1425 lineto 688 -2890 lineto 740 -2890 lineto 740 0 lineto 700 0 lineto closepath } def /radext { 700 0 moveto 700 -600 lineto 740 -600 lineto 740 0 lineto 700 0 lineto closepath } def /int { 245 390 moveto 225 235 195 145 180 100 curveto 165 60 150 50 140 40 curveto 135 30 85 0 50 20 curveto 35 25 25 45 25 55 curveto 20 80 30 100 55 110 curveto 80 125 93 110 110 90 curveto 135 60 155 135 160 195 curveto 165 270 170 340 190 510 curveto 205 650 lineto 220 810 230 890 265 960 curveto 275 980 295 1015 320 1030 curveto 350 1045 380 1030 395 1020 curveto 410 1010 425 980 400 955 curveto 385 940 347 925 330 950 curveto 310 970 290 955 290 940 curveto 280 900 285 745 270 610 curveto 245 390 lineto closepath } def /pathint { 245 390 moveto 225 235 195 145 180 100 curveto 165 60 150 50 140 40 curveto 135 30 85 0 50 20 curveto 35 25 25 45 25 55 curveto 20 80 30 100 55 110 curveto 80 125 93 110 110 90 curveto 135 60 155 135 160 195 curveto 165 270 170 340 190 510 curveto 205 650 lineto 220 810 230 890 265 960 curveto 275 980 295 1015 320 1030 curveto 350 1045 380 1030 395 1020 curveto 410 1010 425 980 400 955 curveto 385 940 347 925 330 950 curveto 310 970 290 955 290 940 curveto 280 900 285 745 270 610 curveto 245 390 lineto fill 20 setlinewidth 220 400 moveto 290 400 340 455 340 500 curveto 340 585 290 640 220 640 curveto 155 640 100 585 100 500 curveto 100 455 155 400 220 400 curveto stroke } def /sum { 330 335 moveto 20 715 lineto 575 715 lineto 600 545 lineto 575 535 lineto 550 625 525 685 395 685 curveto 160 685 lineto 411 370 lineto 160 90 lineto 470 90 lineto 565 90 590 140 610 210 curveto 640 200 lineto 585 5 lineto 15 5 lineto 330 335 lineto closepath } def /prod { 590 90 moveto 590 30 620 20 670 20 curveto 670 0 lineto 410 0 lineto 410 20 lineto 460 20 495 30 495 90 curveto 495 670 lineto 190 670 lineto 190 90 lineto 190 30 225 20 275 20 curveto 275 0 lineto 15 0 lineto 15 20 lineto 65 20 100 30 100 90 curveto 100 615 lineto 100 690 40 700 20 700 curveto 20 715 lineto 670 715 lineto 670 700 lineto 645 700 590 690 590 615 curveto 590 90 lineto closepath } def /cdot { 110 270 moveto 110 300 135 325 165 325 curveto 195 325 220 300 220 270 curveto 220 235 185 215 165 215 curveto 145 215 110 235 110 270 curveto fill 445 270 moveto 445 300 470 325 500 325 curveto 530 325 555 300 555 270 curveto 555 235 520 215 500 215 curveto 480 215 445 235 445 270 curveto fill 775 270 moveto 775 300 800 325 830 325 curveto 860 325 890 300 890 270 curveto 890 235 850 215 830 215 curveto 810 215 775 235 775 270 curveto fill } def /vdot { 220 770 moveto 250 770 280 750 280 720 curveto 280 680 250 660 220 660 curveto 185 660 165 695 165 720 curveto 165 740 185 770 220 770 curveto fill 220 440 moveto 250 440 280 415 280 385 curveto 280 355 250 330 220 330 curveto 185 330 165 365 165 385 curveto 165 405 185 440 220 440 curveto fill 220 105 moveto 250 105 280 80 280 50 curveto 280 20 250 -5 220 -5 curveto 185 -5 165 30 165 50 curveto 165 70 185 105 220 105 curveto fill } def /ddot { 165 720 moveto 205 720 220 700 220 665 curveto 220 630 195 610 165 610 curveto 130 620 110 645 110 665 curveto 110 690 130 720 165 720 curveto fill 500 440 moveto 530 440 555 415 555 385 curveto 555 355 530 330 500 330 curveto 465 330 445 365 445 385 curveto 445 405 465 440 500 440 curveto fill 830 155 moveto 860 155 890 130 890 100 curveto 890 70 860 45 830 45 curveto 795 45 775 80 775 100 curveto 775 120 795 155 830 155 curveto fill } def /hbar { 480 105 moveto 440 55 400 -10 335 -10 curveto 300 -10 290 10 290 35 curveto 290 65 370 350 370 365 curveto 370 380 355 390 340 390 curveto 290 390 205 275 185 250 curveto 140 180 125 125 95 0 curveto 20 0 lineto 150 500 lineto 160 545 180 600 180 615 curveto 180 625 170 640 135 640 curveto 110 640 lineto 100 655 lineto 165 660 215 670 270 685 curveto 275 680 lineto 155 240 lineto 155 235 lineto 190 290 290 440 380 440 curveto 400 440 450 435 450 370 curveto 450 335 365 80 365 55 curveto 365 45 375 40 385 40 curveto 405 40 425 70 465 115 curveto 480 105 lineto fill 55 545 moveto 55 520 lineto 310 560 lineto 310 585 lineto 55 545 lineto fill } def /bigcap { 615 400 moveto 615 575 500 655 340 655 curveto 180 655 70 575 70 400 curveto 70 0 lineto 5 0 lineto 5 400 lineto 5 610 140 720 340 720 curveto 545 720 680 610 680 400 curveto 680 0 lineto 615 0 lineto 615 400 lineto closepath } def /bigcup { 65 320 moveto 65 145 180 65 340 65 curveto 500 65 615 145 615 320 curveto 615 720 lineto 680 720 lineto 680 320 lineto 680 110 540 0 340 0 curveto 140 0 0 110 0 320 curveto 0 720 lineto 65 720 lineto 65 320 lineto closepath } def /parallel { 135 675 moveto 135 -175 lineto 65 -175 lineto 65 675 lineto 135 675 lineto fill 335 675 moveto 335 -175 lineto 265 -175 lineto 265 675 lineto 335 675 lineto fill 400 675 moveto } def end def /BuildChar { 0 begin /char exch def /fontdict exch def /charname fontdict /Encoding get char get def fontdict begin Metrics charname get 0 BBox charname get aload pop setcachedevice CharacterDefs charname get exec eofill end end } def /BuildChar load 0 3 dict put end definefont pop %%EndSetup 0 0 287.218323 72.70694 rectclip 0 0 287.218323 72.70694 rectclip 0 setgray 1 setalpha 0.48 setlinewidth 120.667999 55.950966 moveto 139.955994 55.950966 lineto stroke 0 setgray 1 setalpha /EqBFont 24 selectfont 165.689835 47.310963 moveto <09> [0 0] xyshow 0 setgray 1 setalpha 1 1 scale /Symbol findfont 12 scalefont 47 exch defineuserobject 47 execuserobject setfont 184.286331 60.858963 moveto (\() [0 0] xyshow 1 1 scale 1 1 scale 47 execuserobject setfont 205.181 60.852962 moveto (\)) [0 0] xyshow 1 1 scale 0 setgray 1 setalpha 0.48 setlinewidth 183.222336 55.950962 moveto 216.408997 55.950962 lineto stroke 0 setgray 1 setalpha /EqBFont 24 selectfont 109.892502 10.244995 moveto <09> [0 0] xyshow 0 setgray 1 setalpha 1 1 scale 47 execuserobject setfont 128.488998 24.375267 moveto (\() [0 0] xyshow 1 1 scale 1 1 scale 47 execuserobject setfont 149.383667 24.369267 moveto (\)) [0 0] xyshow 1 1 scale 0 setgray 1 setalpha 1 1.292876 scale 47 execuserobject setfont 160.675659 18.198387 moveto (\() [0 0] xyshow 1 0.773469 scale 1 1.291382 scale 47 execuserobject setfont 250.518326 18.216799 moveto (\)) [0 0] xyshow 1 0.774364 scale 0 setgray 1 setalpha 0.48 setlinewidth 127.425003 18.884995 moveto 255.542343 18.884995 lineto stroke 0 setgray 1 setalpha /Times-BoldItalic findfont 12 scalefont 48 exch defineuserobject 48 execuserobject setfont 280.218323 15.884995 moveto (xnnnancaxnnbbddxcxbxa) [-6.671997 0 -88.248657 -9.118704 39.720657 17.500977 -6 0 -24.293335 0 -5.327988 0 -17.921341 0 71.601334 28.683695 -6.671997 3.814697e-06 -42.77533 -9.118706 -35.752335 9.118702 -36.709335 7.799999 -18.421333 -7.799999 -19.254662 0 -17.921341 0 -5.327995 0 -23.621334 0 -6 0 -23.621334 0 -6 0 0 0 0 0] xyshow /Times-Roman findfont 9 scalefont 49 exch defineuserobject 49 execuserobject setfont 191.969666 11.120996 moveto (3222121223) [52.008652 17.500977 -12.287994 0 -30.293335 0 -80.471985 -26.340969 79.376663 45.90596 -23.57933 -8.839989 -42.966339 25.758694 -91.134659 -7.799999 -29.621334 0 0 0 0 0] xyshow /Symbol findfont 9 scalefont 50 exch defineuserobject 50 execuserobject setfont 115.534004 2.281003 moveto (=\245=\245) [-0.990005 27.523968 56.787331 9.542004 -0.98999 27.523968 0 0 0 0] xyshow 47 execuserobject setfont 237.39032 24.267267 moveto (p---+-+p+=+++) [-27.626648 0 -29.621338 0 -46.853333 0 -40.463669 -8.382272 96.261002 44.865967 -46.463669 -7.799995 -15.454666 7.799999 -15.754662 -7.799999 -18.588005 0 -18.587997 0 -23.249336 0 -29.621332 0 0 0 0 0] xyshow /Times-Roman findfont 12 scalefont 51 exch defineuserobject 51 execuserobject setfont 257.542358 15.884995 moveto (sin612cos143) [4.667999 0 3.335999 0 -100.070694 8.382272 -22.931992 0 -40.463669 -8.382272 116.328995 37.065968 5.328003 0 6 0 -31.395996 7.799999 -46.463669 -7.799995 -24.565338 -8.231415 0 0 0 0] xyshow /Times-BoldItalic findfont 9 scalefont 52 exch defineuserobject 52 execuserobject setfont 153.907669 28.982294 moveto (nnnn) [-43.827667 -26.701292 99.625 63.184994 -43.827667 -26.119022 0 0 0 0] xyshow %%Trailer %%DocumentFonts: Times-BoldItalic %%+ Times-Roman %%+ Symbol %End of Graphics MathPictureEnd end :[font = section; inactive; Cclosed; preserveAspect; startGroup; ] The Dirichlet Kernel :[font = text; inactive; preserveAspect; ] Recall the definition of the Dirichlet kernel: :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 0; pictureWidth = 226; pictureHeight = 44; ] %! %%Creator: Mathematica %%AspectRatio: 0.19469 200 dict begin MathPictureStart /Courier findfont 10 scalefont setfont % Scaling calculations 0 1 0 1 [ [ 0 0 0 0 ] [ 226 44 0 0 ] ] MathScale % Start of Graphics %!PS-Adobe-2.0 EPSF-2.0 %%Title: %%Creator: EquationBuilder %%CreationDate: Mon Aug 3 07:57:48 1992 %%For: gray %%DocumentFonts: (atend) %%Pages: 0 1 %%BoundingBox: 0 0 226 44 %%Translation: 1 17.818001 224.326324 0 %%EndComments %%BeginDocument: /usr/lib/NextStep/printPackage.ps %!PS-Adobe-2.0 %%Title: Printing Package %%EndComments %Version: 2.0 %Copyright: 1988, NeXT, Inc. /__NXdef{1 index where{pop pop pop}{def}ifelse}bind def /__NXbdef{1 index where{pop pop pop}{bind def}ifelse}bind def /UserObjects 10 array __NXdef /defineuserobject{ exch dup 1 add dup UserObjects length gt{ array dup 0 UserObjects putinterval /UserObjects exch def }{pop}ifelse UserObjects exch 3 -1 roll put }__NXbdef /undefineuserobject{UserObjects exch null put}__NXbdef /execuserobject{UserObjects exch get exec}__NXbdef /__NXRectPath{4 2 roll moveto 1 index 0 rlineto 0 exch rlineto neg 0 rlineto closepath}__NXbdef /__NXProcessRectArgs{ 1 index type /arraytype eq{ exch 0 4 2 index length 1 sub{ dup 3 add 1 exch{1 index exch get exch}for 5 1 roll 5 index exec }for pop pop }{exec}ifelse }__NXbdef /rectfill{gsave newpath {__NXRectPath fill} __NXProcessRectArgs grestore}__NXbdef /rectclip{newpath {__NXRectPath} __NXProcessRectArgs clip newpath}__NXbdef /rectstroke{ gsave newpath dup type /arraytype eq{dup length 6 eq}{false}ifelse{ {gsave __NXRectPath null concat stroke grestore} dup length array cvx copy dup 2 4 -1 roll put __NXProcessRectArgs }{{__NXRectPath stroke} __NXProcessRectArgs}ifelse grestore }__NXbdef /xyshow{ 0 1 3 index length 1 sub{ currentpoint 4 index 3 index 1 getinterval show 3 index 3 index 2 mul 1 add get add exch 3 index 3 index 2 mul get add exch moveto pop }for pop pop }__NXbdef /xshow{ 0 1 3 index length 1 sub{ currentpoint 4 index 3 index 1 getinterval show exch 3 index 3 index get add exch moveto pop }for pop pop }__NXbdef /yshow{ 0 1 3 index length 1 sub{ currentpoint 4 index 3 index 1 getinterval show 3 index 3 index get add moveto pop }for pop pop }__NXbdef /arct{arcto pop pop pop pop}__NXbdef /setbbox{pop pop pop pop}__NXbdef /ucache{}__NXbdef /ucachestatus{mark 0 0 0 0 0}__NXbdef /setucacheparams{cleartomark}__NXbdef /uappend{systemdict begin cvx exec end}__NXbdef /ueofill{gsave newpath uappend eofill grestore}__NXbdef /ufill{gsave newpath uappend fill grestore}__NXbdef /ustroke{ gsave newpath dup length 6 eq {exch uappend concat}{uappend}ifelse stroke grestore }__NXbdef /__NXustrokepathMatrix dup where {pop pop}{matrix def}ifelse /ustrokepath{ newpath dup length 6 eq{ exch uappend __NXustrokepathMatrix currentmatrix exch concat strokepath setmatrix }{uappend strokepath}ifelse } __NXbdef /upath{ [exch {/ucache cvx}if pathbbox /setbbox cvx {/moveto cvx}{/lineto cvx}{/curveto cvx}{/closepath cvx}pathforall]cvx } __NXbdef /setstrokeadjust{pop}__NXbdef /currentstrokeadjust{false}__NXbdef /selectfont{exch findfont exch dup type /arraytype eq {makefont}{scalefont}ifelse setfont}__NXbdef /_NXCombineArrays{ counttomark dup 2 add index dup length 3 -1 roll { 2 index length sub dup 4 1 roll 1 index exch 4 -1 roll putinterval exch }repeat pop pop pop }__NXbdef /setcmykcolor{ 1.0 exch sub dup dup 6 -1 roll sub dup 0 lt{pop 0}if 5 1 roll 4 -1 roll sub dup 0 lt{pop 0}if 3 1 roll exch sub dup 0 lt{pop 0}if setrgbcolor }__NXbdef /currentcmykcolor{currentrgbcolor 3{1.0 exch sub 3 1 roll}repeat 0}__NXbdef /flushgraphics{}def /setwindowtype{pop pop}def /currentwindowtype{pop 0}def /setalpha{pop}def /currentalpha{1.0}def /hidecursor{}def /obscurecursor{}def /revealcursor{}def /setcursor{4 {pop}repeat}bind def /showcursor{}def /NextStepEncoding where not{ /NextStepEncoding StandardEncoding 256 array copy def 0 [129/Agrave/Aacute/Acircumflex/Atilde/Adieresis/Aring/Ccedilla/Egrave 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xyshow /Times-BoldItalic findfont 12 scalefont 48 exch defineuserobject 48 execuserobject setfont 218.926331 5.680002 moveto (.uu) [-25.926666 0 -0.689896 25.446499 0 0 -9.254665 11.712001] xyshow %%Trailer %%DocumentFonts: Times-BoldItalic %%+ Times-Bold %%+ Times-Roman %%+ Symbol %End of Graphics MathPictureEnd end :[font = text; inactive; preserveAspect; ] Let us define the Dirichlet kernel in Mathematica: ;[s] 3:0,0;38,1;49,2;51,-1; 3:1,11,8,Times,0,12,0,0,0;1,10,8,Times,2,12,0,0,0;1,11,8,Times,0,12,0,0,0; :[font = input; preserveAspect; ] dirichletkernel[n_,u_]:= If[u!=0,Sin[(n+1/2)u]/(2Pi Sin[u/2]), (2n+1)/(2Pi)] :[font = text; inactive; preserveAspect; ] Here is the plot of dirichletkernel[5,u] ;[s] 2:0,0;21,1;41,-1; 2:1,11,8,Times,0,12,0,0,0;1,10,8,Courier,0,12,0,0,0; :[font = input; preserveAspect; startGroup; ] Plot[dirichletkernel[5,u]//Evaluate, {u,-2Pi,2Pi}, PlotRange->{-0.4,1.9}, PlotPoints->100]; :[font = postscript; PostScript; formatAsPostScript; output; inactive; 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Here is the 3-dimensional plot of the surface dirichletkernel[t,u]: ;[s] 5:0,0;76,1;77,2;139,3;159,4;161,-1; 5:1,11,8,Times,0,12,0,0,0;1,10,8,Times,2,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Courier,0,12,0,0,0;1,11,8,Times,0,12,0,0,0; :[font = input; preserveAspect; startGroup; ] Plot3D[dirichletkernel[t,u]//Evaluate, {t,1,5},{u,-2Pi+.1,2Pi-.1}, PlotRange->{Automatic, Automatic,{-0.4,1.9}}, PlotPoints->{40,60}]; :[font = postscript; PostScript; formatAsPostScript; output; inactive; pageBreakBelow; dontNoPageBreakBelow; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 231; endGroup; ] %! %%Creator: Mathematica %%AspectRatio: 0.82055 MathPictureStart /Courier findfont 10 scalefont setfont % Scaling calculations 0.024936 0.99742 -0.039634 0.99742 [ [(1)] 0.05113 0.25884 1 0.933946 Msboxa [(2)] 0.1926 0.20316 0.96648 1 Msboxa [(3)] 0.34275 0.1441 0.862234 1 Msboxa [(4)] 0.50241 0.08133 0.757988 1 Msboxa [(5)] 0.6725 0.0145 0.653742 1 Msboxa [(-5)] 0.7219 0.0678 -1 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moveto 0.90307 0.37377 lineto stroke grestore gsave 0.001 setlinewidth 0.91588 0.38679 moveto 0.91231 0.3879 lineto stroke grestore gsave 0.001 setlinewidth 0.69735 0.05251 moveto 0.69386 0.05385 lineto stroke grestore gsave 0.001 setlinewidth 0.6842 0.0324 moveto 0.68072 0.03376 lineto stroke grestore gsave 0.001 setlinewidth 0.93393 0.4144 moveto 0.93036 0.41549 lineto stroke grestore gsave 0.001 setlinewidth 0.94275 0.42788 moveto 0.93917 0.42896 lineto stroke grestore grestore gsave gsave 0.002 setlinewidth 0.06024 0.26735 moveto 0.02494 0.49015 lineto stroke grestore gsave 0.002 setlinewidth 0.05447 0.3038 moveto 0.06028 0.30157 lineto stroke grestore [(0)] 0.04283 0.30827 1 -0.38438 Mshowa gsave 0.002 setlinewidth 0.04704 0.35067 moveto 0.05288 0.34848 lineto stroke grestore [(0.5)] 0.03537 0.35505 1 -0.375113 Mshowa gsave 0.002 setlinewidth 0.03937 0.39904 moveto 0.04523 0.3969 lineto stroke grestore [(1)] 0.02766 0.40332 1 -0.365517 Mshowa gsave 0.002 setlinewidth 0.03146 0.44899 moveto 0.03733 0.4469 lineto stroke grestore [(1.5)] 0.01971 0.45317 1 -0.355575 Mshowa gsave 0.001 setlinewidth 0.053 0.31306 moveto 0.05649 0.31172 lineto stroke grestore gsave 0.001 setlinewidth 0.05152 0.32237 moveto 0.05502 0.32104 lineto stroke grestore gsave 0.001 setlinewidth 0.05004 0.33174 moveto 0.05354 0.33042 lineto stroke grestore gsave 0.001 setlinewidth 0.04854 0.34117 moveto 0.05204 0.33986 lineto stroke grestore gsave 0.001 setlinewidth 0.04553 0.36022 moveto 0.04903 0.35891 lineto stroke grestore gsave 0.001 setlinewidth 0.044 0.36983 moveto 0.04751 0.36853 lineto stroke grestore gsave 0.001 setlinewidth 0.04247 0.3795 moveto 0.04598 0.37821 lineto stroke grestore gsave 0.001 setlinewidth 0.04093 0.38924 moveto 0.04444 0.38795 lineto stroke grestore gsave 0.001 setlinewidth 0.03781 0.4089 moveto 0.04133 0.40762 lineto stroke grestore gsave 0.001 setlinewidth 0.03624 0.41883 moveto 0.03976 0.41755 lineto stroke grestore gsave 0.001 setlinewidth 0.03465 0.42882 moveto 0.03817 0.42755 lineto stroke grestore gsave 0.001 setlinewidth 0.03306 0.43887 moveto 0.03658 0.43761 lineto stroke grestore gsave 0.001 setlinewidth 0.05592 0.2946 moveto 0.05941 0.29326 lineto stroke grestore gsave 0.001 setlinewidth 0.05737 0.28546 moveto 0.06086 0.28411 lineto stroke grestore gsave 0.001 setlinewidth 0.05881 0.27638 moveto 0.0623 0.27502 lineto stroke grestore gsave 0.001 setlinewidth 0.02984 0.45918 moveto 0.03337 0.45793 lineto stroke grestore gsave 0.001 setlinewidth 0.02822 0.46943 moveto 0.03175 0.46819 lineto stroke grestore gsave 0.001 setlinewidth 0.02658 0.47976 moveto 0.03011 0.47852 lineto stroke grestore grestore % End of Graphics MathPictureEnd :[font = subsection; inactive; preserveAspect; startGroup; ] Exercise :[font = text; inactive; preserveAspect; ] The Fejer kernel is defined by :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 0; pictureWidth = 220; pictureHeight = 45; ] %! %%Creator: Mathematica %%AspectRatio: 0.204545 200 dict begin MathPictureStart /Courier findfont 10 scalefont setfont % Scaling calculations 0 1 0 1 [ [ 0 0 0 0 ] [ 220 45 0 0 ] ] MathScale % Start of Graphics %!PS-Adobe-2.0 EPSF-2.0 %%Title: %%Creator: EquationBuilder %%CreationDate: Mon Aug 3 08:01:13 1992 %%For: gray %%DocumentFonts: (atend) %%Pages: 0 1 %%BoundingBox: 0 0 220 45 %%Translation: 1 18.277851 218.129333 0 %%EndComments %%BeginDocument: /usr/lib/NextStep/printPackage.ps %!PS-Adobe-2.0 %%Title: Printing Package %%EndComments %Version: 2.0 %Copyright: 1988, NeXT, Inc. /__NXdef{1 index where{pop pop pop}{def}ifelse}bind def /__NXbdef{1 index where{pop pop pop}{bind def}ifelse}bind def /UserObjects 10 array __NXdef /defineuserobject{ exch dup 1 add dup UserObjects length gt{ array dup 0 UserObjects putinterval /UserObjects exch def }{pop}ifelse UserObjects exch 3 -1 roll put }__NXbdef /undefineuserobject{UserObjects exch null put}__NXbdef /execuserobject{UserObjects exch get exec}__NXbdef /__NXRectPath{4 2 roll moveto 1 index 0 rlineto 0 exch rlineto neg 0 rlineto closepath}__NXbdef /__NXProcessRectArgs{ 1 index type /arraytype eq{ exch 0 4 2 index length 1 sub{ dup 3 add 1 exch{1 index exch get exch}for 5 1 roll 5 index exec }for pop pop }{exec}ifelse }__NXbdef /rectfill{gsave newpath {__NXRectPath fill} __NXProcessRectArgs grestore}__NXbdef /rectclip{newpath {__NXRectPath} __NXProcessRectArgs clip newpath}__NXbdef /rectstroke{ gsave newpath dup type /arraytype eq{dup length 6 eq}{false}ifelse{ {gsave __NXRectPath null concat stroke grestore} dup length array cvx copy dup 2 4 -1 roll put __NXProcessRectArgs }{{__NXRectPath stroke} __NXProcessRectArgs}ifelse grestore }__NXbdef /xyshow{ 0 1 3 index length 1 sub{ currentpoint 4 index 3 index 1 getinterval show 3 index 3 index 2 mul 1 add get add exch 3 index 3 index 2 mul get add exch moveto pop }for pop pop }__NXbdef /xshow{ 0 1 3 index length 1 sub{ currentpoint 4 index 3 index 1 getinterval show exch 3 index 3 index get add exch moveto pop }for pop pop }__NXbdef /yshow{ 0 1 3 index length 1 sub{ currentpoint 4 index 3 index 1 getinterval show 3 index 3 index get add moveto pop }for pop pop }__NXbdef /arct{arcto pop pop pop pop}__NXbdef /setbbox{pop pop pop pop}__NXbdef /ucache{}__NXbdef /ucachestatus{mark 0 0 0 0 0}__NXbdef /setucacheparams{cleartomark}__NXbdef /uappend{systemdict begin cvx exec end}__NXbdef /ueofill{gsave newpath uappend eofill grestore}__NXbdef /ufill{gsave newpath uappend fill grestore}__NXbdef /ustroke{ gsave newpath dup length 6 eq {exch uappend concat}{uappend}ifelse stroke grestore }__NXbdef /__NXustrokepathMatrix dup where {pop pop}{matrix def}ifelse /ustrokepath{ newpath dup length 6 eq{ exch uappend __NXustrokepathMatrix currentmatrix exch concat strokepath setmatrix }{uappend strokepath}ifelse } __NXbdef /upath{ [exch {/ucache cvx}if pathbbox /setbbox cvx {/moveto cvx}{/lineto cvx}{/curveto cvx}{/closepath cvx}pathforall]cvx } __NXbdef /setstrokeadjust{pop}__NXbdef /currentstrokeadjust{false}__NXbdef /selectfont{exch findfont exch dup type /arraytype eq {makefont}{scalefont}ifelse setfont}__NXbdef /_NXCombineArrays{ counttomark dup 2 add index dup length 3 -1 roll { 2 index length sub dup 4 1 roll 1 index exch 4 -1 roll putinterval exch }repeat pop pop pop }__NXbdef /setcmykcolor{ 1.0 exch sub dup dup 6 -1 roll sub dup 0 lt{pop 0}if 5 1 roll 4 -1 roll sub dup 0 lt{pop 0}if 3 1 roll exch sub dup 0 lt{pop 0}if setrgbcolor }__NXbdef /currentcmykcolor{currentrgbcolor 3{1.0 exch sub 3 1 roll}repeat 0}__NXbdef /flushgraphics{}def /setwindowtype{pop pop}def /currentwindowtype{pop 0}def /setalpha{pop}def /currentalpha{1.0}def /hidecursor{}def /obscurecursor{}def /revealcursor{}def /setcursor{4 {pop}repeat}bind def /showcursor{}def /NextStepEncoding where not{ /NextStepEncoding StandardEncoding 256 array copy def 0 [129/Agrave/Aacute/Acircumflex/Atilde/Adieresis/Aring/Ccedilla/Egrave /Eacute/Ecircumflex/Edieresis/Igrave/Iacute/Icircumflex/Idieresis /Eth/Ntilde/Ograve/Oacute/Ocircumflex/Otilde/Odieresis/Ugrave/Uacute /Ucircumflex/Udieresis/Yacute/Thorn/mu/multiply/divide/copyright 176/registered 181/brokenbar 190/logicalnot 192/onesuperior 201/twosuperior 204/threesuperior 209/plusminus/onequarter/onehalf/threequarters/agrave /aacute/acircumflex/atilde/adieresis/aring/ccedilla/egrave/eacute /ecircumflex/edieresis/igrave 226/iacute 228/icircumflex/idieresis/eth /ntilde 236/ograve/oacute/ocircumflex/otilde/odieresis 242/ugrave/uacute /ucircumflex 246/udieresis/yacute 252/thorn/ydieresis] {dup type /nametype eq {NextStepEncoding 2 index 2 index put pop 1 add}{exch pop}ifelse }forall pop /NextStepEncoding NextStepEncoding readonly def /_NXfstr 128 string dup 0 (_NX) putinterval def /findfont{ % Because we can never let NextStepEncoding get into % SharedFontDirectory, we cannot reencode a font to NextStepEncoding % if we are in shared mode. So if currentshared is true, % we call the normal findfont and return that /currentshared where {pop currentshared} {false} ifelse {//findfont exec} {dup _NXfstr 3 125 getinterval cvs length 3 add _NXfstr 0 3 -1 roll getinterval cvn exch FontDirectory 2 index known {pop FontDirectory exch get} {//findfont exec dup /Encoding get StandardEncoding eq { dup length dict exch {1 index /FID ne {2 index 3 1 roll put}{pop pop}ifelse}forall dup /Encoding NextStepEncoding put definefont }{exch pop} ifelse }ifelse }ifelse }bind def }{pop}ifelse /_NXProcArray 5 array __NXdef /_NXChannels 0 __NXdef /_NXTotalBytes 0 __NXdef /_NXDoImageOp{ 1 index{dup}{1}ifelse /_NXChannels exch store _NXChannels 2 add 2 roll _NXProcArray 0 _NXChannels getinterval astore pop 5 index 4 index mul 2 index{1 sub 8 idiv 1 add mul}{mul 1 sub 8 idiv 1 add}ifelse 4 index mul /_NXTotalBytes exch store pop exch pop gsave matrix invertmatrix concat 0.5 setgray 0 0 4 2 roll rectfill grestore { 0 1 _NXChannels 1 sub{ _NXProcArray exch get exec length _NXTotalBytes exch sub /_NXTotalBytes exch store}for _NXTotalBytes 0 le{exit}if }loop /_NXProcArray 5 array def }__NXbdef /colorimage{_NXDoImageOp}__NXbdef /alphaimage{1 add _NXDoImageOp}def %%EndDocument % EquationBuilder(TM) Expression % +.\Jr(=:/*V[StRFQAYmdjEEA;1nuN+[co.OZ7L#cq3iK%3:</XJ@:d#6;^G;Td47dud<A3@ % 5%MGToUOR5&d`'TuEDU)B-tbR".R+:nNj/MI2*E+;]U)G>AH,X_XRS2ihA"VLe`-6WLgVDQ[ % K-Oh*sQ3\5X&jqDA/d.K[>m^Q5J[c<T!&,[Tin#Mqf\EPQp`n<3FPaBu@nDhna*n=K8V8]^g % &Ph,W+AU(8/K#JVW;G9-88oJL(Oh['1U:3`L@o]lIuWp?:"^H/d4]%P0!\R-jBZKMU3Y:KKd % MTD;,W$h7Wh'2cVtL)<Ap\cR)V<mrJKAI8@$f5RmGKgm`d(0:`Ys6Dr+EBC6eOGk^:\aHs9I % ]6!l:]bf'7qt6j.JE+II+FX9@(L;Wl/=9al%c/R@!,b?khg,gmVdOQ/D2=j?&[?m,71Bf*"r % h9`OTlgZ]gbM.o;*6'<GZqMLaJ0;_qY5OcCeZjP!$Y?H3%,MgaXZaOoQ2=\:Kj&TKHiIKFNN % rA:WaGYht]iK"#<C&_1;\Q=Fq;b!0M_.@ut/BF0P=!RqDsSH/QfaIE:l\YgPMUl[nZF6^;8P % 748*L.3/n6"PiWJldJc_Y)7q+jE7jJ1I:PQ&o-(dO<3s5JFG`A*#DR^h,=>THcl6(a2Bd3&M % 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pSXbfqR/dBle?Xk'oRR#L0L0ar5_TuQU]jXm9GlP]*=*@d%0mV7@*"E43eW&QA#PCh*ZEF_J % g(c#_c%,@-W2'lJcHB#6P)3X!)E=,&-)\1x % EndExpression N 1349 545 E 9c S 254b2 R f9770706 gsave /__NXbasematrix matrix currentmatrix def grestore %%EndProlog %%BeginSetup /EqBFont 18 dict dup begin /FontType 3 def /PaintType 0 def /FontMatrix [0.001 0 0 0.001 0 0] def /FontBBox [15 -2950 1360 1045] def /Encoding [/.notdef /rad1 /rad2 /rad3 /rad4 /rad5 /radext /int /pathint /sum /prod /cdot /vdot /ddot /hbar /bigcap /bigcup /parallel] def /Metrics 18 dict dup begin /.notdef 0 def /rad1 1000 def /rad2 1000 def /rad3 1000 def /rad4 1000 def /rad5 1048 def /radext 1048 def /int 500 def /pathint 500 def /sum 700 def /prod 700 def /cdot 1000 def /vdot 550 def /ddot 1000 def /hbar 500 def /bigcap 700 def /bigcup 700 def /parallel 500 def end def /BBox 18 dict dup begin /.notdef [0 0 0 0] def /rad1 [110 -1150 1020 0] def /rad2 [110 -1750 1020 0] def /rad3 [110 -2350 1020 0] def /rad4 [110 -2950 1020 0] def /rad5 [128 -2890 740 0] def /radext [700 -600 740 0] def /int [20 0 425 1045] def /pathint [20 0 425 1045] def /sum [15 5 640 715] def /prod [15 0 670 715] def /cdot [110 215 890 325] def /vdot [165 -5 280 770] def /ddot [110 -5 890 770] def /hbar [20 -10 480 685] def /bigcap [0 0 680 720] def /bigcup [0 0 680 720] def /parallel [65 -175 400 675] def end def /CharacterDefs 18 dict dup begin /.notdef { } def /rad1 { 980 0 moveto 464 -1048 lineto 248 -560 lineto 110 -656 lineto 125 -675 lineto 200 -624 lineto 420 -1150 lineto 460 -1150 lineto 1020 0 lineto 980 0 lineto closepath } def /rad2 { 980 0 moveto 460 -1600 lineto 245 -864 lineto 110 -1000 lineto 128 -1023 lineto 200 -950 lineto 424 -1750 lineto 456 -1750 lineto 1020 0 lineto 980 0 lineto closepath } def /rad3 { 980 0 moveto 464 -2144 lineto 248 -1160 lineto 110 -1368 lineto 130 -1390 lineto 192 -1288 lineto 424 -2350 lineto 464 -2350 lineto 1020 0 lineto 980 0 lineto closepath } def /rad4 { 980 0 moveto 880 -488 lineto 672 -1600 lineto 464 -2696 lineto 248 -1464 lineto 110 -1728 lineto 130 -1750 lineto 192 -1616 lineto 424 -2950 lineto 464 -2950 lineto 768 -1312 lineto 1020 0 lineto 980 0 lineto closepath } def /rad5 { 700 0 moveto 700 -2628 lineto 248 -1284 lineto 128 -1564 lineto 150 -1580 lineto 215 -1425 lineto 688 -2890 lineto 740 -2890 lineto 740 0 lineto 700 0 lineto closepath } def /radext { 700 0 moveto 700 -600 lineto 740 -600 lineto 740 0 lineto 700 0 lineto closepath } def /int { 245 390 moveto 225 235 195 145 180 100 curveto 165 60 150 50 140 40 curveto 135 30 85 0 50 20 curveto 35 25 25 45 25 55 curveto 20 80 30 100 55 110 curveto 80 125 93 110 110 90 curveto 135 60 155 135 160 195 curveto 165 270 170 340 190 510 curveto 205 650 lineto 220 810 230 890 265 960 curveto 275 980 295 1015 320 1030 curveto 350 1045 380 1030 395 1020 curveto 410 1010 425 980 400 955 curveto 385 940 347 925 330 950 curveto 310 970 290 955 290 940 curveto 280 900 285 745 270 610 curveto 245 390 lineto closepath } def /pathint { 245 390 moveto 225 235 195 145 180 100 curveto 165 60 150 50 140 40 curveto 135 30 85 0 50 20 curveto 35 25 25 45 25 55 curveto 20 80 30 100 55 110 curveto 80 125 93 110 110 90 curveto 135 60 155 135 160 195 curveto 165 270 170 340 190 510 curveto 205 650 lineto 220 810 230 890 265 960 curveto 275 980 295 1015 320 1030 curveto 350 1045 380 1030 395 1020 curveto 410 1010 425 980 400 955 curveto 385 940 347 925 330 950 curveto 310 970 290 955 290 940 curveto 280 900 285 745 270 610 curveto 245 390 lineto fill 20 setlinewidth 220 400 moveto 290 400 340 455 340 500 curveto 340 585 290 640 220 640 curveto 155 640 100 585 100 500 curveto 100 455 155 400 220 400 curveto stroke } def /sum { 330 335 moveto 20 715 lineto 575 715 lineto 600 545 lineto 575 535 lineto 550 625 525 685 395 685 curveto 160 685 lineto 411 370 lineto 160 90 lineto 470 90 lineto 565 90 590 140 610 210 curveto 640 200 lineto 585 5 lineto 15 5 lineto 330 335 lineto closepath } def /prod { 590 90 moveto 590 30 620 20 670 20 curveto 670 0 lineto 410 0 lineto 410 20 lineto 460 20 495 30 495 90 curveto 495 670 lineto 190 670 lineto 190 90 lineto 190 30 225 20 275 20 curveto 275 0 lineto 15 0 lineto 15 20 lineto 65 20 100 30 100 90 curveto 100 615 lineto 100 690 40 700 20 700 curveto 20 715 lineto 670 715 lineto 670 700 lineto 645 700 590 690 590 615 curveto 590 90 lineto closepath } def /cdot { 110 270 moveto 110 300 135 325 165 325 curveto 195 325 220 300 220 270 curveto 220 235 185 215 165 215 curveto 145 215 110 235 110 270 curveto fill 445 270 moveto 445 300 470 325 500 325 curveto 530 325 555 300 555 270 curveto 555 235 520 215 500 215 curveto 480 215 445 235 445 270 curveto fill 775 270 moveto 775 300 800 325 830 325 curveto 860 325 890 300 890 270 curveto 890 235 850 215 830 215 curveto 810 215 775 235 775 270 curveto fill } def /vdot { 220 770 moveto 250 770 280 750 280 720 curveto 280 680 250 660 220 660 curveto 185 660 165 695 165 720 curveto 165 740 185 770 220 770 curveto fill 220 440 moveto 250 440 280 415 280 385 curveto 280 355 250 330 220 330 curveto 185 330 165 365 165 385 curveto 165 405 185 440 220 440 curveto fill 220 105 moveto 250 105 280 80 280 50 curveto 280 20 250 -5 220 -5 curveto 185 -5 165 30 165 50 curveto 165 70 185 105 220 105 curveto fill } def /ddot { 165 720 moveto 205 720 220 700 220 665 curveto 220 630 195 610 165 610 curveto 130 620 110 645 110 665 curveto 110 690 130 720 165 720 curveto fill 500 440 moveto 530 440 555 415 555 385 curveto 555 355 530 330 500 330 curveto 465 330 445 365 445 385 curveto 445 405 465 440 500 440 curveto fill 830 155 moveto 860 155 890 130 890 100 curveto 890 70 860 45 830 45 curveto 795 45 775 80 775 100 curveto 775 120 795 155 830 155 curveto fill } def /hbar { 480 105 moveto 440 55 400 -10 335 -10 curveto 300 -10 290 10 290 35 curveto 290 65 370 350 370 365 curveto 370 380 355 390 340 390 curveto 290 390 205 275 185 250 curveto 140 180 125 125 95 0 curveto 20 0 lineto 150 500 lineto 160 545 180 600 180 615 curveto 180 625 170 640 135 640 curveto 110 640 lineto 100 655 lineto 165 660 215 670 270 685 curveto 275 680 lineto 155 240 lineto 155 235 lineto 190 290 290 440 380 440 curveto 400 440 450 435 450 370 curveto 450 335 365 80 365 55 curveto 365 45 375 40 385 40 curveto 405 40 425 70 465 115 curveto 480 105 lineto fill 55 545 moveto 55 520 lineto 310 560 lineto 310 585 lineto 55 545 lineto fill } def /bigcap { 615 400 moveto 615 575 500 655 340 655 curveto 180 655 70 575 70 400 curveto 70 0 lineto 5 0 lineto 5 400 lineto 5 610 140 720 340 720 curveto 545 720 680 610 680 400 curveto 680 0 lineto 615 0 lineto 615 400 lineto closepath } def /bigcup { 65 320 moveto 65 145 180 65 340 65 curveto 500 65 615 145 615 320 curveto 615 720 lineto 680 720 lineto 680 320 lineto 680 110 540 0 340 0 curveto 140 0 0 110 0 320 curveto 0 720 lineto 65 720 lineto 65 320 lineto closepath } def /parallel { 135 675 moveto 135 -175 lineto 65 -175 lineto 65 675 lineto 135 675 lineto fill 335 675 moveto 335 -175 lineto 265 -175 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0.13316 -0.567525 1 Mshowa gsave 0.002 setlinewidth 0.87464 0.17656 moveto 0.87115 0.18194 lineto stroke grestore [(5)] 0.8816 0.16578 -0.646259 1 Mshowa gsave 0.001 setlinewidth 0.29338 0.04853 moveto 0.29212 0.05217 lineto stroke grestore gsave 0.001 setlinewidth 0.32574 0.05566 moveto 0.32443 0.05928 lineto stroke grestore gsave 0.001 setlinewidth 0.35789 0.06274 moveto 0.35653 0.06634 lineto stroke grestore gsave 0.001 setlinewidth 0.38984 0.06978 moveto 0.38843 0.07336 lineto stroke grestore gsave 0.001 setlinewidth 0.45313 0.08372 moveto 0.45162 0.08726 lineto stroke grestore gsave 0.001 setlinewidth 0.48448 0.09062 moveto 0.48292 0.09414 lineto stroke grestore gsave 0.001 setlinewidth 0.51562 0.09748 moveto 0.51402 0.10098 lineto stroke grestore gsave 0.001 setlinewidth 0.54657 0.1043 moveto 0.54493 0.10778 lineto stroke grestore gsave 0.001 setlinewidth 0.6079 0.11781 moveto 0.60616 0.12124 lineto stroke grestore gsave 0.001 setlinewidth 0.63827 0.1245 moveto 0.6365 0.12791 lineto stroke grestore gsave 0.001 setlinewidth 0.66846 0.13115 moveto 0.66664 0.13454 lineto stroke grestore gsave 0.001 setlinewidth 0.69846 0.13775 moveto 0.6966 0.14112 lineto stroke grestore gsave 0.001 setlinewidth 0.75791 0.15085 moveto 0.75597 0.15417 lineto stroke grestore gsave 0.001 setlinewidth 0.78736 0.15733 moveto 0.78538 0.16063 lineto stroke grestore gsave 0.001 setlinewidth 0.81663 0.16378 moveto 0.81461 0.16706 lineto stroke grestore gsave 0.001 setlinewidth 0.84572 0.17019 moveto 0.84367 0.17344 lineto stroke grestore gsave 0.001 setlinewidth 0.22803 0.03414 moveto 0.22687 0.03781 lineto stroke grestore gsave 0.001 setlinewidth 0.19503 0.02687 moveto 0.19393 0.03056 lineto stroke grestore gsave 0.001 setlinewidth 0.90338 0.18289 moveto 0.90125 0.1861 lineto stroke grestore gsave 0.001 setlinewidth 0.93194 0.18918 moveto 0.92978 0.19236 lineto stroke grestore grestore gsave gsave 0.002 setlinewidth 0.93668 0.19022 moveto 0.97434 0.43333 lineto stroke grestore gsave 0.002 setlinewidth 0.94257 0.22823 moveto 0.9363 0.22687 lineto stroke grestore [(0)] 0.95511 0.23094 -1 -0.216359 Mshowa gsave 0.002 setlinewidth 0.95479 0.30714 moveto 0.94851 0.30584 lineto stroke grestore [(0.2)] 0.96735 0.30976 -1 -0.208258 Mshowa gsave 0.002 setlinewidth 0.96766 0.39017 moveto 0.96137 0.38891 lineto stroke grestore [(0.4)] 0.98024 0.39268 -1 -0.199712 Mshowa gsave 0.001 setlinewidth 0.94496 0.2437 moveto 0.9412 0.24289 lineto stroke grestore gsave 0.001 setlinewidth 0.94739 0.25932 moveto 0.94362 0.25852 lineto stroke grestore gsave 0.001 setlinewidth 0.94983 0.2751 moveto 0.94606 0.27431 lineto stroke grestore gsave 0.001 setlinewidth 0.9523 0.29104 moveto 0.94853 0.29025 lineto stroke grestore gsave 0.001 setlinewidth 0.95731 0.32341 moveto 0.95354 0.32263 lineto stroke grestore gsave 0.001 setlinewidth 0.95986 0.33984 moveto 0.95609 0.33907 lineto stroke grestore gsave 0.001 setlinewidth 0.96243 0.35645 moveto 0.95866 0.35568 lineto stroke grestore gsave 0.001 setlinewidth 0.96503 0.37322 moveto 0.96126 0.37246 lineto stroke grestore gsave 0.001 setlinewidth 0.9402 0.21291 moveto 0.93644 0.2121 lineto stroke grestore gsave 0.001 setlinewidth 0.93785 0.19775 moveto 0.93409 0.19692 lineto stroke grestore gsave 0.001 setlinewidth 0.97031 0.4073 moveto 0.96653 0.40655 lineto stroke grestore gsave 0.001 setlinewidth 0.97299 0.42461 moveto 0.96921 0.42387 lineto stroke grestore grestore % End of Graphics MathPictureEnd :[font = section; inactive; Cclosed; preserveAspect; startGroup; ] Numerical Series :[font = text; inactive; preserveAspect; ] A well-known result in calculus is that the series :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 0; pictureWidth = 38; pictureHeight = 36; ] %! %%Creator: Mathematica %%AspectRatio: 0.947368 200 dict begin MathPictureStart /Courier findfont 10 scalefont setfont % Scaling calculations 0 1 0 1 [ [ 0 0 0 0 ] [ 38 36 0 0 ] ] MathScale % Start of Graphics %!PS-Adobe-2.0 EPSF-2.0 %%Title: %%Creator: EquationBuilder %%CreationDate: Sun Aug 2 13:31:40 1992 %%For: gray %%DocumentFonts: (atend) %%Pages: 0 1 %%BoundingBox: 0 0 38 36 %%Translation: 1 14.884993 36.378998 0 %%EndComments %%BeginDocument: /usr/lib/NextStep/printPackage.ps %!PS-Adobe-2.0 %%Title: Printing Package %%EndComments %Version: 2.0 %Copyright: 1988, NeXT, Inc. /__NXdef{1 index where{pop pop pop}{def}ifelse}bind def /__NXbdef{1 index where{pop pop pop}{bind def}ifelse}bind def /UserObjects 10 array __NXdef /defineuserobject{ exch dup 1 add dup UserObjects length gt{ array dup 0 UserObjects putinterval /UserObjects exch def }{pop}ifelse UserObjects exch 3 -1 roll put }__NXbdef /undefineuserobject{UserObjects exch null put}__NXbdef /execuserobject{UserObjects exch get exec}__NXbdef /__NXRectPath{4 2 roll moveto 1 index 0 rlineto 0 exch rlineto neg 0 rlineto closepath}__NXbdef /__NXProcessRectArgs{ 1 index type /arraytype eq{ exch 0 4 2 index length 1 sub{ dup 3 add 1 exch{1 index exch get exch}for 5 1 roll 5 index exec }for pop pop }{exec}ifelse }__NXbdef /rectfill{gsave newpath {__NXRectPath fill} __NXProcessRectArgs grestore}__NXbdef /rectclip{newpath {__NXRectPath} __NXProcessRectArgs clip newpath}__NXbdef /rectstroke{ gsave newpath dup type /arraytype eq{dup length 6 eq}{false}ifelse{ {gsave __NXRectPath null concat stroke grestore} dup length array cvx copy dup 2 4 -1 roll put __NXProcessRectArgs }{{__NXRectPath stroke} __NXProcessRectArgs}ifelse grestore }__NXbdef /xyshow{ 0 1 3 index length 1 sub{ currentpoint 4 index 3 index 1 getinterval show 3 index 3 index 2 mul 1 add get add exch 3 index 3 index 2 mul get add exch moveto pop }for pop pop }__NXbdef /xshow{ 0 1 3 index length 1 sub{ currentpoint 4 index 3 index 1 getinterval show exch 3 index 3 index get add exch moveto pop }for pop pop }__NXbdef /yshow{ 0 1 3 index length 1 sub{ currentpoint 4 index 3 index 1 getinterval show 3 index 3 index get add moveto pop }for pop pop }__NXbdef /arct{arcto pop pop pop pop}__NXbdef /setbbox{pop pop pop pop}__NXbdef /ucache{}__NXbdef /ucachestatus{mark 0 0 0 0 0}__NXbdef /setucacheparams{cleartomark}__NXbdef /uappend{systemdict begin cvx exec end}__NXbdef /ueofill{gsave newpath uappend eofill grestore}__NXbdef /ufill{gsave newpath uappend fill grestore}__NXbdef /ustroke{ gsave newpath dup length 6 eq {exch uappend concat}{uappend}ifelse stroke grestore }__NXbdef /__NXustrokepathMatrix dup where {pop pop}{matrix def}ifelse /ustrokepath{ newpath dup length 6 eq{ exch uappend __NXustrokepathMatrix currentmatrix exch concat strokepath setmatrix }{uappend strokepath}ifelse } __NXbdef /upath{ [exch {/ucache cvx}if pathbbox /setbbox cvx {/moveto cvx}{/lineto cvx}{/curveto cvx}{/closepath cvx}pathforall]cvx } __NXbdef /setstrokeadjust{pop}__NXbdef /currentstrokeadjust{false}__NXbdef /selectfont{exch findfont exch dup type /arraytype eq {makefont}{scalefont}ifelse setfont}__NXbdef /_NXCombineArrays{ counttomark dup 2 add index dup length 3 -1 roll { 2 index length sub dup 4 1 roll 1 index exch 4 -1 roll putinterval exch }repeat pop pop pop }__NXbdef /setcmykcolor{ 1.0 exch sub dup dup 6 -1 roll sub dup 0 lt{pop 0}if 5 1 roll 4 -1 roll sub dup 0 lt{pop 0}if 3 1 roll exch sub dup 0 lt{pop 0}if setrgbcolor }__NXbdef /currentcmykcolor{currentrgbcolor 3{1.0 exch sub 3 1 roll}repeat 0}__NXbdef /flushgraphics{}def /setwindowtype{pop pop}def /currentwindowtype{pop 0}def /setalpha{pop}def /currentalpha{1.0}def /hidecursor{}def /obscurecursor{}def /revealcursor{}def /setcursor{4 {pop}repeat}bind def /showcursor{}def /NextStepEncoding where not{ /NextStepEncoding StandardEncoding 256 array copy def 0 [129/Agrave/Aacute/Acircumflex/Atilde/Adieresis/Aring/Ccedilla/Egrave /Eacute/Ecircumflex/Edieresis/Igrave/Iacute/Icircumflex/Idieresis /Eth/Ntilde/Ograve/Oacute/Ocircumflex/Otilde/Odieresis/Ugrave/Uacute /Ucircumflex/Udieresis/Yacute/Thorn/mu/multiply/divide/copyright 176/registered 181/brokenbar 190/logicalnot 192/onesuperior 201/twosuperior 204/threesuperior 209/plusminus/onequarter/onehalf/threequarters/agrave /aacute/acircumflex/atilde/adieresis/aring/ccedilla/egrave/eacute /ecircumflex/edieresis/igrave 226/iacute 228/icircumflex/idieresis/eth /ntilde 236/ograve/oacute/ocircumflex/otilde/odieresis 242/ugrave/uacute /ucircumflex 246/udieresis/yacute 252/thorn/ydieresis] {dup type /nametype eq {NextStepEncoding 2 index 2 index put pop 1 add}{exch pop}ifelse }forall pop /NextStepEncoding NextStepEncoding readonly def /_NXfstr 128 string dup 0 (_NX) putinterval def /findfont{ % Because we can never let NextStepEncoding get into % SharedFontDirectory, we cannot reencode a font to NextStepEncoding % if we are in shared mode. So if currentshared is true, % we call the normal findfont and return that /currentshared where {pop currentshared} {false} ifelse {//findfont exec} {dup _NXfstr 3 125 getinterval cvs length 3 add _NXfstr 0 3 -1 roll getinterval cvn exch FontDirectory 2 index known {pop FontDirectory exch get} {//findfont exec dup /Encoding get StandardEncoding eq { dup length dict exch {1 index /FID ne {2 index 3 1 roll put}{pop pop}ifelse}forall dup /Encoding NextStepEncoding put definefont }{exch pop} ifelse }ifelse }ifelse }bind def }{pop}ifelse /_NXProcArray 5 array __NXdef /_NXChannels 0 __NXdef /_NXTotalBytes 0 __NXdef /_NXDoImageOp{ 1 index{dup}{1}ifelse /_NXChannels exch store _NXChannels 2 add 2 roll _NXProcArray 0 _NXChannels getinterval astore pop 5 index 4 index mul 2 index{1 sub 8 idiv 1 add mul}{mul 1 sub 8 idiv 1 add}ifelse 4 index mul /_NXTotalBytes exch store pop exch pop gsave matrix invertmatrix concat 0.5 setgray 0 0 4 2 roll rectfill grestore { 0 1 _NXChannels 1 sub{ _NXProcArray exch get exec length _NXTotalBytes exch sub /_NXTotalBytes exch store}for _NXTotalBytes 0 le{exit}if }loop /_NXProcArray 5 array def }__NXbdef /colorimage{_NXDoImageOp}__NXbdef /alphaimage{1 add _NXDoImageOp}def %%EndDocument % EquationBuilder(TM) Expression % +.\Jr(=:/*V[StRFQAYmdjEEA;1nuN+[co.OZ7L#cq3iK%3:</XJ@:d#6;^G;Td47dud<A3@ % 5%MGToUOR5&d`'TuEDU)B-tbR".R^]Xs;!!!E7Jg"h0Sid;k!&A0sO@"L$!/a>R!L,$-1s/C % B#Cn*e"h)@2@AXE`Ui2FuJ3`%0`&Rf+8YXkFaW!,slN$o42$fa3D<P)b`0`8d!_h#b$ue*i) % ?EaFa8jYB07tVUj_0#da"Us`Ri5qngBj*p:ja!OajW`'?dji^%RCc*o\m`f)1,g%.PDqG6Bd % !NE62Vj'bgb#0GV9c+[VV9%k#kQ5irrS$&NDEMKmuUaoaJuIYrHi2V0B58BF=G7lRsBN$<2s % &:+?"X[bp/dIdK3+9t;Sh0=mDa'E]aV'dC:&]J-KKE/"WJ17jUo$<$iJYF<CKFEs^M#nq;+! % D%0+W=,MjVOF4X$-6HG`CD)?\fp:86m_rIO5:;Bt2Pp*\ukKg]4>3aB1)W9":RUKW)1a@o,@ % 6Ss8TP(OuoE-;[)LE"%ugISWE>+ELHoGRB*@MR_LuOoQ^50p3X]-]`>+#U5VYGc4\[6e!:"* % 2Fn&fM5Vig#FN-AHE2?B^INR*t0E&J9qZ#O[^9>[3,W^@CY8;"J13+7ghZ]77Io3KY^+rA%8 % F%WhHlD"rRXM!2(IK"p"iT^]GYF,m6Um7M)'i^kk:1zx % EndExpression N 553 229 E 75 S d617 R efbb43b5 gsave /__NXbasematrix matrix currentmatrix def grestore %%EndProlog %%BeginSetup /EqBFont 18 dict dup begin /FontType 3 def /PaintType 0 def /FontMatrix [0.001 0 0 0.001 0 0] def /FontBBox [15 -2950 1360 1045] def /Encoding [/.notdef /rad1 /rad2 /rad3 /rad4 /rad5 /radext /int /pathint /sum /prod /cdot /vdot /ddot /hbar /bigcap /bigcup /parallel] def /Metrics 18 dict dup begin /.notdef 0 def /rad1 1000 def /rad2 1000 def /rad3 1000 def /rad4 1000 def /rad5 1048 def /radext 1048 def /int 500 def /pathint 500 def /sum 700 def /prod 700 def /cdot 1000 def /vdot 550 def /ddot 1000 def /hbar 500 def /bigcap 700 def /bigcup 700 def /parallel 500 def end def /BBox 18 dict dup begin /.notdef [0 0 0 0] def /rad1 [110 -1150 1020 0] def /rad2 [110 -1750 1020 0] def /rad3 [110 -2350 1020 0] def /rad4 [110 -2950 1020 0] def /rad5 [128 -2890 740 0] def /radext [700 -600 740 0] def /int [20 0 425 1045] def /pathint [20 0 425 1045] def /sum [15 5 640 715] def /prod [15 0 670 715] def /cdot [110 215 890 325] def /vdot [165 -5 280 770] def /ddot [110 -5 890 770] def /hbar [20 -10 480 685] def /bigcap [0 0 680 720] def /bigcup [0 0 680 720] def /parallel [65 -175 400 675] def end def /CharacterDefs 18 dict dup begin /.notdef { } def /rad1 { 980 0 moveto 464 -1048 lineto 248 -560 lineto 110 -656 lineto 125 -675 lineto 200 -624 lineto 420 -1150 lineto 460 -1150 lineto 1020 0 lineto 980 0 lineto closepath } def /rad2 { 980 0 moveto 460 -1600 lineto 245 -864 lineto 110 -1000 lineto 128 -1023 lineto 200 -950 lineto 424 -1750 lineto 456 -1750 lineto 1020 0 lineto 980 0 lineto closepath } def /rad3 { 980 0 moveto 464 -2144 lineto 248 -1160 lineto 110 -1368 lineto 130 -1390 lineto 192 -1288 lineto 424 -2350 lineto 464 -2350 lineto 1020 0 lineto 980 0 lineto closepath } def /rad4 { 980 0 moveto 880 -488 lineto 672 -1600 lineto 464 -2696 lineto 248 -1464 lineto 110 -1728 lineto 130 -1750 lineto 192 -1616 lineto 424 -2950 lineto 464 -2950 lineto 768 -1312 lineto 1020 0 lineto 980 0 lineto closepath } def /rad5 { 700 0 moveto 700 -2628 lineto 248 -1284 lineto 128 -1564 lineto 150 -1580 lineto 215 -1425 lineto 688 -2890 lineto 740 -2890 lineto 740 0 lineto 700 0 lineto closepath } def /radext { 700 0 moveto 700 -600 lineto 740 -600 lineto 740 0 lineto 700 0 lineto closepath } def /int { 245 390 moveto 225 235 195 145 180 100 curveto 165 60 150 50 140 40 curveto 135 30 85 0 50 20 curveto 35 25 25 45 25 55 curveto 20 80 30 100 55 110 curveto 80 125 93 110 110 90 curveto 135 60 155 135 160 195 curveto 165 270 170 340 190 510 curveto 205 650 lineto 220 810 230 890 265 960 curveto 275 980 295 1015 320 1030 curveto 350 1045 380 1030 395 1020 curveto 410 1010 425 980 400 955 curveto 385 940 347 925 330 950 curveto 310 970 290 955 290 940 curveto 280 900 285 745 270 610 curveto 245 390 lineto closepath } def /pathint { 245 390 moveto 225 235 195 145 180 100 curveto 165 60 150 50 140 40 curveto 135 30 85 0 50 20 curveto 35 25 25 45 25 55 curveto 20 80 30 100 55 110 curveto 80 125 93 110 110 90 curveto 135 60 155 135 160 195 curveto 165 270 170 340 190 510 curveto 205 650 lineto 220 810 230 890 265 960 curveto 275 980 295 1015 320 1030 curveto 350 1045 380 1030 395 1020 curveto 410 1010 425 980 400 955 curveto 385 940 347 925 330 950 curveto 310 970 290 955 290 940 curveto 280 900 285 745 270 610 curveto 245 390 lineto fill 20 setlinewidth 220 400 moveto 290 400 340 455 340 500 curveto 340 585 290 640 220 640 curveto 155 640 100 585 100 500 curveto 100 455 155 400 220 400 curveto stroke } def /sum { 330 335 moveto 20 715 lineto 575 715 lineto 600 545 lineto 575 535 lineto 550 625 525 685 395 685 curveto 160 685 lineto 411 370 lineto 160 90 lineto 470 90 lineto 565 90 590 140 610 210 curveto 640 200 lineto 585 5 lineto 15 5 lineto 330 335 lineto closepath } def /prod { 590 90 moveto 590 30 620 20 670 20 curveto 670 0 lineto 410 0 lineto 410 20 lineto 460 20 495 30 495 90 curveto 495 670 lineto 190 670 lineto 190 90 lineto 190 30 225 20 275 20 curveto 275 0 lineto 15 0 lineto 15 20 lineto 65 20 100 30 100 90 curveto 100 615 lineto 100 690 40 700 20 700 curveto 20 715 lineto 670 715 lineto 670 700 lineto 645 700 590 690 590 615 curveto 590 90 lineto closepath } def /cdot { 110 270 moveto 110 300 135 325 165 325 curveto 195 325 220 300 220 270 curveto 220 235 185 215 165 215 curveto 145 215 110 235 110 270 curveto fill 445 270 moveto 445 300 470 325 500 325 curveto 530 325 555 300 555 270 curveto 555 235 520 215 500 215 curveto 480 215 445 235 445 270 curveto fill 775 270 moveto 775 300 800 325 830 325 curveto 860 325 890 300 890 270 curveto 890 235 850 215 830 215 curveto 810 215 775 235 775 270 curveto fill } def /vdot { 220 770 moveto 250 770 280 750 280 720 curveto 280 680 250 660 220 660 curveto 185 660 165 695 165 720 curveto 165 740 185 770 220 770 curveto fill 220 440 moveto 250 440 280 415 280 385 curveto 280 355 250 330 220 330 curveto 185 330 165 365 165 385 curveto 165 405 185 440 220 440 curveto fill 220 105 moveto 250 105 280 80 280 50 curveto 280 20 250 -5 220 -5 curveto 185 -5 165 30 165 50 curveto 165 70 185 105 220 105 curveto fill } def /ddot { 165 720 moveto 205 720 220 700 220 665 curveto 220 630 195 610 165 610 curveto 130 620 110 645 110 665 curveto 110 690 130 720 165 720 curveto fill 500 440 moveto 530 440 555 415 555 385 curveto 555 355 530 330 500 330 curveto 465 330 445 365 445 385 curveto 445 405 465 440 500 440 curveto fill 830 155 moveto 860 155 890 130 890 100 curveto 890 70 860 45 830 45 curveto 795 45 775 80 775 100 curveto 775 120 795 155 830 155 curveto fill } def /hbar { 480 105 moveto 440 55 400 -10 335 -10 curveto 300 -10 290 10 290 35 curveto 290 65 370 350 370 365 curveto 370 380 355 390 340 390 curveto 290 390 205 275 185 250 curveto 140 180 125 125 95 0 curveto 20 0 lineto 150 500 lineto 160 545 180 600 180 615 curveto 180 625 170 640 135 640 curveto 110 640 lineto 100 655 lineto 165 660 215 670 270 685 curveto 275 680 lineto 155 240 lineto 155 235 lineto 190 290 290 440 380 440 curveto 400 440 450 435 450 370 curveto 450 335 365 80 365 55 curveto 365 45 375 40 385 40 curveto 405 40 425 70 465 115 curveto 480 105 lineto fill 55 545 moveto 55 520 lineto 310 560 lineto 310 585 lineto 55 545 lineto fill } def /bigcap { 615 400 moveto 615 575 500 655 340 655 curveto 180 655 70 575 70 400 curveto 70 0 lineto 5 0 lineto 5 400 lineto 5 610 140 720 340 720 curveto 545 720 680 610 680 400 curveto 680 0 lineto 615 0 lineto 615 400 lineto closepath } def /bigcup { 65 320 moveto 65 145 180 65 340 65 curveto 500 65 615 145 615 320 curveto 615 720 lineto 680 720 lineto 680 320 lineto 680 110 540 0 340 0 curveto 140 0 0 110 0 320 curveto 0 720 lineto 65 720 lineto 65 320 lineto closepath } def /parallel { 135 675 moveto 135 -175 lineto 65 -175 lineto 65 675 lineto 135 675 lineto fill 335 675 moveto 335 -175 lineto 265 -175 lineto 265 675 lineto 335 675 lineto fill 400 675 moveto } def end def /BuildChar { 0 begin /char exch def /fontdict exch def /charname fontdict /Encoding get char get def fontdict begin Metrics charname get 0 BBox charname get aload pop setcachedevice CharacterDefs charname get exec eofill end end } def /BuildChar load 0 3 dict put end definefont pop %%EndSetup 0 0 37.378998 35.640968 rectclip 0 0 37.378998 35.640968 rectclip 0 setgray 1 setalpha /EqBFont 24 selectfont 1.811499 10.244992 moveto <09> [0 0] xyshow 0 setgray 1 setalpha 0.48 setlinewidth 20.342999 18.884993 moveto 36.378998 18.884993 lineto stroke 0 setgray 1 setalpha /Symbol findfont 9 scalefont 49 exch defineuserobject 49 execuserobject setfont 8.452 2.281 moveto (=\245) [-1.989 27.523968 0 0 0 0] xyshow /Times-BoldItalic findfont 9 scalefont 47 exch defineuserobject 47 execuserobject setfont 1 2.281 moveto (m) [0 0 0 0] xyshow /Times-Roman findfont 9 scalefont 46 exch defineuserobject 46 execuserobject setfont 30.178999 11.120993 moveto (21) [-16.335999 -8.839993 0 0 0 0] xyshow /Times-Roman findfont 12 scalefont 50 exch defineuserobject 50 execuserobject setfont 25.360998 23.684992 moveto (1) [0 0 0 0] xyshow /Times-BoldItalic findfont 12 scalefont 51 exch defineuserobject 51 execuserobject setfont 20.842999 6.766289 moveto (m) [0 0 -1.681558e-44 1.050009e+12] xyshow %%Trailer %%DocumentFonts: Times-BoldItalic %%+ Times-Roman %%+ Symbol %End of Graphics MathPictureEnd end :[font = text; inactive; preserveAspect; ] converges. Its value can be computed using Fourier series. Consider the function defined by :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 0; pictureWidth = 164; pictureHeight = 36; ] %! %%Creator: Mathematica %%AspectRatio: 0.219512 200 dict begin MathPictureStart /Courier findfont 10 scalefont setfont % Scaling calculations 0 1 0 1 [ [ 0 0 0 0 ] [ 164 36 0 0 ] ] MathScale % Start of Graphics %!PS-Adobe-2.0 EPSF-2.0 %%Title: %%Creator: EquationBuilder %%CreationDate: Sun Aug 2 13:33:48 1992 %%For: gray %%DocumentFonts: (atend) %%Pages: 0 1 %%BoundingBox: 0 0 164 36 %%Translation: 1 14.884993 162.047668 0 %%EndComments %%BeginDocument: /usr/lib/NextStep/printPackage.ps %!PS-Adobe-2.0 %%Title: Printing Package %%EndComments %Version: 2.0 %Copyright: 1988, NeXT, Inc. /__NXdef{1 index where{pop pop pop}{def}ifelse}bind def /__NXbdef{1 index where{pop pop pop}{bind def}ifelse}bind def /UserObjects 10 array __NXdef /defineuserobject{ exch dup 1 add dup UserObjects length gt{ array dup 0 UserObjects putinterval /UserObjects exch def }{pop}ifelse UserObjects exch 3 -1 roll put }__NXbdef /undefineuserobject{UserObjects exch null put}__NXbdef /execuserobject{UserObjects exch get exec}__NXbdef /__NXRectPath{4 2 roll moveto 1 index 0 rlineto 0 exch rlineto neg 0 rlineto closepath}__NXbdef /__NXProcessRectArgs{ 1 index type /arraytype eq{ exch 0 4 2 index length 1 sub{ dup 3 add 1 exch{1 index exch get exch}for 5 1 roll 5 index exec }for pop pop }{exec}ifelse }__NXbdef /rectfill{gsave newpath {__NXRectPath fill} __NXProcessRectArgs grestore}__NXbdef /rectclip{newpath {__NXRectPath} __NXProcessRectArgs clip newpath}__NXbdef /rectstroke{ gsave newpath dup type /arraytype eq{dup length 6 eq}{false}ifelse{ {gsave __NXRectPath null concat stroke grestore} dup length array cvx copy dup 2 4 -1 roll put __NXProcessRectArgs }{{__NXRectPath stroke} __NXProcessRectArgs}ifelse grestore }__NXbdef /xyshow{ 0 1 3 index length 1 sub{ currentpoint 4 index 3 index 1 getinterval show 3 index 3 index 2 mul 1 add get add exch 3 index 3 index 2 mul get add exch moveto pop }for pop pop }__NXbdef /xshow{ 0 1 3 index length 1 sub{ currentpoint 4 index 3 index 1 getinterval show exch 3 index 3 index get add exch moveto pop }for pop pop }__NXbdef /yshow{ 0 1 3 index length 1 sub{ currentpoint 4 index 3 index 1 getinterval show 3 index 3 index get add moveto pop }for pop pop }__NXbdef /arct{arcto pop pop pop pop}__NXbdef /setbbox{pop pop pop pop}__NXbdef /ucache{}__NXbdef /ucachestatus{mark 0 0 0 0 0}__NXbdef /setucacheparams{cleartomark}__NXbdef /uappend{systemdict begin cvx exec end}__NXbdef /ueofill{gsave newpath uappend eofill grestore}__NXbdef /ufill{gsave newpath uappend fill grestore}__NXbdef /ustroke{ gsave newpath dup length 6 eq {exch uappend concat}{uappend}ifelse stroke grestore }__NXbdef /__NXustrokepathMatrix dup where {pop pop}{matrix def}ifelse /ustrokepath{ newpath dup length 6 eq{ exch uappend __NXustrokepathMatrix currentmatrix exch concat strokepath setmatrix }{uappend strokepath}ifelse } __NXbdef /upath{ [exch {/ucache cvx}if pathbbox /setbbox cvx {/moveto cvx}{/lineto cvx}{/curveto cvx}{/closepath cvx}pathforall]cvx } __NXbdef /setstrokeadjust{pop}__NXbdef /currentstrokeadjust{false}__NXbdef /selectfont{exch findfont exch dup type /arraytype eq {makefont}{scalefont}ifelse setfont}__NXbdef /_NXCombineArrays{ counttomark dup 2 add index dup length 3 -1 roll { 2 index length sub dup 4 1 roll 1 index exch 4 -1 roll putinterval exch }repeat pop pop pop }__NXbdef /setcmykcolor{ 1.0 exch sub dup dup 6 -1 roll sub dup 0 lt{pop 0}if 5 1 roll 4 -1 roll sub dup 0 lt{pop 0}if 3 1 roll exch sub dup 0 lt{pop 0}if setrgbcolor }__NXbdef /currentcmykcolor{currentrgbcolor 3{1.0 exch sub 3 1 roll}repeat 0}__NXbdef /flushgraphics{}def /setwindowtype{pop pop}def /currentwindowtype{pop 0}def /setalpha{pop}def /currentalpha{1.0}def /hidecursor{}def /obscurecursor{}def /revealcursor{}def /setcursor{4 {pop}repeat}bind def /showcursor{}def /NextStepEncoding where not{ /NextStepEncoding StandardEncoding 256 array copy def 0 [129/Agrave/Aacute/Acircumflex/Atilde/Adieresis/Aring/Ccedilla/Egrave /Eacute/Ecircumflex/Edieresis/Igrave/Iacute/Icircumflex/Idieresis /Eth/Ntilde/Ograve/Oacute/Ocircumflex/Otilde/Odieresis/Ugrave/Uacute /Ucircumflex/Udieresis/Yacute/Thorn/mu/multiply/divide/copyright 176/registered 181/brokenbar 190/logicalnot 192/onesuperior 201/twosuperior 204/threesuperior 209/plusminus/onequarter/onehalf/threequarters/agrave /aacute/acircumflex/atilde/adieresis/aring/ccedilla/egrave/eacute /ecircumflex/edieresis/igrave 226/iacute 228/icircumflex/idieresis/eth /ntilde 236/ograve/oacute/ocircumflex/otilde/odieresis 242/ugrave/uacute /ucircumflex 246/udieresis/yacute 252/thorn/ydieresis] {dup type /nametype eq {NextStepEncoding 2 index 2 index put pop 1 add}{exch pop}ifelse }forall pop /NextStepEncoding NextStepEncoding readonly def /_NXfstr 128 string dup 0 (_NX) putinterval def /findfont{ % Because we can never let NextStepEncoding get into % SharedFontDirectory, we cannot reencode a font to NextStepEncoding % if we are in shared mode. So if currentshared is true, % we call the normal findfont and return that /currentshared where {pop currentshared} {false} ifelse {//findfont exec} {dup _NXfstr 3 125 getinterval cvs length 3 add _NXfstr 0 3 -1 roll getinterval cvn exch FontDirectory 2 index known {pop FontDirectory exch get} {//findfont exec dup /Encoding get StandardEncoding eq { dup length dict exch {1 index /FID ne {2 index 3 1 roll put}{pop pop}ifelse}forall dup /Encoding NextStepEncoding put definefont }{exch pop} ifelse }ifelse }ifelse }bind def }{pop}ifelse /_NXProcArray 5 array __NXdef /_NXChannels 0 __NXdef /_NXTotalBytes 0 __NXdef /_NXDoImageOp{ 1 index{dup}{1}ifelse /_NXChannels exch store _NXChannels 2 add 2 roll _NXProcArray 0 _NXChannels getinterval astore pop 5 index 4 index mul 2 index{1 sub 8 idiv 1 add mul}{mul 1 sub 8 idiv 1 add}ifelse 4 index mul /_NXTotalBytes exch store pop exch pop gsave matrix invertmatrix concat 0.5 setgray 0 0 4 2 roll rectfill grestore { 0 1 _NXChannels 1 sub{ _NXProcArray exch get exec length _NXTotalBytes exch sub /_NXTotalBytes exch store}for _NXTotalBytes 0 le{exit}if }loop /_NXProcArray 5 array def }__NXbdef /colorimage{_NXDoImageOp}__NXbdef /alphaimage{1 add _NXDoImageOp}def %%EndDocument % EquationBuilder(TM) Expression % +.\Jr(=:/*V[StRFQAYmdjEEA;1nuN+[co.OZ7L#cq3iK%3:</XJ@:d#6;^G;Td47dud<A3@ % 5%MGToUOR5&d`'TuEDU)B-tbR".R+:nNN,VT6!O<O^$SqI@LR55(=Tg/YZ`?^*:b*44kjeT] % p^_3=B>@PZ-dtPS8[n?`e+Q,0;)ut^=;<6Hl%@`i0rdKeEK[-&FF=^!e]Qc(1->="q0H2,DK % <o9'!9pdj_3@'!`:tp3GVF^mBYqh6NHc,s9R8dO_FmFMd&!B:>V[0^"iUHa8eJXFLsW'OZ"G % ""Kj-J9iMM8uN)?*S"!V=/UhQ$Ic:c`1>VCh+89TN^Bf?uDN%-hojoBGjL+YUh*^e,?cYo)\ % ^RtQ,`qLPT56(c-J-@"8<!lnN=Q!*-&]-e*ChUbL)rQ:!>^=K.#>,$g&JEo<4b+/9&@+^FJ. % ZgTJ^]>2M'iUTZ3c&QAt2`:^_:aMObOgTjUi_IZm7R1H8DLDgg[b&QX>+_4ssffE"odW,Y`= % SSpjRt(m,4=-;[)LE"%sS@p-s_-'TR5BSph+b4f-qP3=*H09QU9gj4knWF_UFdJ)B(pj!]i1 % :]iPJ.OpLH00?HH5NKO&Zq_c'*YdWfP6/K[UG/un0nZ#OlA:`PeWJ]S0j2Wb2E?)LjDdpnHF % eb6u<VC$Y-OB.;>9gZ!u\n;-:pjX`=bp8tfLU1(G`40O*jjit*t2)V>As+X+7@$AnhWZFX-u % aaku-3QTq>AH4P$LC7VU^)Cc:N4"jS!3$MRXsEJ#)_*Y&`ImNVlaC3f@KQG\/?6kp3aY(t8t % .BM!Vcn-!!6W"+G'e)Z=&<:4@n4lp*r1AQbqi^gbAC2%'58()EM4*r(=8U,lbg<CrOd7o'X" % W`d6_Hl+X7)W'm,F/?1-3_X_/-Wu3;$d-5pu.1\sIJ0[lI@!/ac,hgoc$2o+gOH:roY&jc)n % \uPW:'$-.AY;1rd9V>jW2MOr\F>;E?eP1B996$/`.<LOZ!J+n0LQq%bjbn8V9-62g25u;#q- % (L[Du*l0T^ecdOd`'Ft0mB`F32b+WU]`[un65IZ]`q*FWV?L3*d@5'uZD@XK<<5l`W%!6&;3 % 5X])b_$L+3pgXR0x % EndExpression N 875 36b E 86 S 17af0 R c08e2ed9 gsave /__NXbasematrix matrix currentmatrix def grestore %%EndProlog %%BeginSetup /EqBFont 18 dict dup begin /FontType 3 def /PaintType 0 def /FontMatrix [0.001 0 0 0.001 0 0] def /FontBBox [15 -2950 1360 1045] def /Encoding [/.notdef /rad1 /rad2 /rad3 /rad4 /rad5 /radext /int /pathint /sum /prod /cdot /vdot /ddot /hbar /bigcap /bigcup /parallel] def /Metrics 18 dict dup begin /.notdef 0 def /rad1 1000 def /rad2 1000 def /rad3 1000 def /rad4 1000 def /rad5 1048 def /radext 1048 def /int 500 def /pathint 500 def /sum 700 def /prod 700 def /cdot 1000 def /vdot 550 def /ddot 1000 def /hbar 500 def /bigcap 700 def /bigcup 700 def /parallel 500 def end def /BBox 18 dict dup begin /.notdef [0 0 0 0] def /rad1 [110 -1150 1020 0] def /rad2 [110 -1750 1020 0] def /rad3 [110 -2350 1020 0] def /rad4 [110 -2950 1020 0] def /rad5 [128 -2890 740 0] def /radext [700 -600 740 0] def /int [20 0 425 1045] def /pathint [20 0 425 1045] def /sum [15 5 640 715] def /prod [15 0 670 715] def /cdot [110 215 890 325] def /vdot [165 -5 280 770] def /ddot [110 -5 890 770] def /hbar [20 -10 480 685] def /bigcap [0 0 680 720] def /bigcup [0 0 680 720] def /parallel [65 -175 400 675] def end def /CharacterDefs 18 dict dup begin /.notdef { } def /rad1 { 980 0 moveto 464 -1048 lineto 248 -560 lineto 110 -656 lineto 125 -675 lineto 200 -624 lineto 420 -1150 lineto 460 -1150 lineto 1020 0 lineto 980 0 lineto closepath } def /rad2 { 980 0 moveto 460 -1600 lineto 245 -864 lineto 110 -1000 lineto 128 -1023 lineto 200 -950 lineto 424 -1750 lineto 456 -1750 lineto 1020 0 lineto 980 0 lineto closepath } def /rad3 { 980 0 moveto 464 -2144 lineto 248 -1160 lineto 110 -1368 lineto 130 -1390 lineto 192 -1288 lineto 424 -2350 lineto 464 -2350 lineto 1020 0 lineto 980 0 lineto closepath } def /rad4 { 980 0 moveto 880 -488 lineto 672 -1600 lineto 464 -2696 lineto 248 -1464 lineto 110 -1728 lineto 130 -1750 lineto 192 -1616 lineto 424 -2950 lineto 464 -2950 lineto 768 -1312 lineto 1020 0 lineto 980 0 lineto closepath } def /rad5 { 700 0 moveto 700 -2628 lineto 248 -1284 lineto 128 -1564 lineto 150 -1580 lineto 215 -1425 lineto 688 -2890 lineto 740 -2890 lineto 740 0 lineto 700 0 lineto closepath } def /radext { 700 0 moveto 700 -600 lineto 740 -600 lineto 740 0 lineto 700 0 lineto closepath } def /int { 245 390 moveto 225 235 195 145 180 100 curveto 165 60 150 50 140 40 curveto 135 30 85 0 50 20 curveto 35 25 25 45 25 55 curveto 20 80 30 100 55 110 curveto 80 125 93 110 110 90 curveto 135 60 155 135 160 195 curveto 165 270 170 340 190 510 curveto 205 650 lineto 220 810 230 890 265 960 curveto 275 980 295 1015 320 1030 curveto 350 1045 380 1030 395 1020 curveto 410 1010 425 980 400 955 curveto 385 940 347 925 330 950 curveto 310 970 290 955 290 940 curveto 280 900 285 745 270 610 curveto 245 390 lineto closepath } def /pathint { 245 390 moveto 225 235 195 145 180 100 curveto 165 60 150 50 140 40 curveto 135 30 85 0 50 20 curveto 35 25 25 45 25 55 curveto 20 80 30 100 55 110 curveto 80 125 93 110 110 90 curveto 135 60 155 135 160 195 curveto 165 270 170 340 190 510 curveto 205 650 lineto 220 810 230 890 265 960 curveto 275 980 295 1015 320 1030 curveto 350 1045 380 1030 395 1020 curveto 410 1010 425 980 400 955 curveto 385 940 347 925 330 950 curveto 310 970 290 955 290 940 curveto 280 900 285 745 270 610 curveto 245 390 lineto fill 20 setlinewidth 220 400 moveto 290 400 340 455 340 500 curveto 340 585 290 640 220 640 curveto 155 640 100 585 100 500 curveto 100 455 155 400 220 400 curveto stroke } def /sum { 330 335 moveto 20 715 lineto 575 715 lineto 600 545 lineto 575 535 lineto 550 625 525 685 395 685 curveto 160 685 lineto 411 370 lineto 160 90 lineto 470 90 lineto 565 90 590 140 610 210 curveto 640 200 lineto 585 5 lineto 15 5 lineto 330 335 lineto closepath } def /prod { 590 90 moveto 590 30 620 20 670 20 curveto 670 0 lineto 410 0 lineto 410 20 lineto 460 20 495 30 495 90 curveto 495 670 lineto 190 670 lineto 190 90 lineto 190 30 225 20 275 20 curveto 275 0 lineto 15 0 lineto 15 20 lineto 65 20 100 30 100 90 curveto 100 615 lineto 100 690 40 700 20 700 curveto 20 715 lineto 670 715 lineto 670 700 lineto 645 700 590 690 590 615 curveto 590 90 lineto closepath } def /cdot { 110 270 moveto 110 300 135 325 165 325 curveto 195 325 220 300 220 270 curveto 220 235 185 215 165 215 curveto 145 215 110 235 110 270 curveto fill 445 270 moveto 445 300 470 325 500 325 curveto 530 325 555 300 555 270 curveto 555 235 520 215 500 215 curveto 480 215 445 235 445 270 curveto fill 775 270 moveto 775 300 800 325 830 325 curveto 860 325 890 300 890 270 curveto 890 235 850 215 830 215 curveto 810 215 775 235 775 270 curveto fill } def /vdot { 220 770 moveto 250 770 280 750 280 720 curveto 280 680 250 660 220 660 curveto 185 660 165 695 165 720 curveto 165 740 185 770 220 770 curveto fill 220 440 moveto 250 440 280 415 280 385 curveto 280 355 250 330 220 330 curveto 185 330 165 365 165 385 curveto 165 405 185 440 220 440 curveto fill 220 105 moveto 250 105 280 80 280 50 curveto 280 20 250 -5 220 -5 curveto 185 -5 165 30 165 50 curveto 165 70 185 105 220 105 curveto fill } def /ddot { 165 720 moveto 205 720 220 700 220 665 curveto 220 630 195 610 165 610 curveto 130 620 110 645 110 665 curveto 110 690 130 720 165 720 curveto fill 500 440 moveto 530 440 555 415 555 385 curveto 555 355 530 330 500 330 curveto 465 330 445 365 445 385 curveto 445 405 465 440 500 440 curveto fill 830 155 moveto 860 155 890 130 890 100 curveto 890 70 860 45 830 45 curveto 795 45 775 80 775 100 curveto 775 120 795 155 830 155 curveto fill } def /hbar { 480 105 moveto 440 55 400 -10 335 -10 curveto 300 -10 290 10 290 35 curveto 290 65 370 350 370 365 curveto 370 380 355 390 340 390 curveto 290 390 205 275 185 250 curveto 140 180 125 125 95 0 curveto 20 0 lineto 150 500 lineto 160 545 180 600 180 615 curveto 180 625 170 640 135 640 curveto 110 640 lineto 100 655 lineto 165 660 215 670 270 685 curveto 275 680 lineto 155 240 lineto 155 235 lineto 190 290 290 440 380 440 curveto 400 440 450 435 450 370 curveto 450 335 365 80 365 55 curveto 365 45 375 40 385 40 curveto 405 40 425 70 465 115 curveto 480 105 lineto fill 55 545 moveto 55 520 lineto 310 560 lineto 310 585 lineto 55 545 lineto fill } def /bigcap { 615 400 moveto 615 575 500 655 340 655 curveto 180 655 70 575 70 400 curveto 70 0 lineto 5 0 lineto 5 400 lineto 5 610 140 720 340 720 curveto 545 720 680 610 680 400 curveto 680 0 lineto 615 0 lineto 615 400 lineto closepath } def /bigcup { 65 320 moveto 65 145 180 65 340 65 curveto 500 65 615 145 615 320 curveto 615 720 lineto 680 720 lineto 680 320 lineto 680 110 540 0 340 0 curveto 140 0 0 110 0 320 curveto 0 720 lineto 65 720 lineto 65 320 lineto closepath } def /parallel { 135 675 moveto 135 -175 lineto 65 -175 lineto 65 675 lineto 135 675 lineto fill 335 675 moveto 335 -175 lineto 265 -175 lineto 265 675 lineto 335 675 lineto fill 400 675 moveto } def end def /BuildChar { 0 begin /char exch def /fontdict exch def /charname fontdict /Encoding get char get def fontdict begin Metrics charname get 0 BBox charname get aload pop setcachedevice CharacterDefs charname get exec eofill end end } def /BuildChar load 0 3 dict put end definefont pop %%EndSetup 0 0 163.047668 35.640968 rectclip 0 0 163.047668 35.640968 rectclip 0 setgray 1 setalpha 1 1 scale /Symbol findfont 12 scalefont 54 exch defineuserobject 54 execuserobject setfont 11.56 15.992992 moveto (\() [0 0] xyshow 1 1 scale 1 1 scale 54 execuserobject setfont 23.200001 15.986992 moveto (\)) [0 0] xyshow 1 1 scale 0 setgray 1 setalpha 0.48 setlinewidth 40.978668 18.884993 moveto 54.26667 18.884993 lineto stroke 0 setgray 1 setalpha /EqBFont 24 selectfont 66.999504 10.244992 moveto <09> [0 0] xyshow 0 setgray 1 setalpha 1 1 scale 54 execuserobject setfont 92.595001 23.792992 moveto (\() [0 0] xyshow 1 1 scale 1 1 scale 54 execuserobject setfont 113.489662 23.786991 moveto (\)) [0 0] xyshow 1 1 scale 0 setgray 1 setalpha 0.48 setlinewidth 85.530998 18.884993 moveto 126.715668 18.884993 lineto stroke 0 setgray 1 setalpha /Times-BoldItalic findfont 12 scalefont 51 exch defineuserobject 51 execuserobject setfont 156.047668 15.884993 moveto (xmmxf) [-9.335999 0 -48.106339 -9.118704 -82.245331 9.118704 -15.360001 0 0 0 0 0] xyshow /Times-Roman findfont 12 scalefont 50 exch defineuserobject 50 execuserobject setfont 128.715668 15.884993 moveto (cos1431) [5.328003 0 6 0 -33.394005 7.799999 -20.618668 0 -41.408329 -16.03141 -39.626671 8.231412 0 0 0 0] xyshow 54 execuserobject setfont 97.394997 23.684992 moveto (-+p=) [-40.461662 -7.799999 -15.454666 7.799999 -10.421333 -7.799999 0 0 0 0] xyshow /Times-Roman findfont 9 scalefont 46 exch defineuserobject 46 execuserobject setfont 107.94133 11.120993 moveto (212) [-28.910324 -8.839993 -30.964336 25.758696 0 0 0 0] xyshow /Symbol findfont 9 scalefont 49 exch defineuserobject 49 execuserobject setfont 73.639999 2.281 moveto (=\245) [-1.988998 27.523968 0 0 0 0] xyshow /Times-BoldItalic findfont 9 scalefont 47 exch defineuserobject 47 execuserobject setfont 118.013664 28.400019 moveto (mm) [-51.825661 -26.119019 0 0 0 0] xyshow %%Trailer %%DocumentFonts: Times-BoldItalic %%+ Times-Roman %%+ Symbol %End of Graphics MathPictureEnd end :[font = text; inactive; preserveAspect; ] Also let g1 be the periodic extension of x—>x2 (-p < x < p). ;[s] 14:0,0;10,1;11,2;49,3;50,4;52,5;53,6;55,7;62,8;63,9;67,10;69,11;71,12;72,13;75,-1; 14:1,11,8,Times,0,12,0,0,0;1,10,8,Times,2,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Times,2,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Times,2,12,0,0,0;1,9,7,Times,32,10,0,0,0;1,9,7,Times,0,10,0,0,0;1,0,0,Symbol,0,10,0,0,0;1,9,7,Times,0,10,0,0,0;1,10,8,Times,2,12,0,0,0;1,9,7,Times,0,10,0,0,0;1,0,0,Symbol,0,10,0,0,0;1,9,7,Times,0,10,0,0,0; :[font = text; inactive; preserveAspect; ] Since g1 is continuous, we know that the series f1 converges to g1 everywhere, in particular for x = p . Therefore, ;[s] 12:0,0;8,1;9,2;51,3;52,4;69,5;70,6;71,7;104,8;105,9;108,10;109,11;124,-1; 12:1,11,8,Times,0,12,0,0,0;1,10,8,Times,2,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Times,2,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Times,2,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Times,2,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,0,0,Symbol,0,12,0,0,0;1,11,8,Times,0,12,0,0,0; :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 0; pictureWidth = 315; pictureHeight = 36; ] %! %%Creator: Mathematica %%AspectRatio: 0.114286 200 dict begin MathPictureStart /Courier findfont 10 scalefont setfont % Scaling calculations 0 1 0 1 [ [ 0 0 0 0 ] [ 315 36 0 0 ] ] MathScale % Start of Graphics %!PS-Adobe-2.0 EPSF-2.0 %%Title: %%Creator: EquationBuilder %%CreationDate: Sun Aug 2 14:03:08 1992 %%For: gray %%DocumentFonts: (atend) %%Pages: 0 1 %%BoundingBox: 0 0 315 36 %%Translation: 1 14.884993 313.180023 0 %%EndComments %%BeginDocument: /usr/lib/NextStep/printPackage.ps %!PS-Adobe-2.0 %%Title: Printing Package %%EndComments %Version: 2.0 %Copyright: 1988, NeXT, Inc. /__NXdef{1 index where{pop pop pop}{def}ifelse}bind def /__NXbdef{1 index where{pop pop pop}{bind def}ifelse}bind def /UserObjects 10 array __NXdef /defineuserobject{ exch dup 1 add dup UserObjects length gt{ array dup 0 UserObjects putinterval /UserObjects exch def }{pop}ifelse UserObjects exch 3 -1 roll put }__NXbdef /undefineuserobject{UserObjects exch null put}__NXbdef /execuserobject{UserObjects exch get exec}__NXbdef /__NXRectPath{4 2 roll moveto 1 index 0 rlineto 0 exch rlineto neg 0 rlineto closepath}__NXbdef /__NXProcessRectArgs{ 1 index type /arraytype eq{ exch 0 4 2 index length 1 sub{ dup 3 add 1 exch{1 index exch get exch}for 5 1 roll 5 index exec }for pop pop }{exec}ifelse }__NXbdef /rectfill{gsave newpath {__NXRectPath fill} __NXProcessRectArgs grestore}__NXbdef /rectclip{newpath {__NXRectPath} __NXProcessRectArgs clip newpath}__NXbdef /rectstroke{ gsave newpath dup type /arraytype eq{dup length 6 eq}{false}ifelse{ {gsave __NXRectPath null concat stroke grestore} dup length array cvx copy dup 2 4 -1 roll put __NXProcessRectArgs }{{__NXRectPath stroke} __NXProcessRectArgs}ifelse grestore }__NXbdef /xyshow{ 0 1 3 index length 1 sub{ currentpoint 4 index 3 index 1 getinterval show 3 index 3 index 2 mul 1 add get add exch 3 index 3 index 2 mul get add exch moveto pop }for pop pop }__NXbdef /xshow{ 0 1 3 index length 1 sub{ currentpoint 4 index 3 index 1 getinterval show exch 3 index 3 index get add exch moveto pop }for pop pop }__NXbdef /yshow{ 0 1 3 index length 1 sub{ currentpoint 4 index 3 index 1 getinterval show 3 index 3 index get add moveto pop }for pop pop }__NXbdef /arct{arcto pop pop pop pop}__NXbdef /setbbox{pop pop pop pop}__NXbdef /ucache{}__NXbdef /ucachestatus{mark 0 0 0 0 0}__NXbdef /setucacheparams{cleartomark}__NXbdef /uappend{systemdict begin cvx exec end}__NXbdef /ueofill{gsave newpath uappend eofill grestore}__NXbdef /ufill{gsave newpath uappend fill grestore}__NXbdef /ustroke{ gsave newpath dup length 6 eq {exch uappend concat}{uappend}ifelse stroke grestore }__NXbdef /__NXustrokepathMatrix dup where {pop pop}{matrix def}ifelse /ustrokepath{ newpath dup length 6 eq{ exch uappend __NXustrokepathMatrix currentmatrix exch concat strokepath setmatrix }{uappend strokepath}ifelse } __NXbdef /upath{ [exch {/ucache cvx}if pathbbox /setbbox cvx {/moveto cvx}{/lineto cvx}{/curveto cvx}{/closepath cvx}pathforall]cvx } __NXbdef /setstrokeadjust{pop}__NXbdef /currentstrokeadjust{false}__NXbdef /selectfont{exch findfont exch dup type /arraytype eq {makefont}{scalefont}ifelse setfont}__NXbdef /_NXCombineArrays{ counttomark dup 2 add index dup length 3 -1 roll { 2 index length sub dup 4 1 roll 1 index exch 4 -1 roll putinterval exch }repeat pop pop pop }__NXbdef /setcmykcolor{ 1.0 exch sub dup dup 6 -1 roll sub dup 0 lt{pop 0}if 5 1 roll 4 -1 roll sub dup 0 lt{pop 0}if 3 1 roll exch sub dup 0 lt{pop 0}if setrgbcolor }__NXbdef /currentcmykcolor{currentrgbcolor 3{1.0 exch sub 3 1 roll}repeat 0}__NXbdef /flushgraphics{}def /setwindowtype{pop pop}def /currentwindowtype{pop 0}def /setalpha{pop}def /currentalpha{1.0}def /hidecursor{}def /obscurecursor{}def /revealcursor{}def /setcursor{4 {pop}repeat}bind def /showcursor{}def /NextStepEncoding where not{ /NextStepEncoding StandardEncoding 256 array copy def 0 [129/Agrave/Aacute/Acircumflex/Atilde/Adieresis/Aring/Ccedilla/Egrave /Eacute/Ecircumflex/Edieresis/Igrave/Iacute/Icircumflex/Idieresis /Eth/Ntilde/Ograve/Oacute/Ocircumflex/Otilde/Odieresis/Ugrave/Uacute /Ucircumflex/Udieresis/Yacute/Thorn/mu/multiply/divide/copyright 176/registered 181/brokenbar 190/logicalnot 192/onesuperior 201/twosuperior 204/threesuperior 209/plusminus/onequarter/onehalf/threequarters/agrave /aacute/acircumflex/atilde/adieresis/aring/ccedilla/egrave/eacute /ecircumflex/edieresis/igrave 226/iacute 228/icircumflex/idieresis/eth /ntilde 236/ograve/oacute/ocircumflex/otilde/odieresis 242/ugrave/uacute /ucircumflex 246/udieresis/yacute 252/thorn/ydieresis] {dup type /nametype eq {NextStepEncoding 2 index 2 index put pop 1 add}{exch pop}ifelse }forall pop /NextStepEncoding NextStepEncoding readonly def /_NXfstr 128 string dup 0 (_NX) putinterval def /findfont{ % Because we can never let NextStepEncoding get into % SharedFontDirectory, we cannot reencode a font to NextStepEncoding % if we are in shared mode. So if currentshared is true, % we call the normal findfont and return that /currentshared where {pop currentshared} {false} ifelse {//findfont exec} {dup _NXfstr 3 125 getinterval cvs length 3 add _NXfstr 0 3 -1 roll getinterval cvn exch FontDirectory 2 index known {pop FontDirectory exch get} {//findfont exec dup /Encoding get StandardEncoding eq { dup length dict exch {1 index /FID ne {2 index 3 1 roll put}{pop pop}ifelse}forall dup /Encoding NextStepEncoding put definefont }{exch pop} ifelse }ifelse }ifelse }bind def }{pop}ifelse /_NXProcArray 5 array __NXdef /_NXChannels 0 __NXdef /_NXTotalBytes 0 __NXdef /_NXDoImageOp{ 1 index{dup}{1}ifelse /_NXChannels exch store _NXChannels 2 add 2 roll _NXProcArray 0 _NXChannels getinterval astore pop 5 index 4 index mul 2 index{1 sub 8 idiv 1 add mul}{mul 1 sub 8 idiv 1 add}ifelse 4 index mul /_NXTotalBytes exch store pop exch pop gsave matrix invertmatrix concat 0.5 setgray 0 0 4 2 roll rectfill grestore { 0 1 _NXChannels 1 sub{ _NXProcArray exch get exec length _NXTotalBytes exch sub /_NXTotalBytes exch store}for _NXTotalBytes 0 le{exit}if }loop /_NXProcArray 5 array def }__NXbdef /colorimage{_NXDoImageOp}__NXbdef /alphaimage{1 add _NXDoImageOp}def %%EndDocument % EquationBuilder(TM) Expression % +.\Jr(=:/*V[StRFQAYmdjEEA;1nuN+[co.OZ7L#cq3iK%3:</XJ@:d#6;^G;Td47dud<A3@ % 5%MGToUOR5&d`'TuEDU)B-tbR".R+AdS/,VT6!0M"_^5QDGdj@P.-O?%B4'FH)c.RcZ0TM0' % o;,/#l!%\^3YU,#41L18;Q+I_1"9NTgc2h<DdT6QGhQXOnBCNsA60Psq6cY@ca,dtJ@$1_&! % 4;jU?s($0"q;\\^+spWLJ/UF@H?K@]U6tkaI2sL!]?Y9Q5Q@1.[ZBsN>TL5)-7f0-?X:ZO?L % \i`KuYdTh@Zd\7@@F$AnjLSAGcV&0XomQh*'L4TP0BW4#3\iVd;L2K/jA*u1Bom.&K$"$hp= % \KO?$!<A"&qmdnB.$C)EaX[:HQ4b,TETG,eY_`TmIch<+*d0\'#i9'rg#l^m"(o&O6t^c=Td % 1f<GG0NinqUKZm1IbH&S82/7N6'gn^;E\J[eqb[`+?_MS3+_U4+kpoVn#rCM(JcoJXJ5l[ % ?"!F7I\Rh*-Ul[nZ?#I3;GHk*q,(9SKWSr$&J.bBaZDa_!M^6=;-6\`..HHpNQFPhDYB]6RS % HHn&*G!MFM;b*H3Z0?0e4^`FQW=<@V5@UGae^+gN`d].c)(4`Sr_#2B6+bA@"]M.i$9J-L^I % OZ%(UcfR5W4t0Zp,NFlR/)->qulY`]8"BV*0UU5F/r5c-*4`'W0"?\-4aN[3LQ16*XCGe!(D % K$"qX$r<15?988?W.5GWV7)h%N6uk7LoKe[d0_ak<Y2?ia[,7+!mH6I4lj2"@K<6LV[6l>Ms % )[sqiIlA_JPqP=iJSH]4%4\*"[grP&O*3\V-gfj*:l[nrLrT*^IdHC"+]LR=oBTX2,<2-C'A % j,J/KFbsk*RGIfsr[)B^#qH*kf.b_3S'.aqV4HNR)E7"]n5dJr]YA>d6Kk>[<#;3n6,MJJj> % 32$:CAB<VT8#t8+D*O@,1L%r]O<G>6H8cA$=Bk-GpQ9=aTmufi1J^"Q&mmI8PAXGC_O!s!mt % K[NpPnV?bG4$KLG\Sk_(mQb2^_62hNA)8CE)80sIg`lg<sq77*oC0eeNURleD&Dp2'g!`QlH % CDjmrLp,J0To6#DD#1i9k:9FpBsYjK$@L`b'=+W^3YQ=Kj\#1Al\\AP4nc/I#JVYNVG#?$po % <'Y4Vj2?+c-nrY_'A+PJ5C$M\_*&%i98A[<kn,I't/cp0&i=?$N7g4%#;Oga==O3".P]A/_c % ++MY[*lnKd(hq6ok"-TMMA>1;Q!`+B&ZpB_,]!tmab4iU&"DpSR.,f=V76/El;1pG,"98EF6 % pb8l1.l)O!QRYo-n^X7HWhGu.&]O'84&/bp`a(`W!N?eX,HAV+W=_M(Q'VK#^)6H*8VelH6M % H+4..:=qN;.]+CHdJ(ePkeO8oB]B)tuaO\/e"KuJ_]%I/L0,29=m+UEf28YAEfF>O3p-t&4' % ]VM&S't5:cTp.gP"9B]%L*1F:!1`P6+TMKBx % EndExpression N 1178 49a E 15 S 21069 R ae252ba2 gsave /__NXbasematrix matrix currentmatrix def grestore %%EndProlog %%BeginSetup /EqBFont 18 dict dup begin /FontType 3 def /PaintType 0 def /FontMatrix [0.001 0 0 0.001 0 0] def /FontBBox [15 -2950 1360 1045] def /Encoding [/.notdef /rad1 /rad2 /rad3 /rad4 /rad5 /radext /int /pathint /sum /prod /cdot /vdot /ddot /hbar /bigcap /bigcup /parallel] def /Metrics 18 dict dup begin /.notdef 0 def /rad1 1000 def /rad2 1000 def /rad3 1000 def /rad4 1000 def /rad5 1048 def /radext 1048 def /int 500 def /pathint 500 def /sum 700 def /prod 700 def /cdot 1000 def /vdot 550 def /ddot 1000 def /hbar 500 def /bigcap 700 def /bigcup 700 def /parallel 500 def end def /BBox 18 dict dup begin /.notdef [0 0 0 0] def /rad1 [110 -1150 1020 0] def /rad2 [110 -1750 1020 0] def /rad3 [110 -2350 1020 0] def /rad4 [110 -2950 1020 0] def /rad5 [128 -2890 740 0] def /radext [700 -600 740 0] def /int [20 0 425 1045] def /pathint [20 0 425 1045] def /sum [15 5 640 715] def /prod [15 0 670 715] def /cdot [110 215 890 325] def /vdot [165 -5 280 770] def /ddot [110 -5 890 770] def /hbar [20 -10 480 685] def /bigcap [0 0 680 720] def /bigcup [0 0 680 720] def /parallel [65 -175 400 675] def end def /CharacterDefs 18 dict dup begin /.notdef { } def /rad1 { 980 0 moveto 464 -1048 lineto 248 -560 lineto 110 -656 lineto 125 -675 lineto 200 -624 lineto 420 -1150 lineto 460 -1150 lineto 1020 0 lineto 980 0 lineto closepath } def /rad2 { 980 0 moveto 460 -1600 lineto 245 -864 lineto 110 -1000 lineto 128 -1023 lineto 200 -950 lineto 424 -1750 lineto 456 -1750 lineto 1020 0 lineto 980 0 lineto closepath } def /rad3 { 980 0 moveto 464 -2144 lineto 248 -1160 lineto 110 -1368 lineto 130 -1390 lineto 192 -1288 lineto 424 -2350 lineto 464 -2350 lineto 1020 0 lineto 980 0 lineto closepath } def /rad4 { 980 0 moveto 880 -488 lineto 672 -1600 lineto 464 -2696 lineto 248 -1464 lineto 110 -1728 lineto 130 -1750 lineto 192 -1616 lineto 424 -2950 lineto 464 -2950 lineto 768 -1312 lineto 1020 0 lineto 980 0 lineto closepath } def /rad5 { 700 0 moveto 700 -2628 lineto 248 -1284 lineto 128 -1564 lineto 150 -1580 lineto 215 -1425 lineto 688 -2890 lineto 740 -2890 lineto 740 0 lineto 700 0 lineto closepath } def /radext { 700 0 moveto 700 -600 lineto 740 -600 lineto 740 0 lineto 700 0 lineto closepath } def /int { 245 390 moveto 225 235 195 145 180 100 curveto 165 60 150 50 140 40 curveto 135 30 85 0 50 20 curveto 35 25 25 45 25 55 curveto 20 80 30 100 55 110 curveto 80 125 93 110 110 90 curveto 135 60 155 135 160 195 curveto 165 270 170 340 190 510 curveto 205 650 lineto 220 810 230 890 265 960 curveto 275 980 295 1015 320 1030 curveto 350 1045 380 1030 395 1020 curveto 410 1010 425 980 400 955 curveto 385 940 347 925 330 950 curveto 310 970 290 955 290 940 curveto 280 900 285 745 270 610 curveto 245 390 lineto closepath } def /pathint { 245 390 moveto 225 235 195 145 180 100 curveto 165 60 150 50 140 40 curveto 135 30 85 0 50 20 curveto 35 25 25 45 25 55 curveto 20 80 30 100 55 110 curveto 80 125 93 110 110 90 curveto 135 60 155 135 160 195 curveto 165 270 170 340 190 510 curveto 205 650 lineto 220 810 230 890 265 960 curveto 275 980 295 1015 320 1030 curveto 350 1045 380 1030 395 1020 curveto 410 1010 425 980 400 955 curveto 385 940 347 925 330 950 curveto 310 970 290 955 290 940 curveto 280 900 285 745 270 610 curveto 245 390 lineto fill 20 setlinewidth 220 400 moveto 290 400 340 455 340 500 curveto 340 585 290 640 220 640 curveto 155 640 100 585 100 500 curveto 100 455 155 400 220 400 curveto stroke } def /sum { 330 335 moveto 20 715 lineto 575 715 lineto 600 545 lineto 575 535 lineto 550 625 525 685 395 685 curveto 160 685 lineto 411 370 lineto 160 90 lineto 470 90 lineto 565 90 590 140 610 210 curveto 640 200 lineto 585 5 lineto 15 5 lineto 330 335 lineto closepath } def /prod { 590 90 moveto 590 30 620 20 670 20 curveto 670 0 lineto 410 0 lineto 410 20 lineto 460 20 495 30 495 90 curveto 495 670 lineto 190 670 lineto 190 90 lineto 190 30 225 20 275 20 curveto 275 0 lineto 15 0 lineto 15 20 lineto 65 20 100 30 100 90 curveto 100 615 lineto 100 690 40 700 20 700 curveto 20 715 lineto 670 715 lineto 670 700 lineto 645 700 590 690 590 615 curveto 590 90 lineto closepath } def /cdot { 110 270 moveto 110 300 135 325 165 325 curveto 195 325 220 300 220 270 curveto 220 235 185 215 165 215 curveto 145 215 110 235 110 270 curveto fill 445 270 moveto 445 300 470 325 500 325 curveto 530 325 555 300 555 270 curveto 555 235 520 215 500 215 curveto 480 215 445 235 445 270 curveto fill 775 270 moveto 775 300 800 325 830 325 curveto 860 325 890 300 890 270 curveto 890 235 850 215 830 215 curveto 810 215 775 235 775 270 curveto fill } def /vdot { 220 770 moveto 250 770 280 750 280 720 curveto 280 680 250 660 220 660 curveto 185 660 165 695 165 720 curveto 165 740 185 770 220 770 curveto fill 220 440 moveto 250 440 280 415 280 385 curveto 280 355 250 330 220 330 curveto 185 330 165 365 165 385 curveto 165 405 185 440 220 440 curveto fill 220 105 moveto 250 105 280 80 280 50 curveto 280 20 250 -5 220 -5 curveto 185 -5 165 30 165 50 curveto 165 70 185 105 220 105 curveto fill } def /ddot { 165 720 moveto 205 720 220 700 220 665 curveto 220 630 195 610 165 610 curveto 130 620 110 645 110 665 curveto 110 690 130 720 165 720 curveto fill 500 440 moveto 530 440 555 415 555 385 curveto 555 355 530 330 500 330 curveto 465 330 445 365 445 385 curveto 445 405 465 440 500 440 curveto fill 830 155 moveto 860 155 890 130 890 100 curveto 890 70 860 45 830 45 curveto 795 45 775 80 775 100 curveto 775 120 795 155 830 155 curveto fill } def /hbar { 480 105 moveto 440 55 400 -10 335 -10 curveto 300 -10 290 10 290 35 curveto 290 65 370 350 370 365 curveto 370 380 355 390 340 390 curveto 290 390 205 275 185 250 curveto 140 180 125 125 95 0 curveto 20 0 lineto 150 500 lineto 160 545 180 600 180 615 curveto 180 625 170 640 135 640 curveto 110 640 lineto 100 655 lineto 165 660 215 670 270 685 curveto 275 680 lineto 155 240 lineto 155 235 lineto 190 290 290 440 380 440 curveto 400 440 450 435 450 370 curveto 450 335 365 80 365 55 curveto 365 45 375 40 385 40 curveto 405 40 425 70 465 115 curveto 480 105 lineto fill 55 545 moveto 55 520 lineto 310 560 lineto 310 585 lineto 55 545 lineto fill } def /bigcap { 615 400 moveto 615 575 500 655 340 655 curveto 180 655 70 575 70 400 curveto 70 0 lineto 5 0 lineto 5 400 lineto 5 610 140 720 340 720 curveto 545 720 680 610 680 400 curveto 680 0 lineto 615 0 lineto 615 400 lineto closepath } def /bigcup { 65 320 moveto 65 145 180 65 340 65 curveto 500 65 615 145 615 320 curveto 615 720 lineto 680 720 lineto 680 320 lineto 680 110 540 0 340 0 curveto 140 0 0 110 0 320 curveto 0 720 lineto 65 720 lineto 65 320 lineto closepath } def /parallel { 135 675 moveto 135 -175 lineto 65 -175 lineto 65 675 lineto 135 675 lineto fill 335 675 moveto 335 -175 lineto 265 -175 lineto 265 675 lineto 335 675 lineto fill 400 675 moveto } def end def /BuildChar { 0 begin /char exch def /fontdict exch def /charname fontdict /Encoding get char get def fontdict begin Metrics charname get 0 BBox charname get aload pop setcachedevice CharacterDefs charname get exec eofill end end } def /BuildChar load 0 3 dict put end definefont pop %%EndSetup 0 0 314.180023 35.640968 rectclip 0 0 314.180023 35.640968 rectclip 0 setgray 1 setalpha 1 1 scale /Symbol findfont 12 scalefont 54 exch defineuserobject 54 execuserobject setfont 39.106667 15.992992 moveto (\() [0 0] xyshow 1 1 scale 1 1 scale 54 execuserobject setfont 51.334667 15.986992 moveto (\)) [0 0] xyshow 1 1 scale 0 setgray 1 setalpha 1 1 scale 54 execuserobject setfont 79.67334 15.992992 moveto (\() [0 0] xyshow 1 1 scale 1 1 scale 54 execuserobject setfont 91.901337 15.986992 moveto (\)) [0 0] xyshow 1 1 scale 0 setgray 1 setalpha 0.48 setlinewidth 109.680008 18.884993 moveto 122.96801 18.884993 lineto stroke 0 setgray 1 setalpha /EqBFont 24 selectfont 135.700836 10.244992 moveto <09> [0 0] xyshow 0 setgray 1 setalpha 1 1 scale 54 execuserobject setfont 161.296341 23.792992 moveto (\() [0 0] xyshow 1 1 scale 1 1 scale 54 execuserobject setfont 182.19101 23.786991 moveto (\)) [0 0] xyshow 1 1 scale 0 setgray 1 setalpha 0.48 setlinewidth 154.232346 18.884993 moveto 195.417023 18.884993 lineto stroke 0 setgray 1 setalpha 0.48 setlinewidth 244.591675 18.884993 moveto 257.879669 18.884993 lineto stroke 0 setgray 1 setalpha /EqBFont 24 selectfont 278.612488 10.244992 moveto <09> [0 0] xyshow 0 setgray 1 setalpha 0.48 setlinewidth 297.144012 18.884993 moveto 313.180023 18.884993 lineto stroke 0 setgray 1 setalpha 54 execuserobject setfont 260.546356 15.884993 moveto (+p=p-+p=p=p=p) [-15.454681 7.799999 -10.421341 -7.799999 -9.921326 0 -58.652664 7.799999 -40.46167 -7.799999 -15.454666 7.799999 -10.421333 -7.799999 -15.285339 -9.536743e-07 -25.281334 9.536743e-07 -15.285332 -9.536743e-07 -27.285336 9.536743e-07 -15.621334 0 0 0 0 0] xyshow /Times-Roman findfont 9 scalefont 46 exch defineuserobject 46 execuserobject setfont 306.980011 11.120993 moveto (2122122) [-16.336029 -8.839993 -38.96431 25.758696 -75.036987 -16.918703 -28.910339 -8.839993 -30.96434 25.758696 -109.180008 -7.199999 0 0 0 0] xyshow /Symbol findfont 9 scalefont 49 exch defineuserobject 49 execuserobject setfont 285.252991 2.281 moveto (=\245=\245) [-1.988983 27.523968 -140.922668 -27.523968 -1.988998 27.523968 0 0 0 0] xyshow /Times-BoldItalic findfont 12 scalefont 51 exch defineuserobject 51 execuserobject setfont 297.644012 6.766289 moveto (mmmfg) [-82.231003 9.118704 -48.106323 -9.118704 -98.193352 9.118704 -42.570667 0 0 0 0 0] xyshow /Times-Roman findfont 12 scalefont 50 exch defineuserobject 50 execuserobject setfont 302.162018 23.684992 moveto (143cos14311) [-32.361023 -7.799999 -21.565323 -8.231412 -50.818665 8.231412 5.328003 0 6 0 -33.393997 7.799999 -20.618668 0 -41.40834 -16.03141 -40.214668 8.231412 -40.566669 0 0 0 0 0] xyshow /Times-BoldItalic findfont 9 scalefont 47 exch defineuserobject 47 execuserobject setfont 277.800995 2.281 moveto (mmm) [-91.085983 26.119019 -51.825668 -26.119019 0 0 0 0] xyshow %%Trailer %%DocumentFonts: Times-BoldItalic %%+ Times-Roman %%+ Symbol %End of Graphics MathPictureEnd end :[font = text; inactive; preserveAspect; ] This equation and a little elementary algebra show that :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 0; pictureWidth = 69; pictureHeight = 36; ] %! %%Creator: Mathematica %%AspectRatio: 0.521739 200 dict begin MathPictureStart /Courier findfont 10 scalefont setfont % Scaling calculations 0 1 0 1 [ [ 0 0 0 0 ] [ 69 36 0 0 ] ] MathScale % Start of Graphics %!PS-Adobe-2.0 EPSF-2.0 %%Title: %%Creator: EquationBuilder %%CreationDate: Sun Aug 2 14:06:35 1992 %%For: gray %%DocumentFonts: (atend) %%Pages: 0 1 %%BoundingBox: 0 0 69 36 %%Translation: 1 14.884993 67.921661 0 %%EndComments %%BeginDocument: /usr/lib/NextStep/printPackage.ps %!PS-Adobe-2.0 %%Title: Printing Package %%EndComments %Version: 2.0 %Copyright: 1988, NeXT, Inc. /__NXdef{1 index where{pop pop pop}{def}ifelse}bind def /__NXbdef{1 index where{pop pop pop}{bind def}ifelse}bind def /UserObjects 10 array __NXdef /defineuserobject{ exch dup 1 add dup UserObjects length gt{ array dup 0 UserObjects putinterval /UserObjects exch def }{pop}ifelse UserObjects exch 3 -1 roll put }__NXbdef /undefineuserobject{UserObjects exch null put}__NXbdef /execuserobject{UserObjects exch get exec}__NXbdef /__NXRectPath{4 2 roll moveto 1 index 0 rlineto 0 exch rlineto neg 0 rlineto closepath}__NXbdef /__NXProcessRectArgs{ 1 index type /arraytype eq{ exch 0 4 2 index length 1 sub{ dup 3 add 1 exch{1 index exch get exch}for 5 1 roll 5 index exec }for pop pop }{exec}ifelse }__NXbdef /rectfill{gsave newpath {__NXRectPath fill} __NXProcessRectArgs grestore}__NXbdef /rectclip{newpath {__NXRectPath} __NXProcessRectArgs clip newpath}__NXbdef /rectstroke{ gsave newpath dup type /arraytype eq{dup length 6 eq}{false}ifelse{ {gsave __NXRectPath null concat stroke grestore} dup length array cvx copy dup 2 4 -1 roll put __NXProcessRectArgs }{{__NXRectPath stroke} __NXProcessRectArgs}ifelse grestore }__NXbdef /xyshow{ 0 1 3 index length 1 sub{ currentpoint 4 index 3 index 1 getinterval show 3 index 3 index 2 mul 1 add get add exch 3 index 3 index 2 mul get add exch moveto pop }for pop pop }__NXbdef /xshow{ 0 1 3 index length 1 sub{ currentpoint 4 index 3 index 1 getinterval show exch 3 index 3 index get add exch moveto pop }for pop pop }__NXbdef /yshow{ 0 1 3 index length 1 sub{ currentpoint 4 index 3 index 1 getinterval show 3 index 3 index get add moveto pop }for pop pop }__NXbdef /arct{arcto pop pop pop pop}__NXbdef /setbbox{pop pop pop pop}__NXbdef /ucache{}__NXbdef /ucachestatus{mark 0 0 0 0 0}__NXbdef /setucacheparams{cleartomark}__NXbdef /uappend{systemdict begin cvx exec end}__NXbdef /ueofill{gsave newpath uappend eofill grestore}__NXbdef /ufill{gsave newpath uappend fill grestore}__NXbdef /ustroke{ gsave newpath dup length 6 eq {exch uappend concat}{uappend}ifelse stroke grestore }__NXbdef /__NXustrokepathMatrix dup where {pop pop}{matrix def}ifelse /ustrokepath{ newpath dup length 6 eq{ exch uappend __NXustrokepathMatrix currentmatrix exch concat strokepath setmatrix }{uappend strokepath}ifelse } __NXbdef /upath{ [exch {/ucache cvx}if pathbbox /setbbox cvx {/moveto cvx}{/lineto cvx}{/curveto cvx}{/closepath cvx}pathforall]cvx } __NXbdef /setstrokeadjust{pop}__NXbdef /currentstrokeadjust{false}__NXbdef /selectfont{exch findfont exch dup type /arraytype eq {makefont}{scalefont}ifelse setfont}__NXbdef /_NXCombineArrays{ counttomark dup 2 add index dup length 3 -1 roll { 2 index length sub dup 4 1 roll 1 index exch 4 -1 roll putinterval exch }repeat pop pop pop }__NXbdef /setcmykcolor{ 1.0 exch sub dup dup 6 -1 roll sub dup 0 lt{pop 0}if 5 1 roll 4 -1 roll sub dup 0 lt{pop 0}if 3 1 roll exch sub dup 0 lt{pop 0}if setrgbcolor }__NXbdef /currentcmykcolor{currentrgbcolor 3{1.0 exch sub 3 1 roll}repeat 0}__NXbdef /flushgraphics{}def /setwindowtype{pop pop}def /currentwindowtype{pop 0}def /setalpha{pop}def /currentalpha{1.0}def /hidecursor{}def /obscurecursor{}def /revealcursor{}def /setcursor{4 {pop}repeat}bind def /showcursor{}def /NextStepEncoding where not{ /NextStepEncoding StandardEncoding 256 array copy def 0 [129/Agrave/Aacute/Acircumflex/Atilde/Adieresis/Aring/Ccedilla/Egrave /Eacute/Ecircumflex/Edieresis/Igrave/Iacute/Icircumflex/Idieresis /Eth/Ntilde/Ograve/Oacute/Ocircumflex/Otilde/Odieresis/Ugrave/Uacute /Ucircumflex/Udieresis/Yacute/Thorn/mu/multiply/divide/copyright 176/registered 181/brokenbar 190/logicalnot 192/onesuperior 201/twosuperior 204/threesuperior 209/plusminus/onequarter/onehalf/threequarters/agrave /aacute/acircumflex/atilde/adieresis/aring/ccedilla/egrave/eacute /ecircumflex/edieresis/igrave 226/iacute 228/icircumflex/idieresis/eth /ntilde 236/ograve/oacute/ocircumflex/otilde/odieresis 242/ugrave/uacute /ucircumflex 246/udieresis/yacute 252/thorn/ydieresis] {dup type /nametype eq {NextStepEncoding 2 index 2 index put pop 1 add}{exch pop}ifelse }forall pop /NextStepEncoding NextStepEncoding readonly def /_NXfstr 128 string dup 0 (_NX) putinterval def /findfont{ % Because we can never let NextStepEncoding get into % SharedFontDirectory, we cannot reencode a font to NextStepEncoding % if we are in shared mode. So if currentshared is true, % we call the normal findfont and return that /currentshared where {pop currentshared} {false} ifelse {//findfont exec} {dup _NXfstr 3 125 getinterval cvs length 3 add _NXfstr 0 3 -1 roll getinterval cvn exch FontDirectory 2 index known {pop FontDirectory exch get} {//findfont exec dup /Encoding get StandardEncoding eq { dup length dict exch {1 index /FID ne {2 index 3 1 roll put}{pop pop}ifelse}forall dup /Encoding NextStepEncoding put definefont }{exch pop} ifelse }ifelse }ifelse }bind def }{pop}ifelse /_NXProcArray 5 array __NXdef /_NXChannels 0 __NXdef /_NXTotalBytes 0 __NXdef /_NXDoImageOp{ 1 index{dup}{1}ifelse /_NXChannels exch store _NXChannels 2 add 2 roll _NXProcArray 0 _NXChannels getinterval astore pop 5 index 4 index mul 2 index{1 sub 8 idiv 1 add mul}{mul 1 sub 8 idiv 1 add}ifelse 4 index mul /_NXTotalBytes exch store pop exch pop gsave matrix invertmatrix concat 0.5 setgray 0 0 4 2 roll rectfill grestore { 0 1 _NXChannels 1 sub{ _NXProcArray exch get exec length _NXTotalBytes exch sub /_NXTotalBytes exch store}for _NXTotalBytes 0 le{exit}if }loop /_NXProcArray 5 array def }__NXbdef /colorimage{_NXDoImageOp}__NXbdef /alphaimage{1 add _NXDoImageOp}def %%EndDocument % EquationBuilder(TM) Expression % +.\Jr(=:/*V[StRFQAYmdjEEA;1nuN+[co.OZ7L#cq3iK%3:</XJ@:d#6;^G;Td47dud<A3@ % 5%MGToUOR5&d`'TuEDU)B-tbR".R+HXkhQiI+',c#t=5sa-OjiN\t!?cIG!%AUQ!2,^4&3uZ % 37&-8/GQ:hi<+&?-).pGk:c_Z,!!i_i7[sRrO<f]lS%N0NZOYrT<eJbEdtWl/8WY(t-GUl9_ % >n?RUa]\o9+<"L!&H;Q-O`b1DA4n(ZRKpl^R[UrkUZhCkU1DlUi@&;:*.rF>=8)e)q9D`1Mf % uh0I9L60nY$%7PFs>Ds>+bKdtrGO8ttH5o$n?14T^h!C3GI&'#6T;N\F%MVT(Kp]0C,X'-TB % LAR0'ZiS34d[:^iGD=`TJ9^l%ZCccB#bi"VcPVi'IJLi+#q;%9,)IhCF9g]B#V$)eIK8TP?N % ;udBHi>5e8jPuKJ8K#)]GDNTPGj.2dV@OY1Q*t1c"Pg6)?.[#Bi5T-eg'<7E.u]$i[(KHDS- % LM3Psg:eHt6%7B66%6bV+)t)YtR@$b#k@9qG_*TE7P-@O8KaYa7,^M2TLgc1eJF\bnH04i0+ % TnO6VuTH#pIURR#ls(CP!h+'Se'9/Sg]_(lD0bg??7QcKZrEsV4tDT-^d4`aGP>J7$d,/-<d % sX.R^EHj#E34,=shWKTk_[q2I0TSZf"bCo](t5U`\4S0Ot>J130?*HKBo`fc5`(YMR+ri)Ms % R=Kkh>C-e/#6P8L^u#2mUsXoh/P*][(iJgJ6+lg7d=Mp^Oc]be@Bbm6)#Phu0>.@l5t6M.!" % ^m'!!E<*`>2fI5X])b_$L+3pgXR0x % EndExpression N 655 28f E 16 S 11358 R 63ce791b gsave /__NXbasematrix matrix currentmatrix def grestore %%EndProlog %%BeginSetup /EqBFont 18 dict dup begin /FontType 3 def /PaintType 0 def /FontMatrix [0.001 0 0 0.001 0 0] def /FontBBox [15 -2950 1360 1045] def /Encoding [/.notdef /rad1 /rad2 /rad3 /rad4 /rad5 /radext /int /pathint /sum /prod /cdot /vdot /ddot /hbar /bigcap /bigcup /parallel] def /Metrics 18 dict dup begin /.notdef 0 def /rad1 1000 def /rad2 1000 def /rad3 1000 def /rad4 1000 def /rad5 1048 def /radext 1048 def /int 500 def /pathint 500 def /sum 700 def /prod 700 def /cdot 1000 def /vdot 550 def /ddot 1000 def /hbar 500 def /bigcap 700 def /bigcup 700 def /parallel 500 def end def /BBox 18 dict dup begin /.notdef [0 0 0 0] def /rad1 [110 -1150 1020 0] def /rad2 [110 -1750 1020 0] def /rad3 [110 -2350 1020 0] def /rad4 [110 -2950 1020 0] def /rad5 [128 -2890 740 0] def /radext [700 -600 740 0] def /int [20 0 425 1045] def /pathint [20 0 425 1045] def /sum [15 5 640 715] def /prod [15 0 670 715] def /cdot [110 215 890 325] def /vdot [165 -5 280 770] def /ddot [110 -5 890 770] def /hbar [20 -10 480 685] def /bigcap [0 0 680 720] def /bigcup [0 0 680 720] def /parallel [65 -175 400 675] def end def /CharacterDefs 18 dict dup begin /.notdef { } def /rad1 { 980 0 moveto 464 -1048 lineto 248 -560 lineto 110 -656 lineto 125 -675 lineto 200 -624 lineto 420 -1150 lineto 460 -1150 lineto 1020 0 lineto 980 0 lineto closepath } def /rad2 { 980 0 moveto 460 -1600 lineto 245 -864 lineto 110 -1000 lineto 128 -1023 lineto 200 -950 lineto 424 -1750 lineto 456 -1750 lineto 1020 0 lineto 980 0 lineto closepath } def /rad3 { 980 0 moveto 464 -2144 lineto 248 -1160 lineto 110 -1368 lineto 130 -1390 lineto 192 -1288 lineto 424 -2350 lineto 464 -2350 lineto 1020 0 lineto 980 0 lineto closepath } def /rad4 { 980 0 moveto 880 -488 lineto 672 -1600 lineto 464 -2696 lineto 248 -1464 lineto 110 -1728 lineto 130 -1750 lineto 192 -1616 lineto 424 -2950 lineto 464 -2950 lineto 768 -1312 lineto 1020 0 lineto 980 0 lineto closepath } def /rad5 { 700 0 moveto 700 -2628 lineto 248 -1284 lineto 128 -1564 lineto 150 -1580 lineto 215 -1425 lineto 688 -2890 lineto 740 -2890 lineto 740 0 lineto 700 0 lineto closepath } def /radext { 700 0 moveto 700 -600 lineto 740 -600 lineto 740 0 lineto 700 0 lineto closepath } def /int { 245 390 moveto 225 235 195 145 180 100 curveto 165 60 150 50 140 40 curveto 135 30 85 0 50 20 curveto 35 25 25 45 25 55 curveto 20 80 30 100 55 110 curveto 80 125 93 110 110 90 curveto 135 60 155 135 160 195 curveto 165 270 170 340 190 510 curveto 205 650 lineto 220 810 230 890 265 960 curveto 275 980 295 1015 320 1030 curveto 350 1045 380 1030 395 1020 curveto 410 1010 425 980 400 955 curveto 385 940 347 925 330 950 curveto 310 970 290 955 290 940 curveto 280 900 285 745 270 610 curveto 245 390 lineto closepath } def /pathint { 245 390 moveto 225 235 195 145 180 100 curveto 165 60 150 50 140 40 curveto 135 30 85 0 50 20 curveto 35 25 25 45 25 55 curveto 20 80 30 100 55 110 curveto 80 125 93 110 110 90 curveto 135 60 155 135 160 195 curveto 165 270 170 340 190 510 curveto 205 650 lineto 220 810 230 890 265 960 curveto 275 980 295 1015 320 1030 curveto 350 1045 380 1030 395 1020 curveto 410 1010 425 980 400 955 curveto 385 940 347 925 330 950 curveto 310 970 290 955 290 940 curveto 280 900 285 745 270 610 curveto 245 390 lineto fill 20 setlinewidth 220 400 moveto 290 400 340 455 340 500 curveto 340 585 290 640 220 640 curveto 155 640 100 585 100 500 curveto 100 455 155 400 220 400 curveto stroke } def /sum { 330 335 moveto 20 715 lineto 575 715 lineto 600 545 lineto 575 535 lineto 550 625 525 685 395 685 curveto 160 685 lineto 411 370 lineto 160 90 lineto 470 90 lineto 565 90 590 140 610 210 curveto 640 200 lineto 585 5 lineto 15 5 lineto 330 335 lineto closepath } def /prod { 590 90 moveto 590 30 620 20 670 20 curveto 670 0 lineto 410 0 lineto 410 20 lineto 460 20 495 30 495 90 curveto 495 670 lineto 190 670 lineto 190 90 lineto 190 30 225 20 275 20 curveto 275 0 lineto 15 0 lineto 15 20 lineto 65 20 100 30 100 90 curveto 100 615 lineto 100 690 40 700 20 700 curveto 20 715 lineto 670 715 lineto 670 700 lineto 645 700 590 690 590 615 curveto 590 90 lineto closepath } def /cdot { 110 270 moveto 110 300 135 325 165 325 curveto 195 325 220 300 220 270 curveto 220 235 185 215 165 215 curveto 145 215 110 235 110 270 curveto fill 445 270 moveto 445 300 470 325 500 325 curveto 530 325 555 300 555 270 curveto 555 235 520 215 500 215 curveto 480 215 445 235 445 270 curveto fill 775 270 moveto 775 300 800 325 830 325 curveto 860 325 890 300 890 270 curveto 890 235 850 215 830 215 curveto 810 215 775 235 775 270 curveto fill } def /vdot { 220 770 moveto 250 770 280 750 280 720 curveto 280 680 250 660 220 660 curveto 185 660 165 695 165 720 curveto 165 740 185 770 220 770 curveto fill 220 440 moveto 250 440 280 415 280 385 curveto 280 355 250 330 220 330 curveto 185 330 165 365 165 385 curveto 165 405 185 440 220 440 curveto fill 220 105 moveto 250 105 280 80 280 50 curveto 280 20 250 -5 220 -5 curveto 185 -5 165 30 165 50 curveto 165 70 185 105 220 105 curveto fill } def /ddot { 165 720 moveto 205 720 220 700 220 665 curveto 220 630 195 610 165 610 curveto 130 620 110 645 110 665 curveto 110 690 130 720 165 720 curveto fill 500 440 moveto 530 440 555 415 555 385 curveto 555 355 530 330 500 330 curveto 465 330 445 365 445 385 curveto 445 405 465 440 500 440 curveto fill 830 155 moveto 860 155 890 130 890 100 curveto 890 70 860 45 830 45 curveto 795 45 775 80 775 100 curveto 775 120 795 155 830 155 curveto fill } def /hbar { 480 105 moveto 440 55 400 -10 335 -10 curveto 300 -10 290 10 290 35 curveto 290 65 370 350 370 365 curveto 370 380 355 390 340 390 curveto 290 390 205 275 185 250 curveto 140 180 125 125 95 0 curveto 20 0 lineto 150 500 lineto 160 545 180 600 180 615 curveto 180 625 170 640 135 640 curveto 110 640 lineto 100 655 lineto 165 660 215 670 270 685 curveto 275 680 lineto 155 240 lineto 155 235 lineto 190 290 290 440 380 440 curveto 400 440 450 435 450 370 curveto 450 335 365 80 365 55 curveto 365 45 375 40 385 40 curveto 405 40 425 70 465 115 curveto 480 105 lineto fill 55 545 moveto 55 520 lineto 310 560 lineto 310 585 lineto 55 545 lineto fill } def /bigcap { 615 400 moveto 615 575 500 655 340 655 curveto 180 655 70 575 70 400 curveto 70 0 lineto 5 0 lineto 5 400 lineto 5 610 140 720 340 720 curveto 545 720 680 610 680 400 curveto 680 0 lineto 615 0 lineto 615 400 lineto closepath } def /bigcup { 65 320 moveto 65 145 180 65 340 65 curveto 500 65 615 145 615 320 curveto 615 720 lineto 680 720 lineto 680 320 lineto 680 110 540 0 340 0 curveto 140 0 0 110 0 320 curveto 0 720 lineto 65 720 lineto 65 320 lineto closepath } def /parallel { 135 675 moveto 135 -175 lineto 65 -175 lineto 65 675 lineto 135 675 lineto fill 335 675 moveto 335 -175 lineto 265 -175 lineto 265 675 lineto 335 675 lineto fill 400 675 moveto } def end def /BuildChar { 0 begin /char exch def /fontdict exch def /charname fontdict /Encoding get char get def fontdict begin Metrics charname get 0 BBox charname get aload pop setcachedevice CharacterDefs charname get exec eofill end end } def /BuildChar load 0 3 dict put end definefont pop %%EndSetup 0 0 68.921661 35.640968 rectclip 0 0 68.921661 35.640968 rectclip 0 setgray 1 setalpha 0.48 setlinewidth 1 18.884993 moveto 14.288 18.884993 lineto stroke 0 setgray 1 setalpha /EqBFont 24 selectfont 28.354166 10.244992 moveto <09> [0 0] xyshow 0 setgray 1 setalpha 0.48 setlinewidth 46.885666 18.884993 moveto 62.921665 18.884993 lineto stroke 0 setgray 1 setalpha /Symbol findfont 12 scalefont 54 exch defineuserobject 54 execuserobject setfont 17.621334 15.884993 moveto (=p) [-16.121334 7.799999 0 0 0 0] xyshow /Times-Roman findfont 9 scalefont 46 exch defineuserobject 46 execuserobject setfont 56.721664 11.120993 moveto (212) [-16.335999 -8.839993 -32.297665 25.758696 0 0 0 0] xyshow /Symbol findfont 9 scalefont 49 exch defineuserobject 49 execuserobject setfont 34.994667 2.281 moveto (=\245) [-1.988998 27.523968 0 0 0 0] xyshow /Times-Roman findfont 12 scalefont 50 exch defineuserobject 50 execuserobject setfont 51.903664 23.684992 moveto (16) [-47.259663 -16.03141 0 0 0 0] xyshow /Times-BoldItalic findfont 9 scalefont 47 exch defineuserobject 47 execuserobject setfont 27.542667 2.281 moveto (m) [0 0 -51.825668 -26.119019] xyshow /Times-BoldItalic findfont 12 scalefont 51 exch defineuserobject 51 execuserobject setfont 64.921661 15.884993 moveto (.m) [-17.535995 -9.118704 0 0 0 0] xyshow %%Trailer %%DocumentFonts: Times-BoldItalic %%+ Times-Roman %%+ Symbol %End of Graphics MathPictureEnd end :[font = text; inactive; preserveAspect; ] There are several ways to use Mathematica to illustrate this result. The Mathematica command N means "find the numerical value of". Thus ;[s] 7:0,0;30,1;41,2;73,3;84,4;94,5;96,6;140,-1; 7:1,11,8,Times,0,12,0,0,0;1,10,8,Times,2,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Times,2,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Courier,0,12,0,0,0;1,11,8,Times,0,12,0,0,0; :[font = input; preserveAspect; startGroup; ] N[Pi^2/6] :[font = output; output; inactive; pageBreakBelow; dontNoPageBreakBelow; preserveAspect; endGroup; ] 1.644934066848226 ;[o] 1.64493 :[font = text; inactive; preserveAspect; ] Similarly, :[font = input; preserveAspect; startGroup; ] N[Sum[1/m^2,{m,1,20}]] :[font = output; output; inactive; preserveAspect; endGroup; ] 1.596163243913023 ;[o] 1.59616 :[font = input; preserveAspect; startGroup; ] N[Sum[1/m^2,{m,1,40}]] :[font = output; output; inactive; preserveAspect; endGroup; ] 1.620243963006935 ;[o] 1.62024 :[font = text; inactive; preserveAspect; ] So the partial sums :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 0; pictureWidth = 38; pictureHeight = 37; ] %! %%Creator: Mathematica %%AspectRatio: 0.973684 200 dict begin MathPictureStart /Courier findfont 10 scalefont setfont % Scaling calculations 0 1 0 1 [ [ 0 0 0 0 ] [ 38 37 0 0 ] ] MathScale % Start of Graphics %!PS-Adobe-2.0 EPSF-2.0 %%Title: %%Creator: EquationBuilder %%CreationDate: Sun Aug 2 14:15:00 1992 %%For: gray %%DocumentFonts: (atend) %%Pages: 0 1 %%BoundingBox: 0 0 38 37 %%Translation: 1 14.884993 36.378998 0 %%EndComments %%BeginDocument: /usr/lib/NextStep/printPackage.ps %!PS-Adobe-2.0 %%Title: Printing Package %%EndComments %Version: 2.0 %Copyright: 1988, NeXT, Inc. /__NXdef{1 index where{pop pop pop}{def}ifelse}bind def /__NXbdef{1 index where{pop pop pop}{bind def}ifelse}bind def /UserObjects 10 array __NXdef /defineuserobject{ exch dup 1 add dup UserObjects length gt{ array dup 0 UserObjects putinterval /UserObjects exch def }{pop}ifelse UserObjects exch 3 -1 roll put }__NXbdef /undefineuserobject{UserObjects exch null put}__NXbdef /execuserobject{UserObjects exch get exec}__NXbdef /__NXRectPath{4 2 roll moveto 1 index 0 rlineto 0 exch rlineto neg 0 rlineto closepath}__NXbdef /__NXProcessRectArgs{ 1 index type /arraytype eq{ exch 0 4 2 index length 1 sub{ dup 3 add 1 exch{1 index exch get exch}for 5 1 roll 5 index exec }for pop pop }{exec}ifelse }__NXbdef /rectfill{gsave newpath {__NXRectPath fill} __NXProcessRectArgs grestore}__NXbdef /rectclip{newpath {__NXRectPath} __NXProcessRectArgs clip newpath}__NXbdef /rectstroke{ gsave newpath dup type /arraytype eq{dup length 6 eq}{false}ifelse{ {gsave __NXRectPath null concat stroke grestore} dup length array cvx copy dup 2 4 -1 roll put __NXProcessRectArgs }{{__NXRectPath stroke} __NXProcessRectArgs}ifelse grestore }__NXbdef /xyshow{ 0 1 3 index length 1 sub{ currentpoint 4 index 3 index 1 getinterval show 3 index 3 index 2 mul 1 add get add exch 3 index 3 index 2 mul get add exch moveto pop }for pop pop }__NXbdef /xshow{ 0 1 3 index length 1 sub{ currentpoint 4 index 3 index 1 getinterval show exch 3 index 3 index get add exch moveto pop }for pop pop }__NXbdef /yshow{ 0 1 3 index length 1 sub{ currentpoint 4 index 3 index 1 getinterval show 3 index 3 index get add moveto pop }for pop pop }__NXbdef /arct{arcto pop pop pop pop}__NXbdef /setbbox{pop pop pop pop}__NXbdef /ucache{}__NXbdef /ucachestatus{mark 0 0 0 0 0}__NXbdef /setucacheparams{cleartomark}__NXbdef /uappend{systemdict begin cvx exec end}__NXbdef /ueofill{gsave newpath uappend eofill grestore}__NXbdef /ufill{gsave newpath uappend fill grestore}__NXbdef /ustroke{ gsave newpath dup length 6 eq {exch uappend concat}{uappend}ifelse stroke grestore }__NXbdef /__NXustrokepathMatrix dup where {pop pop}{matrix def}ifelse /ustrokepath{ newpath dup length 6 eq{ exch uappend __NXustrokepathMatrix currentmatrix exch concat strokepath setmatrix }{uappend strokepath}ifelse } __NXbdef /upath{ [exch {/ucache cvx}if pathbbox /setbbox cvx {/moveto cvx}{/lineto cvx}{/curveto cvx}{/closepath cvx}pathforall]cvx } __NXbdef /setstrokeadjust{pop}__NXbdef /currentstrokeadjust{false}__NXbdef /selectfont{exch findfont exch dup type /arraytype eq {makefont}{scalefont}ifelse setfont}__NXbdef /_NXCombineArrays{ counttomark dup 2 add index dup length 3 -1 roll { 2 index length sub dup 4 1 roll 1 index exch 4 -1 roll putinterval exch }repeat pop pop pop }__NXbdef /setcmykcolor{ 1.0 exch sub dup dup 6 -1 roll sub dup 0 lt{pop 0}if 5 1 roll 4 -1 roll sub dup 0 lt{pop 0}if 3 1 roll exch sub dup 0 lt{pop 0}if setrgbcolor }__NXbdef /currentcmykcolor{currentrgbcolor 3{1.0 exch sub 3 1 roll}repeat 0}__NXbdef /flushgraphics{}def /setwindowtype{pop pop}def /currentwindowtype{pop 0}def /setalpha{pop}def /currentalpha{1.0}def /hidecursor{}def /obscurecursor{}def /revealcursor{}def /setcursor{4 {pop}repeat}bind def /showcursor{}def /NextStepEncoding where not{ /NextStepEncoding StandardEncoding 256 array copy def 0 [129/Agrave/Aacute/Acircumflex/Atilde/Adieresis/Aring/Ccedilla/Egrave /Eacute/Ecircumflex/Edieresis/Igrave/Iacute/Icircumflex/Idieresis /Eth/Ntilde/Ograve/Oacute/Ocircumflex/Otilde/Odieresis/Ugrave/Uacute /Ucircumflex/Udieresis/Yacute/Thorn/mu/multiply/divide/copyright 176/registered 181/brokenbar 190/logicalnot 192/onesuperior 201/twosuperior 204/threesuperior 209/plusminus/onequarter/onehalf/threequarters/agrave /aacute/acircumflex/atilde/adieresis/aring/ccedilla/egrave/eacute /ecircumflex/edieresis/igrave 226/iacute 228/icircumflex/idieresis/eth /ntilde 236/ograve/oacute/ocircumflex/otilde/odieresis 242/ugrave/uacute /ucircumflex 246/udieresis/yacute 252/thorn/ydieresis] {dup type /nametype eq {NextStepEncoding 2 index 2 index put pop 1 add}{exch pop}ifelse }forall pop /NextStepEncoding NextStepEncoding readonly def /_NXfstr 128 string dup 0 (_NX) putinterval def /findfont{ % Because we can never let NextStepEncoding get into % SharedFontDirectory, we cannot reencode a font to NextStepEncoding % if we are in shared mode. So if currentshared is true, % we call the normal findfont and return that /currentshared where {pop currentshared} {false} ifelse {//findfont exec} {dup _NXfstr 3 125 getinterval cvs length 3 add _NXfstr 0 3 -1 roll getinterval cvn exch FontDirectory 2 index known {pop FontDirectory exch get} {//findfont exec dup /Encoding get StandardEncoding eq { dup length dict exch {1 index /FID ne {2 index 3 1 roll put}{pop pop}ifelse}forall dup /Encoding NextStepEncoding put definefont }{exch pop} ifelse }ifelse }ifelse }bind def }{pop}ifelse /_NXProcArray 5 array __NXdef /_NXChannels 0 __NXdef /_NXTotalBytes 0 __NXdef /_NXDoImageOp{ 1 index{dup}{1}ifelse /_NXChannels exch store _NXChannels 2 add 2 roll _NXProcArray 0 _NXChannels getinterval astore pop 5 index 4 index mul 2 index{1 sub 8 idiv 1 add mul}{mul 1 sub 8 idiv 1 add}ifelse 4 index mul /_NXTotalBytes exch store pop exch pop gsave matrix invertmatrix concat 0.5 setgray 0 0 4 2 roll rectfill grestore { 0 1 _NXChannels