Education Sampler 1992 [NeXTSTEP]/Education_1992_Sampler.iso/Mathematics/Notebooks/PartialDifferentials/legendre.ma

Text File | 1992-08-04 | 235KB | 6,610 lines

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Legendre Polynomials
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The Definitions of the Legendre Polynomials
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        An important class of polynomials that we encounter when we
study Laplace's Equation are the Legendre Polynomials.  The  Legendre
Polynomial  is a polynomial of degree  n;  it can be defined in several ways.
For example, if we require that
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:[font = text; inactive; preserveAspect; ]
then the 's are determined uniquely.  However, there are more explicit ways
to find the Legendre Polynomials.  There is a recursion relation to get  Pn(x).
Temporarily, we shall denote the Legendre Polynomial  in Mathematica as
LP[n,x].  In Mathematica we type in the commands
;[s]
12:0,0;149,1;150,2;151,3;152,4;153,5;213,6;224,7;228,8;236,9;241,10;252,11;277,-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,Times,2,12,0,0,0;1,11,8,Times,0,12,0,0,0;
:[font = input; preserveAspect; ]
LP[0,x_]:=1
    LP[1,x_]:=x
    LP[n_,x_]:= LP[n,x]=
      (1/n)((2n-1)x LP[n-1,x]-(n-1)LP[n-2,x])
:[font = text; inactive; preserveAspect; ]
(We could have used
    LP[n_,x_]:=
        (1/n)((2n-1)x LP[n-1,x]-(n-1)LP[n-2,x])
but this would have been much slower.)
;[s]
3:0,0;24,1;85,2;123,-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 = text; inactive; preserveAspect; ]
We get a function  LP  of the two variables   n and  x.  For example, we
type
;[s]
7:0,0;19,1;22,2;46,3;48,4;53,5;54,6;77,-1;
7:1,11,8,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;
:[font = input; preserveAspect; startGroup; ]
LP[5,x]//Simplify
:[font = output; output; inactive; pageBreakBelow; noPageBreak; dontNoPageBreakBelow; preserveAspect; endGroup; ]
(x*(15 - 70*x^2 + 63*x^4))/8
;[o]
            2       4
x (15 - 70 x  + 63 x )
----------------------
          8
:[font = text; inactive; preserveAspect; ]
To graph  LP[5,x]  from  -1  to  1 we type
;[s]
7:0,0;10,1;18,2;26,3;27,4;33,5;34,6;42,-1;
7:1,11,8,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;
:[font = input; preserveAspect; startGroup; ]
Plot[LP[5,x],{x,-1,1}];
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The most straightforward command, namely
Integrate[LP[n,x]^2,{x,-1,1}]
doesn't work because "LP[n,x]" is too general for Mathematica.  However,
the following command works:
;[s]
5:0,0;41,1;70,2;121,3;132,4;173,-1;
5:1,11,8,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;
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Integrate[LP[5,x]^2,{x,-1,1}]
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2/11
;[o]
2
--
11
:[font = text; inactive; preserveAspect; ]
Integrating  LP[n,x]^2 for a few other values of  leads us to the heuristic
conclusion that
;[s]
3:0,0;13,1;22,2;91,-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;
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} def /rad3 {
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} def /rad4 {
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} 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
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    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 {
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} def /prod {
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} def /cdot {
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} def /vdot {
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} def /ddot {
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} def /hbar {
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for general  n.  This is in fact the correct answer.
;[s]
3:0,0;13,1;14,2;52,-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;
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        Actually Mathematica has a built-in function for the  Legendre
Polynomials, namely  LegendreP[n,x].  Computations using
LegendreP[n,x] are much faster than those using  LP[n,x].
This can be checked with the commands
;[s]
9:0,0;17,1;28,2;92,3;107,4;129,5;143,6;178,7;185,8;225,-1;
9:1,11,8,Times,0,12,0,0,0;1,10,8,Times,2,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Courier,0,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Courier,0,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Courier,0,12,0,0,0;1,11,8,Times,0,12,0,0,0;
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LP[10,x]//Simplify//Timing
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{3.416666666666666*Second,
  (-63 + 3465*x^2 - 30030*x^4 + 90090*x^6 -
     109395*x^8 + 46189*x^10)/256}
;[o]
                              2          4
{3.41667 Second, (-63 + 3465 x  - 30030 x  +

            6           8          10
     90090 x  - 109395 x  + 46189 x  ) / 256}
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LegendreP[10,x]//Timing
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{0.0833333333333333*Second,
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     109395*x^8 + 46189*x^10)/256}
;[o]
{0.0833333 Second,

               2          4          6
  (-63 + 3465 x  - 30030 x  + 90090 x  -

             8          10
     109395 x  + 46189 x  ) / 256}
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This function may be plotted using the command
:[font = input; preserveAspect; startGroup; ]
Plot[LegendreP[10,x],{x,-1,1}];
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Exercise
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List several properties of the Legendre polynomials suggested by the graphs.
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The Legendre Differential Equation
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        Each Legendre polynomial satisfies a second order linear differential
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called the  Legendre differential equation.  Mathematica can be used to verify
that  Legendre[20,x] is indeed a solution of the Legendre Equation for
n = 20.  For this we will need to use the differentiation operator  D.  To find
the exact syntax type ?D.  Next define a function  y[x]  by
;[s]
11:0,0;45,1;56,2;85,3;99,4;151,5;152,6;219,7;220,8;252,9;255,10;291,-1;
11:1,11,8,Times,0,12,0,0,0;1,10,8,Times,2,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,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,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 = input; preserveAspect; ]
y[x_]:=LegendreP[20,x]
:[font = text; inactive; pageBreakBelow; preserveAspect; ]
The first and second derivatives of  y[x]  are  D[y[x],x] and
D[y[x],x,2].  Then  we compute
;[s]
7:0,0;37,1;42,2;48,3;57,4;62,5;73,6;92,-1;
7:1,11,8,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;
:[font = input; preserveAspect; startGroup; ]
lde=(1-x^2)D[y[x],{x,2}]-2 x D[y[x],x] +
 20 21 y[x]
:[font = output; output; inactive; preserveAspect; endGroup; ]
((1 - x^2)*(-19399380 + 4015671660*x^2 -
       133855722000*x^4 +
       1686582097200*x^6 -
       10480903032600*x^8 +
       36100888223400*x^10 -
       72201776446800*x^12 +
       83309742054000*x^14 -
       51374340933300*x^16 +
       13095420237900*x^18))/262144 -
  (x*(-19399380*x + 1338557220*x^3 -
       26771144400*x^5 + 240940299600*x^7 -
       1164544781400*x^9 +
       3281898929400*x^11 -
       5553982803600*x^13 +
       5553982803600*x^15 -
       3022020054900*x^17 +
       689232644100*x^19))/131072 +
  (105*(46189 - 9699690*x^2 +
       334639305*x^4 - 4461857400*x^6 +
       30117537450*x^8 - 116454478140*x^10 +
       273491577450*x^12 -
       396713057400*x^14 +
       347123925225*x^16 -
       167890003050*x^18 + 34461632205*x^20))
    /65536
;[o]
       2                           2
((1 - x ) (-19399380 + 4015671660 x  -

                     4                  6
       133855722000 x  + 1686582097200 x  -

                       8
       10480903032600 x  +

                       10
       36100888223400 x   -

                       12
       72201776446800 x   +

                       14
       83309742054000 x   -

                       16
       51374340933300 x   +

                       18
       13095420237900 x  )) / 262144 -

                                3
  (x (-19399380 x + 1338557220 x  -

                    5                 7
       26771144400 x  + 240940299600 x  -

                      9
       1164544781400 x  +

                      11
       3281898929400 x   -

                      13
       5553982803600 x   +

                      15
       5553982803600 x   -

                      17                 19
       3022020054900 x   + 689232644100 x  ))

     / 131072 + (105

                       2              4
     (46189 - 9699690 x  + 334639305 x  -

                   6                8
       4461857400 x  + 30117537450 x  -

                     10                 12
       116454478140 x   + 273491577450 x   -

                     14                 16
       396713057400 x   + 347123925225 x   -

                     18                20
       167890003050 x   + 34461632205 x  )) \

    / 65536
:[font = text; inactive; preserveAspect; ]
The result is complicated, but it reduces to 0 when we apply
the command Simplify.
;[s]
3:0,0;73,1;81,2;83,-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; ]
lde//Simplify
:[font = output; output; inactive; pageBreakBelow; dontNoPageBreakBelow; preserveAspect; endGroup; endGroup; ]
0
;[o]
0
:[font = section; inactive; Cclosed; preserveAspect; startGroup; ]
Checking Formulas for Legendre Polynomials
with Mathematica
;[s]
2:0,0;49,1;60,-1;
2:1,16,12,Times,1,18,0,0,0;1,16,12,Times,2,18,0,0,0;
:[font = text; inactive; preserveAspect; ]
There are several equivalent ways to define the Legendre polynomials in
addition to the recursion formula definition.  For example we can use the
so-called generating function method.  Let
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It turns out that the coefficients coincide precisely with the Legendre functions.
It is easy to use Mathematica to verify this fact for the first few coefficients:
;[s]
3:0,0;101,1;112,2;165,-1;
3:1,11,8,Times,0,12,0,0,0;1,10,8,Times,2,12,0,0,0;1,11,8,Times,0,12,0,0,0;
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Series[1/(1-2 x r +r^2)^(1/2),
{r,0,4}]//Simplify
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SeriesData[r, 0, {1, x, (-1 + 3*x^2)/2,
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   (3 - 30*x^2 + 35*x^4)/8}, 0, 5, 1]
;[o]
                   2   2
          (-1 + 3 x ) r
1 + x r + -------------- +
                2

             2   3
  x (-3 + 5 x ) r
  ---------------- +
         2

           2       4   4
  (3 - 30 x  + 35 x ) r        5
  ---------------------- + O[r]
            8
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Another way to generate the Legendre polynomials is by means of
 Rodrigues's formula:
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:[font = text; inactive; preserveAspect; ]
To check that the first few  Pn's given by this formula coincide with the
Legendre polynomials we again use Mathematica.  First we define
;[s]
6:0,0;29,1;30,2;31,3;108,4;119,5;138,-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; ]
ff[n_,x_]:=1/(2^n n!)D[(x^2-1)^n,{x,n}]
:[font = text; inactive; preserveAspect; ]
Then the following command generates the first few Legendre polynomials
:[font = input; preserveAspect; startGroup; ]
Table[ff[n,x],{n,0,5}]//Simplify
:[font = output; output; inactive; preserveAspect; endGroup; ]
{1, x, (-1 + 3*x^2)/2, (x*(-3 + 5*x^2))/2,
  (3 - 30*x^2 + 35*x^4)/8,
  (x*(15 - 70*x^2 + 63*x^4))/8}
;[o]
               2             2
       -1 + 3 x   x (-3 + 5 x )
{1, x, ---------, -------------,
           2            2

          2       4              2       4
  3 - 30 x  + 35 x   x (15 - 70 x  + 63 x )
  -----------------, ----------------------}
          8                    8
:[font = text; inactive; preserveAspect; ]
    Finally the Legendre polynomials are given by the following integral
formula:
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:[font = text; inactive; preserveAspect; ]
We can check this formula by first defining
:[font = input; preserveAspect; ]
gg[n_,x_]:=(1/Pi)Integrate[
   (x +(x^2-1)^(1/2)Cos[phi])^n,{phi,0,Pi}]
:[font = text; inactive; preserveAspect; ]
That we again get the Legendre polynomials is confirmed by:
:[font = input; preserveAspect; startGroup; ]
Table[gg[n,x],{n,0,5}]//Simplify
:[font = output; output; inactive; preserveAspect; endGroup; endGroup; ]
{1, x, (-1 + 3*x^2)/2, (x*(-3 + 5*x^2))/2,
  (3 - 30*x^2 + 35*x^4)/8,
  (x*(15 - 70*x^2 + 63*x^4))/8}
;[o]
               2             2
       -1 + 3 x   x (-3 + 5 x )
{1, x, ---------, -------------,
           2            2

          2       4              2       4
  3 - 30 x  + 35 x   x (15 - 70 x  + 63 x )
  -----------------, ----------------------}
          8                    8
^*)