Education Sampler 1992 [NeXTSTEP]

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To obtain the first few terms in the power series expansion for at the origin we can use, for example, ;[s] 5:0,0;40,1;51,2;56,3;69,4;173,-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; ] Series[BesselJ[0,x],{x,0,7}] :[font = output; output; inactive; preserveAspect; endGroup; ] SeriesData[x, 0, {1, 0, -1/4, 0, 1/64, 0, -1/2304, 0, 1/147456}, 0, 9, 1] ;[o] 2 4 6 8 x x x x 9 1 - -- + -- - ---- + ------ + O[x] 4 64 2304 147456 :[font = text; inactive; pageBreakBelow; preserveAspect; ] Bessel functions can be plotted in the same way as trigonometric functions and Legendre polynomials. In particular, Options[Plot] lists all of the options associated with Plot together with their default values. Changing the default value of 25 of the option PlotPoints is useful for getting accurate graphs; for a complicated function such as BesselJ[5,x] the number 25 should be increased to 40. Thus, for example, J5(x) is nicely plotted with the Mathematica command ;[s] 18:0,0;117,1;131,2;173,3;177,4;262,5;272,6;348,7;360,8;423,9;424,10;425,11;426,12;427,13;428,14;429,15;456,16;467,17;476,-1; 18:1,11,8,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;1,10,8,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,1,12,0,0,0;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; ] Plot[BesselJ[5,x],{x,-50,50},PlotPoints->40]; :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174; endGroup; ] %! %%Creator: Mathematica %%AspectRatio: 0.61803 MathPictureStart /Courier findfont 10 scalefont 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MathPictureEnd :[font = text; inactive; preserveAspect; ] The graphing facilities of Mathematica can be used very effectively to find the zeros of Bessel functions. Let us find the first zero of J1(x), that is, the first number ;[s] 8:0,0;27,1;38,2;139,3;140,4;141,5;142,6;143,7;172,-1; 8:1,11,8,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;1,10,8,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; :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 0; pictureWidth = 243; pictureHeight = 20; ] %! %%Creator: Mathematica %%AspectRatio: 0.082305 200 dict begin MathPictureStart /Courier findfont 10 scalefont setfont % Scaling calculations 0 1 0 1 [ [ 0 0 0 0 ] [ 243 20 0 0 ] ] MathScale % Start of Graphics %!PS-Adobe-2.0 EPSF-2.0 %%Title: %%Creator: EquationBuilder %%CreationDate: Mon Aug 3 12:32:14 1992 %%For: gray %%DocumentFonts: (atend) %%Pages: 0 1 %%BoundingBox: 0 0 243 20 %%Translation: 1 5.203605 241.787338 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'K'JKOf0M'9TV$2,7,X_XRS2ihA"VLe`-6WLgVDQ[ % K-Oh*sQ3\5X&jqDA/d.K[>m^Q5J[c;a)NmMYCQg\nDUo-lO_PFgDq`RC,cPeD^''h!5o:_]J % 2B%2DQ*X=8=.%!T#`(=`_@:1Gn)4!ZcR26^7@LuTXn]hkIq;#jCohCY\M,lLS-[+S$ca%YX& % :D6qDjTV4[lp1)W=\`;h#U?,sjQ@KRfcfVE5FL<e!%XW&K4'KZ<^(q2po5*^N+.Ho1\BLem^ % @LPVYg]p)de"+?JF9i3d^k)Y<#f$t$JDq])&d".]$=-1.iDRnAVd=cPi`@6Gr"6dQJ-$@^+" % WS.+pi\#.+]I_#tt@i(h8e]SnfIGR,<@cbeW!b4L-?0O!CA3jH)gMAtOnUOglY[6oOb-O\ZJ % ^UQe)Cg#S8l!nh.MP$I6^Lfu,c-(/@sR:_[UG3NG'4[LM]P1jp(iKm]!E[RCU"]372<n9:hT % DJdQ#,--JM5Hni?AA`KKfYYHU0LUMka@d;.4:aK@Tm5/F>2E\T;$'*MlDFt$KuV@Y\L/.OiF % -[WM+WaCTF24$#6AWqH>>G@:nfdY\pHhk,u&)\a6Sj-G*iZbS'6CZ(HD(^aOl[BgA`JC6o9K % W&XLQOTDQ3X^_ci)sN3?rWiOV!D'_.!"Ip\iIhdM@NgIEG.$,:NLajeZ6X/i09Z0/+:/+4Zn % $]dTV]S]7NAs9gaKp(=$WPnq*=J?\2'T<FYHcFXfg$:>V*97],U+(*FB11Rkd3dF0c.i9JWf % 6as-/fWF"c-d)d%D&K*(m3N]O53mI6@#<tp=Z;T$(FM(Q(h"-c`,.fGCP!"kJ_Ug4k@ff*+! % QRGE4(^Cs+2=i:^iRe,ot)W,ab;!;85b-e>#8HVZKVu7[No0)2tibLS6b-a,WZ]$]'@RTL?& % ^9mAq5ug<6/&&,*&B^].moaG+qDcFJe'/*hp8@S!db)_uH9T,XIS3'+gMn!gnCauCRTEu0(, % pW4$?0XB<h&lQp(-6)h<cd>8D6L6r1!QmWRiAO(EEm4'.*'6'e'@3E@PmJiEW3sZBI5$b5!W % =T8`HWl?+@TaJE'o*tYeN*g:C)Io6PASo#4[YG57/gs!>>`r5i@QB5l^rjJT(ns]HRWO!!"D % E@0ci5x % EndExpression N 982 3d6 E fd S 1b713 R e9ce61fb 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 0 242.787338 19.308605 rectclip 0 0 242.787338 19.308605 rectclip 0 setgray 1 setalpha 1 1 scale /Symbol findfont 9 scalefont 46 exch defineuserobject 46 execuserobject setfont 7.423 12.247104 moveto ($) [0 0] xyshow 1 1 scale 1 1 scale 46 execuserobject setfont 16.153 12.242603 moveto ($) [0 0] xyshow 1 1 scale 0 setgray 1 setalpha 1 1 scale 46 execuserobject setfont 101.165001 12.247104 moveto ($) [0 0] xyshow 1 1 scale 1 1 scale 46 execuserobject setfont 109.895004 12.242603 moveto ($) [0 0] xyshow 1 1 scale 0 setgray 1 setalpha 1 1 scale 46 execuserobject setfont 198.845673 12.247104 moveto ($) [0 0] xyshow 1 1 scale 1 1 scale 46 execuserobject setfont 207.575668 12.242603 moveto ($) [0 0] xyshow 1 1 scale 0 setgray 1 setalpha 1 1.582485 scale /Symbol findfont 12 scalefont 44 exch defineuserobject 44 execuserobject setfont 187.622665 2.923917 moveto ($) [0 0] xyshow 1 0.631917 scale 1 1.580656 scale 44 execuserobject setfont 213.008667 2.924649 moveto ($) [0 0] xyshow 1 0.632649 scale 0 setgray 1 setalpha /Times-BoldItalic findfont 12 scalefont 48 exch defineuserobject 48 execuserobject setfont 238.787338 6.203605 moveto (.xJxx) [-46.36467 0 -17.352005 0 -80.328659 0 -93.742004 0 0 0 0 0] xyshow 46 execuserobject setfont 62.230675 18.908699 moveto () [0 0] xyshow /Times-Roman findfont 9 scalefont 45 exch defineuserobject 45 execuserobject setfont 198.422668 2.519604 moveto (1111111) [4.022995 9.6465 -21.087006 -7.762475 -80.616653 -1.884025 4.023003 9.6465 -97.765007 -9.6465 4.023001 9.6465 0 0 0 0] xyshow /Times-Roman findfont 12 scalefont 52 exch defineuserobject 52 execuserobject setfont 230.787338 6.203605 moveto (0dna0tahthcus) [-73.716675 0 -6 0 -5.327988 0 -18.000008 0 -52.33667 0 -5.327995 0 -6 0 -3.336002 0 -10 0 -5.327999 0 -6 0 -4.667999 0 0 0 0 1.401298e-45] xyshow 44 execuserobject setfont 220.865997 6.203605 moveto (=>) [-103.044662 0 0 0 0 0] xyshow %%Trailer %%DocumentFonts: Times-BoldItalic %%+ Times-Roman %%+ Symbol %End of Graphics MathPictureEnd end :[font = text; inactive; preserveAspect; ] To do this we draw successive graphs of that narrow in on the zero. We shall find the zero correct to 4 decimal places. First we define a plot-valued function jj by ;[s] 3:0,0;167,1;170,2;174,-1; 3:1,11,8,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; pageBreakBelow; dontNoPageBreakBelow; preserveAspect; ] jj[a_,b_]:=Plot[BesselJ[1,x],{x,a,b}] :[font = text; inactive; preserveAspect; ] Then jj[0,5] gives us the first graph: ;[s] 3:0,0;6,1;14,2;40,-1; 3:1,11,8,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; ] jj[0,5]; :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174; endGroup; ] %! %%Creator: Mathematica %%AspectRatio: 0.61803 MathPictureStart /Courier findfont 10 scalefont setfont % Scaling calculations 0.02381 0.190476 0.226728 0.647213 [ [(1)] 0.21429 0.22673 0 2 Msboxa [(2)] 0.40476 0.22673 0 2 Msboxa [(3)] 0.59524 0.22673 0 2 Msboxa [(4)] 0.78571 0.22673 0 2 Msboxa [(5)] 0.97619 0.22673 0 2 Msboxa [(-0.2)] 0.01131 0.09729 1 0 Msboxa [(0.2)] 0.01131 0.35617 1 0 Msboxa [(0.4)] 0.01131 0.48561 1 0 Msboxa [(0.6)] 0.01131 0.61506 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 0.61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath %%Object: Graphics [ ] 0 setdash 0 setgray gsave gsave 0.002 setlinewidth 0.21429 0.22673 moveto 0.21429 0.23298 lineto stroke grestore [(1)] 0.21429 0.22673 0 2 Mshowa gsave 0.002 setlinewidth 0.40476 0.22673 moveto 0.40476 0.23298 lineto stroke grestore [(2)] 0.40476 0.22673 0 2 Mshowa gsave 0.002 setlinewidth 0.59524 0.22673 moveto 0.59524 0.23298 lineto stroke grestore [(3)] 0.59524 0.22673 0 2 Mshowa gsave 0.002 setlinewidth 0.78571 0.22673 moveto 0.78571 0.23298 lineto stroke grestore [(4)] 0.78571 0.22673 0 2 Mshowa gsave 0.002 setlinewidth 0.97619 0.22673 moveto 0.97619 0.23298 lineto stroke grestore [(5)] 0.97619 0.22673 0 2 Mshowa gsave 0.001 setlinewidth 0.0619 0.22673 moveto 0.0619 0.23048 lineto stroke grestore gsave 0.001 setlinewidth 0.1 0.22673 moveto 0.1 0.23048 lineto stroke grestore gsave 0.001 setlinewidth 0.1381 0.22673 moveto 0.1381 0.23048 lineto stroke grestore gsave 0.001 setlinewidth 0.17619 0.22673 moveto 0.17619 0.23048 lineto stroke grestore gsave 0.001 setlinewidth 0.25238 0.22673 moveto 0.25238 0.23048 lineto stroke grestore gsave 0.001 setlinewidth 0.29048 0.22673 moveto 0.29048 0.23048 lineto stroke grestore gsave 0.001 setlinewidth 0.32857 0.22673 moveto 0.32857 0.23048 lineto stroke grestore gsave 0.001 setlinewidth 0.36667 0.22673 moveto 0.36667 0.23048 lineto stroke grestore gsave 0.001 setlinewidth 0.44286 0.22673 moveto 0.44286 0.23048 lineto stroke grestore gsave 0.001 setlinewidth 0.48095 0.22673 moveto 0.48095 0.23048 lineto stroke grestore gsave 0.001 setlinewidth 0.51905 0.22673 moveto 0.51905 0.23048 lineto stroke grestore gsave 0.001 setlinewidth 0.55714 0.22673 moveto 0.55714 0.23048 lineto stroke grestore gsave 0.001 setlinewidth 0.63333 0.22673 moveto 0.63333 0.23048 lineto stroke grestore gsave 0.001 setlinewidth 0.67143 0.22673 moveto 0.67143 0.23048 lineto stroke grestore gsave 0.001 setlinewidth 0.70952 0.22673 moveto 0.70952 0.23048 lineto stroke grestore gsave 0.001 setlinewidth 0.74762 0.22673 moveto 0.74762 0.23048 lineto stroke grestore gsave 0.001 setlinewidth 0.82381 0.22673 moveto 0.82381 0.23048 lineto stroke grestore gsave 0.001 setlinewidth 0.8619 0.22673 moveto 0.8619 0.23048 lineto stroke grestore gsave 0.001 setlinewidth 0.9 0.22673 moveto 0.9 0.23048 lineto stroke grestore gsave 0.001 setlinewidth 0.9381 0.22673 moveto 0.9381 0.23048 lineto stroke grestore gsave 0.002 setlinewidth 0 0.22673 moveto 1 0.22673 lineto stroke grestore gsave 0.002 setlinewidth 0.02381 0.09729 moveto 0.03006 0.09729 lineto stroke grestore [(-0.2)] 0.01131 0.09729 1 0 Mshowa gsave 0.002 setlinewidth 0.02381 0.35617 moveto 0.03006 0.35617 lineto stroke grestore [(0.2)] 0.01131 0.35617 1 0 Mshowa gsave 0.002 setlinewidth 0.02381 0.48561 moveto 0.03006 0.48561 lineto stroke grestore [(0.4)] 0.01131 0.48561 1 0 Mshowa gsave 0.002 setlinewidth 0.02381 0.61506 moveto 0.03006 0.61506 lineto stroke grestore [(0.6)] 0.01131 0.61506 1 0 Mshowa gsave 0.001 setlinewidth 0.02381 0.12317 moveto 0.02756 0.12317 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.14906 moveto 0.02756 0.14906 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.17495 moveto 0.02756 0.17495 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.20084 moveto 0.02756 0.20084 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.25262 moveto 0.02756 0.25262 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.27851 moveto 0.02756 0.27851 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.30439 moveto 0.02756 0.30439 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.33028 moveto 0.02756 0.33028 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.38206 moveto 0.02756 0.38206 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.40795 moveto 0.02756 0.40795 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.43384 moveto 0.02756 0.43384 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.45973 moveto 0.02756 0.45973 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.5115 moveto 0.02756 0.5115 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.53739 moveto 0.02756 0.53739 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.56328 moveto 0.02756 0.56328 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.58917 moveto 0.02756 0.58917 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.0714 moveto 0.02756 0.0714 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.04551 moveto 0.02756 0.04551 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.01962 moveto 0.02756 0.01962 lineto stroke grestore gsave 0.002 setlinewidth 0.02381 0 moveto 0.02381 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 0.22673 moveto 0.06349 0.29378 lineto 0.10317 0.35866 lineto 0.14286 0.41927 lineto 0.18254 0.47366 lineto 0.22222 0.52012 lineto 0.2619 0.55721 lineto 0.28175 0.57189 lineto 0.30159 0.58383 lineto 0.32143 0.59298 lineto 0.33135 0.59648 lineto 0.34127 0.59926 lineto 0.35119 0.60132 lineto 0.35615 0.60208 lineto 0.35863 0.60239 lineto 0.36111 0.60266 lineto 0.36359 0.60288 lineto 0.36607 0.60306 lineto 0.36731 0.60313 lineto 0.36855 0.60319 lineto 0.36979 0.60324 lineto 0.37103 0.60327 lineto 0.37227 0.6033 lineto 0.37351 0.60332 lineto 0.37475 0.60332 lineto 0.37599 0.60331 lineto 0.37723 0.60329 lineto 0.37847 0.60326 lineto 0.38095 0.60317 lineto 0.38219 0.6031 lineto 0.38343 0.60303 lineto 0.38591 0.60284 lineto 0.39087 0.60234 lineto 0.39583 0.60166 lineto 0.40079 0.60081 lineto 0.41071 0.59856 lineto 0.42063 0.59562 lineto 0.44048 0.58769 lineto 0.46032 0.5771 lineto 0.5 0.54845 lineto 0.53968 0.51088 lineto 0.57937 0.46588 lineto 0.61905 0.41516 lineto 0.65873 0.36065 lineto 0.69841 0.30433 lineto 0.7381 0.24823 lineto 0.77778 0.19434 lineto 0.81746 0.14448 lineto 0.85714 0.10033 lineto 0.89683 0.0633 lineto Mistroke 0.91667 0.04782 lineto 0.93651 0.0345 lineto 0.97619 0.01472 lineto Mfstroke grestore grestore % End of Graphics MathPictureEnd :[font = text; inactive; preserveAspect; ] Now we have a good idea that the first zero of J1(x) lies somewhere between 3.5 and 4. So next we type jj[3.5,4] and we get ;[s] 8:0,0;50,1;51,2;52,3;53,4;54,5;111,6;120,7;132,-1; 8:1,11,8,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; :[font = input; preserveAspect; plain; fontName = "Courier"; startGroup; ] jj[3.5,4]; ;[s] 1:0,0;10,-1; 1:1,10,8,Courier,1,12,0,0,0; :[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 -6.642857 1.904762 0.205813 2.893527 [ [(3.5)] 0.02381 0.20581 0 2 Msboxa [(3.6)] 0.21429 0.20581 0 2 Msboxa [(3.7)] 0.40476 0.20581 0 2 Msboxa [(3.8)] 0.59524 0.20581 0 2 Msboxa [(3.9)] 0.78571 0.20581 0 2 Msboxa [(-0.05)] 0.96369 0.06114 1 0 Msboxa [(0.05)] 0.96369 0.35049 1 0 Msboxa [(0.1)] 0.96369 0.49517 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 0.61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath %%Object: Graphics [ ] 0 setdash 0 setgray gsave gsave 0.002 setlinewidth 0.02381 0.20581 moveto 0.02381 0.21206 lineto stroke grestore [(3.5)] 0.02381 0.20581 0 2 Mshowa gsave 0.002 setlinewidth 0.21429 0.20581 moveto 0.21429 0.21206 lineto stroke grestore [(3.6)] 0.21429 0.20581 0 2 Mshowa gsave 0.002 setlinewidth 0.40476 0.20581 moveto 0.40476 0.21206 lineto stroke grestore [(3.7)] 0.40476 0.20581 0 2 Mshowa gsave 0.002 setlinewidth 0.59524 0.20581 moveto 0.59524 0.21206 lineto stroke grestore [(3.8)] 0.59524 0.20581 0 2 Mshowa gsave 0.002 setlinewidth 0.78571 0.20581 moveto 0.78571 0.21206 lineto stroke grestore [(3.9)] 0.78571 0.20581 0 2 Mshowa gsave 0.001 setlinewidth 0.0619 0.20581 moveto 0.0619 0.20956 lineto stroke grestore gsave 0.001 setlinewidth 0.1 0.20581 moveto 0.1 0.20956 lineto stroke grestore gsave 0.001 setlinewidth 0.1381 0.20581 moveto 0.1381 0.20956 lineto stroke grestore gsave 0.001 setlinewidth 0.17619 0.20581 moveto 0.17619 0.20956 lineto stroke grestore gsave 0.001 setlinewidth 0.25238 0.20581 moveto 0.25238 0.20956 lineto stroke grestore gsave 0.001 setlinewidth 0.29048 0.20581 moveto 0.29048 0.20956 lineto stroke grestore gsave 0.001 setlinewidth 0.32857 0.20581 moveto 0.32857 0.20956 lineto stroke grestore gsave 0.001 setlinewidth 0.36667 0.20581 moveto 0.36667 0.20956 lineto stroke grestore gsave 0.001 setlinewidth 0.44286 0.20581 moveto 0.44286 0.20956 lineto stroke grestore gsave 0.001 setlinewidth 0.48095 0.20581 moveto 0.48095 0.20956 lineto stroke grestore gsave 0.001 setlinewidth 0.51905 0.20581 moveto 0.51905 0.20956 lineto stroke grestore gsave 0.001 setlinewidth 0.55714 0.20581 moveto 0.55714 0.20956 lineto stroke grestore gsave 0.001 setlinewidth 0.63333 0.20581 moveto 0.63333 0.20956 lineto stroke grestore gsave 0.001 setlinewidth 0.67143 0.20581 moveto 0.67143 0.20956 lineto stroke grestore gsave 0.001 setlinewidth 0.70952 0.20581 moveto 0.70952 0.20956 lineto stroke grestore gsave 0.001 setlinewidth 0.74762 0.20581 moveto 0.74762 0.20956 lineto stroke grestore gsave 0.001 setlinewidth 0.82381 0.20581 moveto 0.82381 0.20956 lineto stroke grestore gsave 0.001 setlinewidth 0.8619 0.20581 moveto 0.8619 0.20956 lineto stroke grestore gsave 0.001 setlinewidth 0.9 0.20581 moveto 0.9 0.20956 lineto stroke grestore gsave 0.001 setlinewidth 0.9381 0.20581 moveto 0.9381 0.20956 lineto stroke grestore gsave 0.002 setlinewidth 0 0.20581 moveto 1 0.20581 lineto stroke grestore gsave 0.002 setlinewidth 0.97619 0.06114 moveto 0.98244 0.06114 lineto stroke grestore [(-0.05)] 0.96369 0.06114 1 0 Mshowa gsave 0.002 setlinewidth 0.97619 0.35049 moveto 0.98244 0.35049 lineto stroke grestore [(0.05)] 0.96369 0.35049 1 0 Mshowa gsave 0.002 setlinewidth 0.97619 0.49517 moveto 0.98244 0.49517 lineto stroke grestore [(0.1)] 0.96369 0.49517 1 0 Mshowa gsave 0.001 setlinewidth 0.97619 0.09007 moveto 0.97994 0.09007 lineto stroke grestore gsave 0.001 setlinewidth 0.97619 0.11901 moveto 0.97994 0.11901 lineto stroke grestore gsave 0.001 setlinewidth 0.97619 0.14794 moveto 0.97994 0.14794 lineto stroke grestore gsave 0.001 setlinewidth 0.97619 0.17688 moveto 0.97994 0.17688 lineto stroke grestore gsave 0.001 setlinewidth 0.97619 0.23475 moveto 0.97994 0.23475 lineto stroke grestore gsave 0.001 setlinewidth 0.97619 0.26368 moveto 0.97994 0.26368 lineto stroke grestore gsave 0.001 setlinewidth 0.97619 0.29262 moveto 0.97994 0.29262 lineto stroke grestore gsave 0.001 setlinewidth 0.97619 0.32155 moveto 0.97994 0.32155 lineto stroke grestore gsave 0.001 setlinewidth 0.97619 0.37942 moveto 0.97994 0.37942 lineto stroke grestore gsave 0.001 setlinewidth 0.97619 0.40836 moveto 0.97994 0.40836 lineto stroke grestore gsave 0.001 setlinewidth 0.97619 0.4373 moveto 0.97994 0.4373 lineto stroke grestore gsave 0.001 setlinewidth 0.97619 0.46623 moveto 0.97994 0.46623 lineto stroke grestore gsave 0.001 setlinewidth 0.97619 0.0322 moveto 0.97994 0.0322 lineto stroke grestore gsave 0.001 setlinewidth 0.97619 0.00327 moveto 0.97994 0.00327 lineto stroke grestore gsave 0.001 setlinewidth 0.97619 0.5241 moveto 0.97994 0.5241 lineto stroke grestore gsave 0.001 setlinewidth 0.97619 0.55304 moveto 0.97994 0.55304 lineto stroke grestore gsave 0.001 setlinewidth 0.97619 0.58197 moveto 0.97994 0.58197 lineto stroke grestore gsave 0.001 setlinewidth 0.97619 0.61091 moveto 0.97994 0.61091 lineto stroke grestore gsave 0.002 setlinewidth 0.97619 0 moveto 0.97619 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 0.57804 lineto 0.10317 0.55275 lineto 0.14286 0.52748 lineto 0.18254 0.50223 lineto 0.22222 0.477 lineto 0.2619 0.45182 lineto 0.30159 0.42667 lineto 0.34127 0.40159 lineto 0.38095 0.37656 lineto 0.42063 0.35161 lineto 0.46032 0.32674 lineto 0.5 0.30196 lineto 0.53968 0.27728 lineto 0.57937 0.25271 lineto 0.61905 0.22825 lineto 0.65873 0.20392 lineto 0.69841 0.17972 lineto 0.7381 0.15566 lineto 0.77778 0.13175 lineto 0.81746 0.10799 lineto 0.85714 0.0844 lineto 0.89683 0.06099 lineto 0.93651 0.03776 lineto 0.97619 0.01472 lineto stroke grestore grestore % End of Graphics MathPictureEnd :[font = text; inactive; preserveAspect; ] Consequently the first zero of J1(x) lies between 3.8 and 3.85. So for the next approximation we type ;[s] 6:0,0;34,1;35,2;36,3;37,4;38,5;109,-1; 6:1,11,8,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; :[font = input; preserveAspect; startGroup; ] jj[3.8,3.85]; :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174; endGroup; ] %! %%Creator: Mathematica %%AspectRatio: 0.61803 MathPictureStart /Courier findfont 10 scalefont setfont % Scaling calculations -72.357143 19.047619 0.229196 29.180435 [ [(3.81)] 0.21429 0.2292 0 2 Msboxa [(3.82)] 0.40476 0.2292 0 2 Msboxa [(3.83)] 0.59524 0.2292 0 2 Msboxa [(3.84)] 0.78571 0.2292 0 2 Msboxa [(3.85)] 0.97619 0.2292 0 2 Msboxa [(-0.005)] 0.01131 0.08329 1 0 Msboxa [(0.005)] 0.01131 0.3751 1 0 Msboxa [(0.01)] 0.01131 0.521 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 0.61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath %%Object: Graphics [ ] 0 setdash 0 setgray gsave gsave 0.002 setlinewidth 0.21429 0.2292 moveto 0.21429 0.23545 lineto stroke grestore [(3.81)] 0.21429 0.2292 0 2 Mshowa gsave 0.002 setlinewidth 0.40476 0.2292 moveto 0.40476 0.23545 lineto stroke grestore [(3.82)] 0.40476 0.2292 0 2 Mshowa gsave 0.002 setlinewidth 0.59524 0.2292 moveto 0.59524 0.23545 lineto stroke grestore [(3.83)] 0.59524 0.2292 0 2 Mshowa gsave 0.002 setlinewidth 0.78571 0.2292 moveto 0.78571 0.23545 lineto stroke grestore [(3.84)] 0.78571 0.2292 0 2 Mshowa gsave 0.002 setlinewidth 0.97619 0.2292 moveto 0.97619 0.23545 lineto stroke grestore [(3.85)] 0.97619 0.2292 0 2 Mshowa gsave 0.001 setlinewidth 0.0619 0.2292 moveto 0.0619 0.23295 lineto stroke grestore gsave 0.001 setlinewidth 0.1 0.2292 moveto 0.1 0.23295 lineto stroke grestore gsave 0.001 setlinewidth 0.1381 0.2292 moveto 0.1381 0.23295 lineto stroke grestore gsave 0.001 setlinewidth 0.17619 0.2292 moveto 0.17619 0.23295 lineto stroke grestore gsave 0.001 setlinewidth 0.25238 0.2292 moveto 0.25238 0.23295 lineto stroke grestore gsave 0.001 setlinewidth 0.29048 0.2292 moveto 0.29048 0.23295 lineto stroke grestore gsave 0.001 setlinewidth 0.32857 0.2292 moveto 0.32857 0.23295 lineto stroke grestore gsave 0.001 setlinewidth 0.36667 0.2292 moveto 0.36667 0.23295 lineto stroke grestore gsave 0.001 setlinewidth 0.44286 0.2292 moveto 0.44286 0.23295 lineto stroke grestore gsave 0.001 setlinewidth 0.48095 0.2292 moveto 0.48095 0.23295 lineto stroke grestore gsave 0.001 setlinewidth 0.51905 0.2292 moveto 0.51905 0.23295 lineto stroke grestore gsave 0.001 setlinewidth 0.55714 0.2292 moveto 0.55714 0.23295 lineto stroke grestore gsave 0.001 setlinewidth 0.63333 0.2292 moveto 0.63333 0.23295 lineto stroke grestore gsave 0.001 setlinewidth 0.67143 0.2292 moveto 0.67143 0.23295 lineto stroke grestore gsave 0.001 setlinewidth 0.70952 0.2292 moveto 0.70952 0.23295 lineto stroke grestore gsave 0.001 setlinewidth 0.74762 0.2292 moveto 0.74762 0.23295 lineto stroke grestore gsave 0.001 setlinewidth 0.82381 0.2292 moveto 0.82381 0.23295 lineto stroke grestore gsave 0.001 setlinewidth 0.8619 0.2292 moveto 0.8619 0.23295 lineto stroke grestore gsave 0.001 setlinewidth 0.9 0.2292 moveto 0.9 0.23295 lineto stroke grestore gsave 0.001 setlinewidth 0.9381 0.2292 moveto 0.9381 0.23295 lineto stroke grestore gsave 0.002 setlinewidth 0 0.2292 moveto 1 0.2292 lineto stroke grestore gsave 0.002 setlinewidth 0.02381 0.08329 moveto 0.03006 0.08329 lineto stroke grestore [(-0.005)] 0.01131 0.08329 1 0 Mshowa gsave 0.002 setlinewidth 0.02381 0.3751 moveto 0.03006 0.3751 lineto stroke grestore [(0.005)] 0.01131 0.3751 1 0 Mshowa gsave 0.002 setlinewidth 0.02381 0.521 moveto 0.03006 0.521 lineto stroke grestore [(0.01)] 0.01131 0.521 1 0 Mshowa gsave 0.001 setlinewidth 0.02381 0.11247 moveto 0.02756 0.11247 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.14166 moveto 0.02756 0.14166 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.17084 moveto 0.02756 0.17084 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.20002 moveto 0.02756 0.20002 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.25838 moveto 0.02756 0.25838 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.28756 moveto 0.02756 0.28756 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.31674 moveto 0.02756 0.31674 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.34592 moveto 0.02756 0.34592 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.40428 moveto 0.02756 0.40428 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.43346 moveto 0.02756 0.43346 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.46264 moveto 0.02756 0.46264 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.49182 moveto 0.02756 0.49182 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.05411 moveto 0.02756 0.05411 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.02493 moveto 0.02756 0.02493 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.55018 moveto 0.02756 0.55018 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.57936 moveto 0.02756 0.57936 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.60854 moveto 0.02756 0.60854 lineto stroke grestore gsave 0.002 setlinewidth 0.02381 0 moveto 0.02381 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 0.57865 lineto 0.10317 0.55399 lineto 0.14286 0.52934 lineto 0.18254 0.50471 lineto 0.22222 0.48008 lineto 0.2619 0.45547 lineto 0.30159 0.43088 lineto 0.34127 0.40629 lineto 0.38095 0.38172 lineto 0.42063 0.35716 lineto 0.46032 0.33261 lineto 0.5 0.30808 lineto 0.53968 0.28356 lineto 0.57937 0.25905 lineto 0.61905 0.23456 lineto 0.65873 0.21007 lineto 0.69841 0.18561 lineto 0.7381 0.16115 lineto 0.77778 0.13671 lineto 0.81746 0.11229 lineto 0.85714 0.08787 lineto 0.89683 0.06347 lineto 0.93651 0.03909 lineto 0.97619 0.01472 lineto stroke grestore grestore % End of Graphics MathPictureEnd :[font = text; inactive; pageBreakBelow; preserveAspect; ] Continuing in is this way we find that the first zero of J1(x) is slightly more than 3.8317. ;[s] 6:0,0;59,1;60,2;61,3;62,4;63,5;96,-1; 6:1,11,8,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; :[font = text; inactive; preserveAspect; ] The command FindRoot can also be used to find zeros of Bessel functions. This command attempts to solve equations numerically, but it must be given a hint. For example the commands ;[s] 3:0,0;11,1;20,2;182,-1; 3:1,11,8,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; ] FindRoot[BesselJ[1,x],{x,3}] :[font = output; output; inactive; preserveAspect; endGroup; ] {x -> 3.831705970207511} ;[o] {x -> 3.83171} :[font = input; preserveAspect; startGroup; ] FindRoot[BesselJ[1,x],{x,7}] :[font = output; output; inactive; preserveAspect; endGroup; ] {x -> 7.015586669815619} ;[o] {x -> 7.01559} :[font = text; inactive; preserveAspect; ] tell us the first and second zeros of J1(x). However, FindRoot can sometimes give the wrong answer. For example, if we type in ;[s] 8:0,0;38,1;39,2;40,3;41,4;42,5;55,6;63,7;129,-1; 8:1,11,8,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; :[font = input; preserveAspect; startGroup; ] FindRoot[BesselJ[1,x],{x,2}] :[font = output; output; inactive; preserveAspect; endGroup; ] {x -> 10.17346813506272} ;[o] {x -> 10.1735} :[font = text; inactive; pageBreakBelow; preserveAspect; ] But 10.1735 is the third zero of J1(x), whereas the nearest zero to 2 is 3.83171. Since Mathematica's internal algorithm for FindRoot does not always give the right answer, the output of FindRoot should always be checked by plotting. ;[s] 12:0,0;36,1;37,2;38,3;39,4;40,5;93,6;106,7;130,8;138,9;192,10;200,11;239,-1; 12:1,11,8,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,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; ] It is possible to see just how well the function :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 0; pictureWidth = 84; pictureHeight = 31; ] %! %%Creator: Mathematica %%AspectRatio: 0.369048 200 dict begin MathPictureStart /Courier findfont 10 scalefont setfont % Scaling calculations 0 1 0 1 [ [ 0 0 0 0 ] [ 84 31 0 0 ] ] MathScale % Start of Graphics %!PS-Adobe-2.0 EPSF-2.0 %%Title: %%Creator: EquationBuilder %%CreationDate: Mon Aug 3 15:51:46 1992 %%For: gray %%DocumentFonts: (atend) %%Pages: 0 1 %%BoundingBox: 0 0 84 31 %%Translation: 1 8.633711 82.749336 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".RJBA*j!!#t8"9`(0Sid;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*iR % K6<qbQ-(F07tWmaJ<eh7SY#BM!8_jS7umh:]j&SD*Xm*Rl#!1d!:I/nS6(8>_<BF3<j-B7c] % 6=ajEh$eVup&(Q_e>EJb+#jCok$n+2E(d$0"Y>7rCDGmOcgRWS<NP6"WZ$iiWEHl7YEBki$e % ct`ZZ&;!MaXe9;t.d[u?SpdZl(D.Y_%EMY&rdg4KJcN:c!.tacJ;59Q[MQ`2[=IfYL_sm*8r % 3SA-mFMc`@+7W1hJ]l!'^Gbb)JFL#XA_GFKm&%%tIPGI\ek:>_N.bf[qb[VE!RiKJYYq$kYZ % @jbm)Aabkuo:8uAh_rZ)?&T,<<cuZ07\RgL&G0(jT>_&dt0p5'4AM=mg8P""2NHPMOXM(TBP % n"9X`Z'![-VC'I#!e&pN@cA[Cecb:N1j])8r,!H.NlZCV4Dd3@VojpBUg'ff)h+pD9?C>&k[ % c#WXF.,8sYFl>2b0?[Mf_eG)?T$R3@<qi3=XVX5^Aba@Ke@2H&9SO#dX=@ureOOA@f"'YCH' % HL[iI&9Z&K`uB%1b1KJ(3nrmKH=[-b!bgsU<$)Mo@"<k;:k>*./P(Fp(iD;=J\Z+6/Atn+,X % Fp;[uK@>!Xq.!0dqTL]-F^L!:_cD+^[Q$6t6A9#lpLj@gE&7x % EndExpression N 670 29e E 10 S 10c2e R cbd68ad8 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 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section; inactive; Cclosed; preserveAspect; startGroup; ] Gibbs' Phenomenon for Bessel Functions :[font = text; inactive; preserveAspect; ] There is a Gibbs phenomenon for Bessel functions slightly different from that of trigonometric functions. To see it let us graph some of the partial sums of the Fourier-Bessel expansion of the function defined by :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 0; pictureWidth = 139; pictureHeight = 41; ] %! %%Creator: Mathematica %%AspectRatio: 0.294964 200 dict begin MathPictureStart /Courier findfont 10 scalefont setfont % Scaling calculations 0 1 0 1 [ [ 0 0 0 0 ] [ 139 41 0 0 ] ] MathScale % Start of Graphics %!PS-Adobe-2.0 EPSF-2.0 %%Title: %%Creator: EquationBuilder %%CreationDate: Mon Aug 3 19:34:23 1992 %%For: gray %%DocumentFonts: (atend) %%Pages: 0 1 %%BoundingBox: 0 0 139 41 %%Translation: 1 16.444 137.880005 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&l.51c&0M'9:V1XA]=NZgT+TapkbQ1TiO8qO##pFg % SV27Z%C%),,7&[ne(c'<W+g@4!;?.76d#C!iJ0@Ju`AW"Zk%:1$+0PmBo;2s;99LJh%SVCE) % MA.D#"V,>l4t:'\cc%f,u5I7KGQRBN*lp[0QL,ShBY)pi(fsC&eZ=/.B!659En<Z0`[#-KJ" % ,Lgjhp52JLm@U%;/?*SQWm'>bd*F,\J!5"R)D41?A2^4?AOD@1YW4GM.Z%pga9n,%\<kQ_); % 5Cai+oG>;m!=Aln$7U)E++R!97hGhPZ(Vb"DHI-9D<:SI*e+@?7nA!GojAgaV-pghDlAV,<^ % LM'LRf]pPW+,*1OuB[A?qG46!IR7+9QlJ793'9o'rV!HgV%KK7eh&0EiG)N)>=fOpu]2L@_P % A+Xf,ta%M[cDYT]VVJtcGQrb%`VNte:8][5bL7'?7B"j*KDHIfKAt<m5@O7U\$F$Ft62E#XI % tFt_S9!TEBqC]K,$E\?b8'm:UV91n@?PZ$)/#CI*kA,:I9rB*RWO_Xkc(;bl)CALc8k0jVQ/ % <!T<$[,BVi7o8nO5?/B:Ad^3-AfEMJJ+l(P%n8k:GK'$K7;K[;L6K1<ooj[L7fq'g1_GkRnG % L%<,tX2qukCLlSIWopW!aUk)JlP8n.P6jKOLDr3U-uH:57`<0VF".145Hj550#2Vd(n&7N<P % Y@<hiTGB"6WQDbGuI5I&j90FUT\3WH+L`S%+3#EliL)m.%6<\ib#1e1t4+aI"/DD(P!FK(Gt % 1JgcQ!0;Ar]cR[>q1=AUtgum"p)GTkJmlGl:d]O52cdaBM?q\_,#_@-04cA.'ZaUKN[1`3H1 % 9A\[K4:pX&b2MFf2Mj2*40VD*V\@7>-$E&@hYRhfMIduCd]nSNN*d"4+<tAJ5LF('T?aQ7N% % U+1-c.[U]Cf'1D6q2$u'gW?N>h21;LS#QNI!Kr8kD[J-=jA@"/D.BO:%e,@!5P$4Kde*-8b) % )e63G[aWp,r07R:#dh"[:jk$ZJ-#'i5XfcE[3.nJ@CY8;"Q-[#^(>s[OTZ\55W=FE$DbFcqk % +'I+RVMq_*8KtkY(ur&-P`.J.qo9"98E%x % EndExpression N 945 3b1 E 63 S 1a5e9 R 82c544c4 gsave /__NXbasematrix matrix currentmatrix def grestore %%EndProlog %%BeginSetup %%EndSetup 0 0 138.880005 40.888 rectclip 0 0 138.880005 40.888 rectclip 0 setgray 1 setalpha 1 1 scale /Symbol findfont 12 scalefont 48 exch defineuserobject 48 execuserobject setfont 5.56 17.552 moveto ($) [0 0] xyshow 1 1 scale 1 1 scale 48 execuserobject setfont 17.200001 17.546 moveto ($) [0 0] xyshow 1 1 scale 0 setgray 1 setalpha 0.48 setlinewidth 124.980003 32.304001 moveto 130.480011 32.304001 lineto stroke 0 setgray 1 setalpha 0.48 setlinewidth 87.470673 8.500002 moveto 92.970673 8.500002 lineto stroke 0 setgray 1 setalpha 48 execuserobject setfont 1 0.153813 scale 36.01067 90.456581 moveto <ef> show 36.01067 166.368423 moveto <ef> show 1 6.501393 scale 36.01067 3.764956 moveto <eeedec> [0 11.579044 0 11.507046 0 0] xyshow 0 setgray 1 setalpha /Times-BoldItalic findfont 12 scalefont 49 exch defineuserobject 49 execuserobject setfont 131.480011 5.500001 moveto (.xxxf) [-25.254677 0 -0.5 23.803999 -95.365334 -11.860001 -9.360001 0 0 0 -1.681558e-44 14937.355469] xyshow 48 execuserobject setfont 115.55867 5.500001 moveto <a33ca3a33d> [-19.254669 0 18.754669 23.803999 -19.254669 0 -70.746666 -11.860001 0 0 0 0] xyshow /Times-Roman findfont 12 scalefont 50 exch defineuserobject 50 execuserobject setfont 125.480003 5.500001 moveto (1rof0,0rof1) [-56.005333 0 -6 0 -3.996002 0 -16 0 89.001343 23.803999 -46.009338 0 -15.996002 0 -6 1.907349e-06 -3.996002 -1.907349e-06 -18 0 0 0 0 0] xyshow /Times-Roman findfont 9 scalefont 51 exch defineuserobject 51 execuserobject setfont 87.970673 1.000001 moveto (2121) [0 9 37.509331 14.803999 0 9 0 0 0 0] xyshow %%Trailer %%DocumentFonts: Times-BoldItalic %%+ Times-Roman %%+ Symbol %End of Graphics MathPictureEnd end :[font = text; inactive; preserveAspect; ] The Fourier-Bessel expansion of f(x) is ;[s] 5:0,0;34,1;35,2;36,3;37,4;42,-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 = 141; pictureHeight = 36; ] %! %%Creator: Mathematica %%AspectRatio: 0.255319 200 dict begin MathPictureStart /Courier findfont 10 scalefont setfont % Scaling calculations 0 1 0 1 [ [ 0 0 0 0 ] [ 141 36 0 0 ] ] MathScale % Start of Graphics %!PS-Adobe-2.0 EPSF-2.0 %%Title: %%Creator: EquationBuilder %%CreationDate: Mon Aug 3 16:22:41 1992 %%For: gray %%DocumentFonts: (atend) %%Pages: 0 1 %%BoundingBox: 0 0 141 36 %%Translation: 1 14.884993 139.253265 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&m,VT6!0M"`eV$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 % &Pgo:u,"O8/K#JVW;G9-88oJL(Oh['1U:3`L@o]lIuWp?:"^H/d3!JP0!\R-jBZKMU3Y:IN& % hI_^8*I1j!'<VA%_SCdPN3Cl[JBd&4p[B\PF5C3ORr`D%).LrU'Po=dkoi_<cVJGT=3fa%!K % OT#7Xq'QFJ.AY=EIf1jP*(::3`.g`$\M5*QZAH%lH2/0d+!KnUf>Di^1%7RsEbP*"CPfl.+a % 0LaYaGpUIHMFu(;?-oL)i`_$?DRt7O*fnCc4Y>O53\YL:lKT?3"jca%DIam\V.R*#?!jLb_S % 3,`<JU7qI\]`8D]"kERe2.0#7jV3uuFON*ak-lAZj-/oWtFO`s'?#9YM6:d/t'dFb#S'fS'K % 29QS7N-0OQ3X"B+X>hDYB]f/1/[18AsL4^Yok1r'[[+)"&4#SVP$VN?@#_Fb,((hIT]o,C2+ % Y/2NfmA8MlT4"(E^AMAH7BKGa<J2WFC$a>/fr1MN[gH@tL[*N"FY1lk6*.[$fPC.bOeBj4f. % [`V2J/@EK$ME6oN,7-AnTED>Bc>L6@V\.%4DOKq"\8Qcn##O3*!<*e-_>WUgTRdAlFW7ZI8I % ?`MO23WSP(_J.D8iGsgYoWNASJE_jfU'Q%0*'BO!O9e@ZMf2r#.,E_uBJ!5KWbi,leaNC[JE % gnN=(QNZ$3MaZj9>FN55NN*MZ<PH6C7ke^N#"HjlM#m6>U0V6hb8UtEqV2-_-_"Q`BY^c^#4 % CE^F$Bj=R@&#)mRPkEI/C9DSY]lBK*6i3GNs./Da/k##\DSB.9kJbs8:lC<*(U,4L_*5=pX! % '+^j'i;';DiZSfch]]VUZn\&l"[%YH#R8>:\]/n"\2k-\&Y#dB?2+=sdce+!*A2LJ7*iJnXk % ]"(QL/W'%u5AmXaCC23k%[-<(#nUH:?]#EeThTKTqmkOkJ.'C\pX>3!/iC&UiNdIl9JbT)RS % $*p4;!`!EIZ)&(u+<W(]^iF0MQthL]W<-#<C#e7$)H5:>UaN^179Z#Z/\cL=-;2,esEG=TB2 % ,UQQB`5quee_g3FZ_D[WT#_=-d''Km]k)<_3$>AB+@gIrW'Pp!"i#WG\E!\i7?jo?Z"9:F?% % mp<sx % EndExpression N 1039 40f E f5 S 1c1d3 R b19aebcd 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 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Working with Bessel functions is more complicated than working with trigonometric functions because the zeros do not occur at regular intervals. The zeros either have to be computed explicitly (for example, using FindRoot) or looked up in a table. ;[s] 14:0,0;7,1;8,2;9,3;18,4;19,5;21,6;60,7;61,8;62,9;63,10;64,11;285,12;293,13;320,-1; 14:1,11,8,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,32,12,0,0,0;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; :[font = text; inactive; preserveAspect; ] We define a list X in Mathematica whose elements are the first 20 positive zeros of J0(x): ;[s] 8:0,0;26,1;37,2;89,3;90,4;91,5;92,6;93,7;96,-1; 8:1,11,8,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;1,10,8,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; :[font = input; preserveAspect; startGroup; ] X={2.405,5.520,8.654,11.792,14.931, 18.071,21.212,24.352,27.493,30.635, 33.776,36.972,40.058,43.200,46.483, 49.482,52.624,55.766,58.907,62.049} :[font = output; output; inactive; preserveAspect; endGroup; ] {2.405, 5.52, 8.654, 11.792, 14.931, 18.071, 21.212, 24.352, 27.493, 30.635, 33.776, 36.972, 40.058, 43.20000000000001, 46.483, 49.482, 52.62400000000001, 55.766, 58.907, 62.04900000000001} ;[o] {2.405, 5.52, 8.654, 11.792, 14.931, 18.071, 21.212, 24.352, 27.493, 30.635, 33.776, 36.972, 40.058, 43.2, 46.483, 49.482, 52.624, 55.766, 58.907, 62.049} :[font = text; inactive; preserveAspect; ] Then X[[k]] is the kth element of X, that is, xk. Next we define a function of two variables f[x,n] that represents the partial sum for the Fourier-Bessel expansion of f(x): ;[s] 17:0,0;6,1;13,2;21,3;22,4;24,5;38,6;39,7;52,8;53,9;54,10;101,11;108,12;179,13;180,14;181,15;182,16;185,-1; 17:1,11,8,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,32,12,0,0,0;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,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; :[font = input; pageBreakBelow; dontNoPageBreakBelow; preserveAspect; ] f[x_,n_]:=Sum[ BesselJ[1,Evaluate[X[[k]]]/2]* BesselJ[0,x Evaluate[X[[k]]]]/ (Evaluate[X[[k]]]* BesselJ[1,Evaluate[X[[k]]]]^2), {k,n}] :[font = text; inactive; preserveAspect; ] We plot two of these partial sums: :[font = input; preserveAspect; startGroup; ] 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0.88843 0.07385 lineto 0.88919 0.07359 lineto 0.88996 0.07341 lineto 0.89072 0.0733 lineto 0.89148 0.07329 lineto 0.89225 0.07338 lineto 0.89301 0.07359 lineto 0.89377 0.07392 lineto 0.89454 0.07439 lineto 0.8953 0.07497 lineto 0.89683 0.07652 lineto 0.89988 0.08087 lineto 0.90293 0.08615 lineto 0.90598 0.09125 lineto 0.90751 0.0934 lineto 0.90904 0.09517 lineto 0.9098 0.09589 lineto 0.91056 0.0965 lineto Mistroke 0.91132 0.09699 lineto 0.91209 0.09738 lineto 0.91285 0.09766 lineto 0.91361 0.09785 lineto 0.91438 0.09795 lineto 0.91514 0.09798 lineto 0.9159 0.09794 lineto 0.91667 0.09786 lineto 0.91819 0.0976 lineto 0.91896 0.09747 lineto 0.91972 0.09734 lineto 0.92048 0.09725 lineto 0.92125 0.0972 lineto 0.92201 0.09721 lineto 0.92277 0.09728 lineto 0.92353 0.09743 lineto 0.9243 0.09767 lineto 0.92506 0.098 lineto 0.92582 0.09842 lineto 0.92735 0.09956 lineto 0.95177 0.12084 lineto 0.97619 0.14182 lineto Mfstroke grestore grestore % End of Graphics MathPictureEnd :[font = text; inactive; pageBreakBelow; preserveAspect; ] The Gibbs phenomenon is evident from these graphs, but with a new twist: the graph of the of the first 10 terms goes up at 0, but the graph of the first 20 terms goes down! :[font = subsection; inactive; preserveAspect; startGroup; ] Exercises :[font = subsubsection; inactive; preserveAspect; ] #1 Use Mathematica to graph the following Bessel functions: ;[s] 3:0,0;8,1;19,2;61,-1; 3:1,10,8,Times,1,12,0,0,0;1,10,8,Times,3,12,0,0,0;1,10,8,Times,1,12,0,0,0; :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 0; pictureWidth = 140; pictureHeight = 83; ] %! %%Creator: Mathematica %%AspectRatio: 0.592857 200 dict begin MathPictureStart /Courier findfont 10 scalefont setfont % Scaling calculations 0 1 0 1 [ [ 0 0 0 0 ] [ 140 83 0 0 ] ] MathScale % Start of Graphics %!PS-Adobe-2.0 EPSF-2.0 %%Title: %%Creator: EquationBuilder %%CreationDate: Mon Aug 3 19:37:34 1992 %%For: gray %%DocumentFonts: (atend) 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/.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 0 139.775208 82.669975 rectclip 0 0 139.775208 82.669975 rectclip 0 setgray 1 setalpha 1 1 scale /Symbol findfont 12 scalefont 48 exch defineuserobject 48 execuserobject 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execuserobject setfont 123.623878 18.291973 moveto ($) [0 0] xyshow 1 1 scale 0 setgray 1 setalpha 1 1 scale 48 execuserobject setfont 18.145599 3.297997 moveto ($) [0 0] xyshow 1 1 scale 1 1 scale 48 execuserobject setfont 29.785601 3.291996 moveto ($) [0 0] xyshow 1 1 scale 0 setgray 1 setalpha 1 1 scale 48 execuserobject setfont 67.474541 3.297997 moveto ($) [0 0] xyshow 1 1 scale 1 1 scale 48 execuserobject setfont 123.623878 3.291996 moveto ($) [0 0] xyshow 1 1 scale 0 setgray 1 setalpha /Times-BoldItalic findfont 12 scalefont 49 exch defineuserobject 49 execuserobject setfont 128.147873 3.189996 moveto (.xxYxxJxxJxxJxxJxxJ) [-36.618668 0 -68.583603 0 -18.972 0 87.555603 14.999975 -65.610001 1.907349e-06 -24.919201 -1.907349e-06 98.15654 13.806 -74.770935 1.907349e-06 -21.852001 0 96.62294 13.805998 -77.020935 0 -17.352001 3.814697e-06 94.37294 13.805996 -77.020935 0 -17.352001 3.814697e-06 94.37294 13.806004 -77.020935 -7.629395e-06 -17.352001 7.629395e-06 0 0 0 0] xyshow /Symbol findfont 9 scalefont 60 exch defineuserobject 60 execuserobject setfont 0 0 moveto () [-1.159154e-41 1.680567e-18] xyshow /Times-Roman findfont 9 scalefont 51 exch defineuserobject 51 execuserobject setfont 11.881601 1.390021 moveto (02105210) [2.973598 14.999977 -7.5672 0 6.0336 13.733974 -4.5 0 2.250002 13.878027 0 13.806 0 13.806 0 0 0 0] xyshow 48 execuserobject setfont 100.862534 3.189996 moveto (<<<<<<-<<-<<-<<-) [-19.254662 0 19.254662 14.999975 -19.254662 0 26.881996 13.806 -19.254669 0 -24.587997 0 43.842667 13.806 -19.254669 0 -24.587997 0 43.842667 13.806 -19.254669 0 -24.587997 0 43.842667 13.806007 -19.254669 0 -24.587997 0 0 0 0 0] xyshow /Times-Roman findfont 12 scalefont 50 exch defineuserobject 50 execuserobject setfont 116.783875 3.189996 moveto (050,050,0505,0505,0505,0505) [-6 0 -38.509338 0 55.873337 14.999975 -11.363998 0 -6 0 -38.509338 0 63.500671 13.806 -11.364006 0 -6 0 -38.509331 0 -6 0 61.873337 13.806004 -11.364006 -3.814697e-06 -6 0 -38.509331 0 -6 0 61.873337 13.806004 -11.364006 -3.814697e-06 -6 0 -38.509331 0 -6 0 61.873337 13.806007 -11.364006 0 -6 0 -38.509331 0 -6 0 0 0 0 0] xyshow %%Trailer %%DocumentFonts: Times-BoldItalic %%+ Times-Roman %%+ Symbol %End of Graphics MathPictureEnd end :[font = subsubsection; inactive; preserveAspect; ] #2 :[font = text; inactive; preserveAspect; ] Use Mathematica to find the second positive zero of J0, in other words, the second number y such that y > 0 and J0(y) = 0. Do this by drawing successive graphs of J0 that narrow in on y. Find y correct to 3 decimal places. ;[s] 17:0,0;4,1;15,2;53,3;54,4;55,5;92,6;94,7;117,8;118,9;119,10;120,11;121,12;169,13;171,14;172,15;174,16;240,-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,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,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,Times,2,12,0,0,0;1,10,8,Times,66,12,0,0,0;1,10,8,Times,2,12,0,0,0;1,11,8,Times,0,12,0,0,0; :[font = subsubsection; inactive; preserveAspect; ] #3 :[font = text; inactive; pageBreakBelow; preserveAspect; endGroup; endGroup; ] Use Mathematica to find the second positive zero of J0 using FindRoot. ;[s] 8:0,0;4,1;15,2;53,3;54,4;56,5;63,6;71,7;73,-1; 8:1,11,8,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;1,10,8,Times,66,12,0,0,0;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 = section; inactive; Cclosed; preserveAspect; startGroup; ] The Circular Drumhead :[font = text; inactive; preserveAspect; ] Mathematica can be used to draw 3-dimensional graphs of the vibrating drum. The solutions of the polar coordinate version ;[s] 2:0,0;11,1;122,-1; 2:1,10,8,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 = 126; pictureHeight = 31; ] %! %%Creator: Mathematica %%AspectRatio: 0.246032 200 dict begin MathPictureStart /Courier findfont 10 scalefont setfont % Scaling calculations 0 1 0 1 [ [ 0 0 0 0 ] [ 126 31 0 0 ] ] MathScale % Start of Graphics %!PS-Adobe-2.0 EPSF-2.0 %%Title: %%Creator: EquationBuilder %%CreationDate: Mon Aug 3 16:59:13 1992 %%For: gray %%DocumentFonts: (atend) %%Pages: 0 1 %%BoundingBox: 0 0 126 31 %%Translation: 1 11.878704 124.049332 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 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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 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/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 0 125.049332 30.284416 rectclip 0 0 125.049332 30.284416 rectclip 0 setgray 1 setalpha 0.48 setlinewidth 73.224998 15.878704 moveto 87.037994 15.878704 lineto stroke 0 setgray 1 setalpha 0.48 setlinewidth 98.959335 15.878704 moveto 118.685333 15.878704 lineto stroke 0 setgray 1 setalpha 1 2.585978 scale /Symbol findfont 12 scalefont 47 exch defineuserobject 47 execuserobject setfont 38.749668 3.248308 moveto ($) [0 0] xyshow 1 0.386701 scale 1 2.582989 scale 47 execuserobject setfont 119.525337 3.249415 moveto ($) [0 0] xyshow 1 0.387148 scale 0 setgray 1 setalpha /Times-BoldItalic findfont 12 scalefont 44 exch defineuserobject 44 execuserobject setfont 99.459335 21.37468 moveto (uuucu) [-25.734337 0.054001 -30.175331 -8.549976 -16.392 0 -26.157667 0 0 0 0 0] xyshow /Symbol findfont 9 scalefont 45 exch defineuserobject 45 execuserobject setfont 111.558334 19.574705 moveto (jjrrr) [-5.427002 0 -25.734337 0.053999 -25.234329 -8.549976 -4.940998 0 0 0 0 0] xyshow /Times-BoldItalic findfont 9 scalefont 69 exch defineuserobject 69 execuserobject setfont 10.174 11.078729 moveto (tt) [-2.502 0 0 0 0 0] xyshow 47 execuserobject setfont 102.678337 3.76 moveto (r+r+=) [-12.973671 9.118704 -12.867165 -8.231412 -12.867168 8.231412 -46.734001 0 0 0 0 0] xyshow /Times-Roman findfont 9 scalefont 46 exch defineuserobject 46 execuserobject setfont 109.266335 8.114704 moveto (22) [-76.78067 9.718704 0 0 0 0] xyshow %%Trailer %%DocumentFonts: Times-BoldItalic %%+ Times-Roman %%+ Symbol %End of Graphics MathPictureEnd end :[font = text; inactive; preserveAspect; ] of the 2-dimensional wave equation are of the form :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 0; pictureWidth = 219; pictureHeight = 20; ] %! %%Creator: Mathematica %%AspectRatio: 0.091324 200 dict begin MathPictureStart /Courier findfont 10 scalefont setfont % Scaling calculations 0 1 0 1 [ [ 0 0 0 0 ] [ 219 20 0 0 ] ] MathScale % Start of Graphics %!PS-Adobe-2.0 EPSF-2.0 %%Title: %%Creator: EquationBuilder %%CreationDate: Mon Aug 3 17:07:49 1992 %%For: gray %%DocumentFonts: (atend) %%Pages: 0 1 %%BoundingBox: 0 0 219 20 %%Translation: 1 5.203605 217.172668 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'?'JKOf0M'9NV$2,7,X_XRS2ihA"VLe`-6WLgVDQ[ % K-Oh*sQ3\5X&jqDA/d.K[>m^Q5J[c<T!&,[Tin#Mqf\EPQp`n<3FPaBu@nDhna*n=K8V8]^g % &Ph$W+?>=8/K#JVW;G9-88oJL(Oh['1U:3`L@o]lIuWp?:"^H/d3!JP0!\R-jBZK8pD@DIN& % hI_^8*I1j!'<VA%_SCdPN3:hAs\OJ^'oBS.e=]gh`lO&(Vi958YOho/;)O!k8/8LS`:i1(9, % rH^iu!PfY%KZGM\$co%)R!RE<#,$;G)AnSTaLU;;KC&!V)Z]EQ(VgLe?<CEd!cHYaE-)rf@# % =CC!m3_W"um`\/dB*1."VSpUOY?+n,jdm#2g_fJcLm!$=2Tb,7/&97gK2GZ(Vb"DX]%.Ikna % Y@=]Kni1OEc4<XqIBt2Po*n++:a`Y!#Yh]`N4Z3_7)Fiqa&l'cM6lU(RA>NdW@^tEBFr'kLJ % s$6tih.W=.9mfHf#_>WOr,F#16OckALU\=(dKS-4Z8:E!c05OS;pT:VgqsB\-d8p4JlkG"40 % @?U>)b1j,"B,JK1SJjm":nLk_[?UB]Qf.>o13F/+^TmMP#He,G/uO[ptTeSVf;>A-EB/A#[Y % P&qq*Z0+*f."(9l.'AEog'Fg@6<6aTOOb1mA<MQBTSt,T3#PmdAY,#rVi/]Rm!Jnon\uQ`,0 % V")h`1lOWOUa[Qos5CR;@\>E@bdhKt?h-I/:@b/r(n!)l0i(k)1mTf^eq";TQ*f0o<M'b"K\ % (M>(__NWC8/XVs8J5f^MA0F#U]/3U;7-c,rN@oBdM/#!G+m:\+;9@EL;8P:7Yf,q2$M?V>?> % ]7j\;QF`ErGE#9fEkLDW%KH=W.?3(Qn8UKl7*47"tCS[GMCNZ4gViRm]5E`!LM'.K8MVT6Jn % Vai,2m9.54.1Em4'.CPfn$fM5@$)]rFTJ:`;THOSF0!soeuhZ?b:Vc!n#-l.OVo0L`M)nuam % (`>ol&OphObbr8HaY+/s+H?O8";'%"IgH.h"MkBESG6S"x % EndExpression N 900 384 E 8c S 186ce R 3b33190f 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 gsave 0 0 218.172668 19.308605 rectclip 0 0 218.172668 19.308605 rectclip 0 setgray 1 setalpha 1 1.001111 scale /Symbol findfont 12 scalefont 47 exch defineuserobject 47 execuserobject setfont 8.236 6.301391 moveto ($) [0 0] xyshow 1 0.99889 scale 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