{VERSION 4 0 "IBM INTEL NT" "4.0" }
{USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 
1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 
0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }
{CSTYLE "" -1 256 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 
257 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 0 1 0 0 
0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 1 0 0 0 0 0 
0 0 0 1 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }
{CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 
262 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 263 "" 0 1 0 0 
0 0 1 0 0 0 0 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Tim
es" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }
{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 
2 2 2 2 1 1 1 1 }1 1 0 0 6 6 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 2" -1 
4 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 
0 4 4 1 0 1 0 2 2 0 1 }{PSTYLE "R3 Font 0" -1 256 1 {CSTYLE "" -1 -1 "
Times" 1 12 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }
{PSTYLE "R3 Font 2" -1 257 1 {CSTYLE "" -1 -1 "Courier" 1 12 0 0 0 1 
2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 
258 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 
0 0 0 0 1 0 1 0 2 2 0 1 }}
{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "Unit 3: Ordinary Different
ial Equations - Part 2" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 34 "Chapter 16: The Eigenvalue Problem" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 "Section 16.3:  Legendre
's equation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "
" {TEXT -1 9 "Copyright" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 48 "Copyright * 2001 by Addison Wesley Longman, Inc." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 302 "All righ
ts reserved.  No part of this publication may be reproduced, stored in
 a retrieval system, or transmitted, in any form or by any means, elec
tronic, mechanical, photocopying, recording, or otherwise, without the
 prior written permission of the publisher. Printed in the United Stat
es of America." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 
0 "" {TEXT -1 15 "Initializations" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 31 "with(student):\nwith(orthopoly):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "interface(showassumed=0);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "
" {TEXT -1 45 "A Singular Sturm-Liouville Eigenvalue Problem" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "Legendre
's equation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "(1-x^2)*`y''`(x)-2*x*`y'`(x)+lambda*y(x
)=0" "6#/,(*&,&\"\"\"F'*$%\"xG\"\"#!\"\"F'-%$y''G6#F)F'F'*(F*F'F)F'-%#
y'G6#F)F'F+*&%'lambdaGF'-%\"yG6#F)F'F'\"\"!" }{TEXT -1 1 " " }}{PARA 
0 "" 0 "" {TEXT -1 8 "that is," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "q := (1-x^2)*diff(y(x),x,x)-
2*x*diff(y(x),x)+lambda*y(x)=0;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "on the interval " }
{XPPEDIT 18 0 "-1<=x" "6#1,$\"\"\"!\"\"%\"xG" }{XPPEDIT 18 0 "``<=1" "
6#1%!G\"\"\"" }{TEXT -1 83 " typically arises in physical problems bei
ng solved in spherical coordinates.  With" }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 258 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "p(x)=1-x^2" "6
#/-%\"pG6#%\"xG,&\"\"\"F)*$F'\"\"#!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 
0 "" {TEXT -1 52 "it's clear the equation is in the self-adjoint form \+
" }}{PARA 258 "" 0 "" {TEXT -1 2 " (" }{XPPEDIT 18 0 "p*`y'`" "6#*&%\"
pG\"\"\"%#y'GF%" }{TEXT -1 1 ")" }{XPPEDIT 18 0 "`'`" "6#%\"'G" }
{TEXT -1 3 " + " }{XPPEDIT 18 0 "(q+lambda*w)*y=0" "6#/*&,&%\"qG\"\"\"
*&%'lambdaGF'%\"wGF'F'F'%\"yGF'\"\"!" }}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "q=0" "6#/%\"qG\"
\"!" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "w=1" "6#/%\"wG\"\"\"" }{TEXT 
-1 27 ".  Although the values for " }{XPPEDIT 18 0 "q" "6#%\"qG" }
{TEXT -1 28 " and the weighting function " }{XPPEDIT 18 0 "w" "6#%\"wG
" }{TEXT -1 118 " are as simple as they could be, solving this equatio
n presents unique challenges.  In fact, we now realize that both " }
{XPPEDIT 18 0 "x=-1" "6#/%\"xG,$\"\"\"!\"\"" }{TEXT -1 5 " and " }
{XPPEDIT 18 0 "x=1" "6#/%\"xG\"\"\"" }{TEXT -1 95 " are regular singul
ar points of the differential equation, and the problem is singular be
cause " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" }{TEXT -1 33 " is not \+
positive on the interval " }{XPPEDIT 18 0 "[-1,1]" "6#7$,$\"\"\"!\"\"F
%" }{TEXT -1 84 ".  Hence, there are no boundary conditions at the end
points!  Instead, the solution " }{XPPEDIT 18 0 "y(x)" "6#-%\"yG6#%\"x
G" }{TEXT -1 70 " must be bounded throughout the interval, especially \+
at each endpoint." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 61 "Our boundary value problem, then is the differential equa
tion" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 2 "
 (" }{XPPEDIT 18 0 "((1-x^2)*`y'`)" "6#*&,&\"\"\"F%*$%\"xG\"\"#!\"\"F%
%#y'GF%" }{TEXT -1 1 ")" }{XPPEDIT 18 0 "`'`+lambda*y=0" "6#/,&%\"'G\"
\"\"*&%'lambdaGF&%\"yGF&F&\"\"!" }}{PARA 0 "" 0 "" {TEXT -1 36 "along \+
with the \"boundary\" conditions" }}{PARA 258 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "y(-1)" "6#-%\"yG6#,$\"\"\"!\"\"" }{TEXT -1 7 " finite" 
}}{PARA 258 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y(1)" "6#-%\"yG6#\"
\"\"" }{TEXT -1 7 " finite" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 
-1 33 "Solving the Differential Equation" }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 45 "The eigenvalue problem base
d on the equation " }{XPPEDIT 18 0 "`y''`=lambda*y" "6#/%$y''G*&%'lamb
daG\"\"\"%\"yGF'" }{TEXT -1 685 " presented some new challenges, but s
ince the solutions were the familiar trig and exponential functions, w
e ultimately determined both the eigenvalues and eigenfunctions.  The \+
Bessel equation presented some additional challenges since the functio
ns which satisfy the equation, the Bessel functions, are not part of t
he suite of elementary functions studied in calculus.  Legendre's equa
tion poses further challenges because, unlike the Bessel equation, the
re are not \"Legendre functions\" with which we can ultimately constru
ct the solution.  Each of these three classic BVPs differ in key ways,
 making it seem to the beginning student that there is no coherence in
 the subject at all." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 218 "We avoided grinding out the power series solution of B
essel's equation because we ultimately relied on Maple to generate the
 functions for us.  We cannot do that with the solutions of Legendre's
 equation since Maple's " }{TEXT 256 6 "dsolve" }{TEXT -1 131 " comman
d does not yield a useful solution to this equation.  In fact, the two
 linearly independent solutions Maple provides, namely" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "q1 := \+
dsolve(q,y(x),output=basis):\nY1 := q1[1];\nY2 := q1[2];" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 206 "are unweildy, and contain the Legendre and associated Legendre
 functions which Maple does not readily manipulate.  For example, Mapl
e generates  an imposing series expansion for the first (simpler) solu
tion" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 13 "series(Y1,x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 69 "Therefore, we now reve
rt to a series solution at a fundamental level." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 
1 {PARA 4 "" 0 "" {TEXT -1 21 "Power Series Solution" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 6 "About " }
{XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" }{TEXT -1 44 ", seek a Taylor seri
es solution of the form " }{XPPEDIT 18 0 "y(x)=Sum(a[k]*x^k,k=0..infin
ity)" "6#/-%\"yG6#%\"xG-%$SumG6$*&&%\"aG6#%\"kG\"\"\")F'F/F0/F/;\"\"!%
)infinityG" }{TEXT -1 77 ".  A seventh-order finite approximating sum \+
satisfying the initial conditions" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 258 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y(0)=A" "6#/-%\"yG6#
\"\"!%\"AG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 3 "and" }}
{PARA 258 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`y'`(0)=B" "6#/-%#y'G
6#\"\"!%\"BG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 2 "is" }}
{PARA 258 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(-lambda*(lambda-6)*(
lambda-20)*x^6/720+lambda*(lambda-6)*x^4/24-lambda*x^2/2+1)*A" "6#*&,*
*,%'lambdaG\"\"\",&F&F'\"\"'!\"\"F',&F&F'\"#?F*F'%\"xGF)\"$?(F*F***F&F
',&F&F'F)F*F'F-\"\"%\"#CF*F'*(F&F'*$F-\"\"#F'F5F*F*F'F'F'%\"AGF'" }
{TEXT -1 37 "                                     " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 29 "                       \+
      " }{XPPEDIT 18 0 "``+(-(lambda-12)*(lambda-2)*(lambda-30)*x^7/50
40+(lambda-2)*(lambda-12)*x^5/120+(2-lambda)*x^3/6+x)*B" "6#,&%!G\"\"
\"*&,**,,&%'lambdaGF%\"#7!\"\"F%,&F*F%\"\"#F,F%,&F*F%\"#IF,F%%\"xG\"\"
(\"%S]F,F,**,&F*F%F.F,F%,&F*F%F+F,F%F1\"\"&\"$?\"F,F%*(,&F.F%F*F,F%*$F
1\"\"$F%\"\"'F,F%F1F%F%%\"BGF%F%" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 62 "The solution appears to split into a series of even powers of \+
" }{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 14 " (multiplying " }
{XPPEDIT 18 0 "A=y(0)" "6#/%\"AG-%\"yG6#\"\"!" }{TEXT -1 31 ") and a s
eries of od powers of " }{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 14 " (m
ultiplying " }{XPPEDIT 18 0 "B=`y'`(0)" "6#/%\"BG-%#y'G6#\"\"!" }
{TEXT -1 13 ").  Also, if " }{XPPEDIT 18 0 "lambda" "6#%'lambdaG" }
{TEXT -1 23 " is an integer such as " }{XPPEDIT 18 0 "2,6,12" "6%\"\"#
\"\"'\"#7" }{TEXT -1 249 ", or 30, coefficients will be zero, and the \+
possibility exists that instead of an infinite series, the solution is
 just a polynomial.  However, verification of these suspicions require
s knowing more about the general form of the general coefficient " }
{XPPEDIT 18 0 "a[k])" "6#&%\"aG6#%\"kG" }{TEXT -1 1 "." }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 88 "These calculations c
an be implemented in Maple if we first set to 8 the system variable " 
}{TEXT 257 5 "Order" }{TEXT -1 27 ", whose default value is 6," }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 11 "Order := 8;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 67 "then impose the initial conditions
 y(0) = A and y'(0) = B, yielding" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "q2 := dsolve(\{q,y(0)=A,D(y)
(0)=B\},y(x),series);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 64 "In order to manipulate the resul
ting series expansion, the term " }{XPPEDIT 18 0 "O(x^8)" "6#-%\"OG6#*
$%\"xG\"\")" }{TEXT -1 38 " needs to be removed.  Thus, we obtain" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 31 "q3 := convert(rhs(q2),polynom);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 89 "To find a pat
tern in this solution, group terms with respect to y(0) and y'(0), obt
aining" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 33 "q4 := collect(q3,[A,B,x],factor);" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 18 "R
ecursion Relation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 87 "To obtain a general recursion formula for the coef
ficients, seek a solution of the form" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 258 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y = y(0)" "6#/%\"
yG-F$6#\"\"!" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(a[2*n]*x^(2*n),n=0..
infinity) " "6#-%$SumG6$*&&%\"aG6#*&\"\"#\"\"\"%\"nGF,F,)%\"xG*&F+F,F-
F,F,/F-;\"\"!%)infinityG" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``+`y'`(0)" 
"6#,&%!G\"\"\"-%#y'G6#\"\"!F%" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(a[2
*n+1]*x^(2*n+1),n=0..infinity)" "6#-%$SumG6$*&&%\"aG6#,&*&\"\"#\"\"\"%
\"nGF-F-F-F-F-)%\"xG,&*&F,F-F.F-F-F-F-F-/F.;\"\"!%)infinityG" }{TEXT 
-1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
33 "As seen in Section 14.1, the sum " }{XPPEDIT 18 0 "Sum(a[n]*x^n,n=
k..k+2)" "6#-%$SumG6$*&&%\"aG6#%\"nG\"\"\")%\"xGF*F+/F*;%\"kG,&F0F+\"
\"#F+" }{TEXT -1 64 " suffices, and substitution into the differential
 equation gives" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "(1-x^2)*(sum(a[n]*x^n,n=k..k+2))^`''`" 
"6#*&,&\"\"\"F%*$%\"xG\"\"#!\"\"F%)-%$sumG6$*&&%\"aG6#%\"nGF%)F'F2F%/F
2;%\"kG,&F6F%F(F%%#''GF%" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``-2*x*(sum(
a[n]*x^n,n=k..k+2))^`'`+lambda" "6#,(%!G\"\"\"*(\"\"#F%%\"xGF%)-%$sumG
6$*&&%\"aG6#%\"nGF%)F(F1F%/F1;%\"kG,&F5F%F'F%%\"'GF%!\"\"%'lambdaGF%" 
}{TEXT -1 1 " " }{XPPEDIT 18 0 "sum(a[n]*x^n,n=k..k+2)=0" "6#/-%$sumG6
$*&&%\"aG6#%\"nG\"\"\")%\"xGF+F,/F+;%\"kG,&F1F,\"\"#F,\"\"!" }{TEXT 
-1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
33 "Differentiating termwise leads to" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 258 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[k]*(k^2-k)*x^(k
-2)+a[k+1]*(k+k^2)*x^(k-1)+(a[k+2]*(k^2+3*k+2)-a[k]*(k^2+k-lambda))*x^
k" "6#,(*(&%\"aG6#%\"kG\"\"\",&*$F(\"\"#F)F(!\"\"F))%\"xG,&F(F)F,F-F)F
)*(&F&6#,&F(F)F)F)F),&F(F)*$F(F,F)F))F/,&F(F)F)F-F)F)*&,&*&&F&6#,&F(F)
F,F)F),(*$F(F,F)*&\"\"$F)F(F)F)F,F)F)F)*&&F&6#F(F),(*$F(F,F)F(F)%'lamb
daGF-F)F-F))F/F(F)F)" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{PARA 258 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``+a[k+1]*(lambda-3
*k-k^2-2)*x^(k+1)+a[k+2]*(lambda-6-k^2-5*k)*x^(k+2)=0" "6#/,(%!G\"\"\"
*(&%\"aG6#,&%\"kGF&F&F&F&,*%'lambdaGF&*&\"\"$F&F,F&!\"\"*$F,\"\"#F1F3F
1F&)%\"xG,&F,F&F&F&F&F&*(&F)6#,&F,F&F3F&F&,*F.F&\"\"'F1*$F,F3F1*&\"\"&
F&F,F&F1F&)F5,&F,F&F3F&F&F&\"\"!" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 40 "where the coefficients of
 each power of " }{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 61 " must sepa
rately vanish.  Setting to zero the coefficient of " }{XPPEDIT 18 0 "x
^k" "6#)%\"xG%\"kG" }{TEXT -1 7 " yields" }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 258 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[k+2]*(k^2+3*
k+2)-a[k]*(k^2+k-lambda)=0" "6#/,&*&&%\"aG6#,&%\"kG\"\"\"\"\"#F+F+,(*$
F*F,F+*&\"\"$F+F*F+F+F,F+F+F+*&&F'6#F*F+,(*$F*F,F+F*F+%'lambdaG!\"\"F+
F7\"\"!" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 16 "and solving for " }{XPPEDIT 18 0 "a[k+2]" "6#&%\"aG6
#,&%\"kG\"\"\"\"\"#F(" }{TEXT -1 41 " gives the desired recursion rela
tionship" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT 
-1 1 " " }{XPPEDIT 18 0 "a[k+2]=(k*(k+1)-lambda)/(k+2)/(k+1)" "6#/&%\"
aG6#,&%\"kG\"\"\"\"\"#F)*(,&*&F(F),&F(F)F)F)F)F)%'lambdaG!\"\"F),&F(F)
F*F)F0,&F(F)F)F)F0" }{TEXT -1 1 " " }{XPPEDIT 18 0 "a[k]=g(k)*a[k]" "6
#/&%\"aG6#%\"kG*&-%\"gG6#F'\"\"\"&F%6#F'F," }{TEXT -1 1 " " }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 56 "To implement these calculations in Maple, define the su
m" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 33 "Y := Sum(a[n] * x^n, n=k .. k+2);" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 59 "and
 substitute it into the differential equation, obtaining" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "q6 :
= subs(y(x) = Y, q);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "Differentiating termwise leads to
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 26 "q7 := simplify(value(q6));" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "where the \+
coefficient of each power of " }{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 
61 " must separately vanish.  Setting to zero the coefficient of " }
{XPPEDIT 18 0 "x^k" "6#)%\"xG%\"kG" }{TEXT -1 7 " yields" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "q8 :
= map(coeff,q7,x^k);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "and solving for " }{XPPEDIT 18 0 
"a[k+2]" "6#&%\"aG6#,&%\"kG\"\"\"\"\"#F(" }{TEXT -1 41 " gives the des
ired recursion relationship" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "factor(isolate(q8,a[k+2]));
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 29 "where we write the factor of " }{XPPEDIT 18 0 "a[k]
" "6#&%\"aG6#%\"kG" }{TEXT -1 29 " on the right as the function" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 39 "g := k -> (k*(k+1)-lambda)/(k+2)/(k+1);" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 0 "" }}}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 32 "Apply
ing the Boundary Conditions" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 24 "The recurrence relation " }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[
k+2] = a[k]*(k*(k+1)-lambda)/(k+2)/(k+1)" "6#/&%\"aG6#,&%\"kG\"\"\"\"
\"#F)**&F%6#F(F),&*&F(F),&F(F)F)F)F)F)%'lambdaG!\"\"F),&F(F)F*F)F2,&F(
F)F)F)F2" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
14 "suggests that " }{XPPEDIT 18 0 "a[0]" "6#&%\"aG6#\"\"!" }{TEXT -1 
53 " determines all other even-indexed coefficients, and " }{XPPEDIT 
18 0 "a[1]" "6#&%\"aG6#\"\"\"" }{TEXT -1 62 " determines all other odd
-indexed coefficients.  Moreover, if " }{XPPEDIT 18 0 "lambda" "6#%'la
mbdaG" }{TEXT -1 16 " is the integer " }{XPPEDIT 18 0 "k*(k+1)" "6#*&%
\"kG\"\"\",&F$F%F%F%F%" }{TEXT -1 6 ", then" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[k+2]=0" "
6#/&%\"aG6#,&%\"kG\"\"\"\"\"#F)\"\"!" }{TEXT -1 3 " = " }{XPPEDIT 18 
0 "a[k+4]=a[k+6]" "6#/&%\"aG6#,&%\"kG\"\"\"\"\"%F)&F%6#,&F(F)\"\"'F)" 
}{TEXT -1 3 " = " }{XPPEDIT 18 0 "`...`" "6#%$...G" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "so the series starting wi
th " }{XPPEDIT 18 0 "a[0]" "6#&%\"aG6#\"\"!" }{TEXT -1 5 " (if " }
{XPPEDIT 18 0 "k" "6#%\"kG" }{TEXT -1 18 " is even) or with " }
{XPPEDIT 18 0 "a[1]" "6#&%\"aG6#\"\"\"" }{TEXT -1 5 " (if " }{XPPEDIT 
18 0 "k" "6#%\"kG" }{TEXT -1 47 " is odd) becomes a simple polynomial,
 called a " }{TEXT 262 8 "Legendre" }{TEXT -1 1 " " }{TEXT 263 10 "pol
ynomial" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 23 "To determine the large-" }{XPPEDIT 18 0 "k" "6#%\"kG
" }{TEXT -1 64 " behavior of the coefficients in either infinite serie
s, express" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT 
-1 1 " " }{XPPEDIT 18 0 "a[2]" "6#&%\"aG6#\"\"#" }{TEXT -1 13 " in ter
ms of " }{XPPEDIT 18 0 "a[0]" "6#&%\"aG6#\"\"!" }{TEXT -1 1 " " }}
{PARA 0 "" 0 "" {TEXT -1 4 "then" }}{PARA 258 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "a[4]" "6#&%\"aG6#\"\"%" }{TEXT -1 13 " in terms of " }
{XPPEDIT 18 0 "a[2]" "6#&%\"aG6#\"\"#" }{TEXT -1 11 " and hence " }
{XPPEDIT 18 0 "a[0]" "6#&%\"aG6#\"\"!" }{TEXT -1 1 " " }}{PARA 0 "" 0 
"" {TEXT -1 4 "etc." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 65 "In Maple, we can do this by writing the iteration as th
e function" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "f := k-> if k<2 t
hen a[0] else f(k-2)*'g'(k-2) fi;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 11 "and, since " 
}{XPPEDIT 18 0 "f(2*k)=a[2*k]" "6#/-%\"fG6#*&\"\"#\"\"\"%\"kGF)&%\"aG6
#*&F(F)F*F)" }{TEXT -1 5 " for " }{XPPEDIT 18 0 "k=1,2,`...`" "6%/%\"k
G\"\"\"\"\"#%$...G" }{TEXT -1 57 ", the first few even-indexed coeffic
ients fit the pattern" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "for k from 0 to 5 do\na[2*k] = f(2*
k);\nod;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 11 "Thus, with " }{XPPEDIT 18 0 "a[0]=1" "6#/
&%\"aG6#\"\"!\"\"\"" }{TEXT -1 3 ",  " }{XPPEDIT 18 0 "a[2*m]" "6#&%\"
aG6#*&\"\"#\"\"\"%\"mGF(" }{TEXT -1 15 " is the product" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "q9 := \+
Product(g(2*n),n=0..m-1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 
"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "A similar result holds when \+
" }{XPPEDIT 18 0 "a[k+2]" "6#&%\"aG6#,&%\"kG\"\"\"\"\"#F(" }{TEXT -1 
13 " is based on " }{XPPEDIT 18 0 "a[1]" "6#&%\"aG6#\"\"\"" }{TEXT -1 
1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "E
valuating the expression for " }{XPPEDIT 18 0 "a[2*m]" "6#&%\"aG6#*&\"
\"#\"\"\"%\"mGF(" }{TEXT -1 33 " gives the closed-form expression" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 45 "assume(m,posint);\nq10 := simplify(value(q9));" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "f
rom which we can obtain" }}{PARA 258 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 
18 0 "limit(a[2*m]/[1/m],m = infinity) = c;" "6#/-%&limitG6$*&&%\"aG6#
*&\"\"#\"\"\"%\"mGF-F-7#*&F-F-F.!\"\"F1/F.%)infinityG%\"cG" }}{PARA 0 
"" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "c*`>`*0" "6#*(%\"cG\"\"\"%
\">GF%\"\"!F%" }{TEXT -1 16 " is the constant" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 258 "" 0 "" {XPPEDIT 18 0 "c = sqrt(Pi)/(GAMMA(1
/4+1*sqrt(1+4*lambda)/4)*GAMMA(1/4-1*sqrt(1+4*lambda)/4));" "6#/%\"cG*
&-%%sqrtG6#%#PiG\"\"\"*&-%&GAMMAG6#,&*&F*F*\"\"%!\"\"F**(F*F*-F'6#,&F*
F**&F1F*%'lambdaGF*F*F*F1F2F*F*-F-6#,&*&F*F*F1F2F**(F*F*-F'6#,&F*F**&F
1F*F8F*F*F*F1F2F2F*F2" }{TEXT -1 2 "  " }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 33 "To obtain this in Maple, replace " }
{XPPEDIT 18 0 "sqrt(1+4*lambda);" "6#-%%sqrtG6#,&\"\"\"F'*&\"\"%F'%'la
mbdaGF'F'" }{TEXT -1 39 " with a single symbol, say X, to obtain" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 36 "q11 := subs(sqrt(1+4*lambda)=X,q10);" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "Then, compu
te the limit" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 35 "q12 := limit(q11/(1/m),m=infinity);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 40 "and reverse the substitution, leading to" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "c = subs(X=
sqrt(1+4*lambda),q12);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 88 "Thus, the coefficients grow at \+
the same rate as the coefficients in the harmonic series " }{XPPEDIT 
18 0 "Sum(1/k,k=1..infinity)" "6#-%$SumG6$*&\"\"\"F'%\"kG!\"\"/F(;F'%)
infinityG" }{TEXT -1 155 ", a series known to diverge.  (See Section 7
.2.)  Hence, if this limit is true, solutions of Legendre's equation w
hich are infinite series will diverge at " }{XPPEDIT 18 0 "x=-1" "6#/%
\"xG,$\"\"\"!\"\"" }{TEXT -1 138 " and 1.  The only solutions which co
uld then be bounded at the endpoints would be the polynomial solutions
, the ones arising by choice of " }{XPPEDIT 18 0 "lambda" "6#%'lambdaG
" }{TEXT -1 24 " as one of the integers " }{XPPEDIT 18 0 "k*(k+1)" "6#
*&%\"kG\"\"\",&F$F%F%F%F%" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 66 "We can obtain the following exact \+
solution for specific values of " }{XPPEDIT 18 0 "lambda;" "6#%'lambda
G" }{TEXT -1 7 ", say, " }{XPPEDIT 18 0 "lambda = 3;" "6#/%'lambdaG\"
\"$" }{TEXT -1 88 ", thereby gaining additional insight into the natur
e of the divergence at the endpoints." }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "F := rhs(dsolve(\{subs(
lambda=3,q),y(0)=0, D(y)(0)=1\},y(x)));" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "If we evaluat
e this solution very near " }{XPPEDIT 18 0 "x = 1;" "6#/%\"xG\"\"\"" }
{TEXT -1 41 ", we obtain, with extra digits, the value" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "fnorma
l(evalf(eval(F,x=.9999999999999999999999999999999999999999999),50));" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 4 "for " }{XPPEDIT 18 0 "y;" "6#%\"yG" }{TEXT -1 102 ".  Th
e divergence is slow, because the behavior is analogous to the diverge
nce of the harmonic series." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "
" {TEXT -1 17 "Numeric Solutions" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 118 "Figure 16.7, generated below in M
aple, shows as a solid (black( curve, a numeric solution of Legendre's
 equation with " }{XPPEDIT 18 0 "lambda=3" "6#/%'lambdaG\"\"$" }{TEXT 
-1 57 " (which is not an eigenvalue) and the initial conditions " }
{XPPEDIT 18 0 "y(0)=0" "6#/-%\"yG6#\"\"!F'" }{TEXT -1 32 ", y'(0) = 1,
 and the derivative " }{XPPEDIT 18 0 "`y'`(x)" "6#-%#y'G6#%\"xG" }
{TEXT -1 27 " as the dotted (red) curve." }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 59 "To produce this figure, execute th
e following Maple command" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "F := dsolve(\{subs(lambda=3,q),y(0)
=0, D(y)(0)=1\},y(x),numeric);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 50 "then extract the numer
ically determined functions " }{XPPEDIT 18 0 "y(x)" "6#-%\"yG6#%\"xG" 
}{TEXT -1 8 " and y'(" }{TEXT 258 1 "x" }{TEXT -1 18 ") using the synt
ax" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 62 "Y := z -> subs(F(z),y(x));\nYP := z -> subs(F(z),diff
(y(x),x));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 50 "Then the  following graph is Figure 16.7,
 showing " }{XPPEDIT 18 0 "y(x)" "6#-%\"yG6#%\"xG" }{TEXT -1 35 " as t
he solid (black) curve and y'(" }{TEXT 259 1 "x" }{TEXT -1 28 ") as th
e dotted (red) curve." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 147 "plot([Y,YP],-1..1, color=[black,re
d], linestyle=[1,2], xtickmarks=3, ytickmarks=4, labels=[x,``],labelfo
nt=[TIMES,ITALIC,12], view=[-1..1,-10..2]);" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "The solid \+
(black) curve suggests the values of " }{XPPEDIT 18 0 "y(-1)" "6#-%\"y
G6#,$\"\"\"!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y(1)" "6#-%\"yG6
#\"\"\"" }{TEXT -1 210 " are finite.  They are supposed to be unbounde
d.  However, the graph of the derivative shows that the slope at the e
ndpoints is becoming large, and hence, there well may be vertical asym
ptotes at the endpoints." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 11 "Evaluating " }{XPPEDIT 18 0 "y(x)" "6#-%\"yG6#%\"x
G" }{TEXT -1 55 " near an endpoint reveals the computational difficult
y." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 91 "Y(.99);\nY(.999);\nY(.9999);\nY(.99999);\nY(.999999);
\nY(.9999999);\nY(.99999999);\nY(.999999999);" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 299 "The gra
ph does not capture the actual endpoint behavior.  Moreover, computing
 the numeric solution of the differential equation near an endpoint be
comes increasingly difficult.  This observation is consistent with the
 slow convergence of the harmonic series, the series which models the \+
behavior of " }{XPPEDIT 18 0 "y(x)" "6#-%\"yG6#%\"xG" }{TEXT -1 262 " \+
at the endpoint.  The computational tools are just inadequate for conv
incingly demonstrating the unbounded behavior at the endpoint.  Here i
s an example, therefore, where symbolic analysis and theoretical argum
ent cannot be displaced by brute-force computation." }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}
{SECT 1 {PARA 3 "" 0 "" {TEXT -1 20 "Legendre Polynomials" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "In Section \+
10.7 we showed that a function " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"x
G" }{TEXT -1 36 " can be represented on the interval " }{XPPEDIT 18 0 
"[-1,1]" "6#7$,$\"\"\"!\"\"F%" }{TEXT -1 60 " by a Fourier-Legendre se
ries built of Legendre polynomials " }{XPPEDIT 18 0 "p[k](x)" "6#-&%\"
pG6#%\"kG6#%\"xG" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "p[k](x)" "6#-
&%\"pG6#%\"kG6#%\"xG" }{TEXT -1 70 ", the solution of Legendre's equat
ion corresponding to the eigenvalue " }{XPPEDIT 18 0 "lambda=k*(k+1)" 
"6#/%'lambdaG*&%\"kG\"\"\",&F&F'F'F'F'" }{TEXT -1 19 ", is normalized \+
so " }{XPPEDIT 18 0 "p[k](1)=1." "6#/-&%\"pG6#%\"kG6#\"\"\"$F*\"\"!" }
{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 45 "These polynomials can be accessed in Maple's " }{TEXT 
260 9 "orthopoly" }{TEXT -1 11 " package as" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "for k from 0 to 5
 do\np[k] := P(k,x);\nod;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 
"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 155 "That these polynomials are \+
solutions of Legendre's equation is established by direct substitution
 into Legendre's equation, using the following Maple loop." }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "for
 k from 0 to 5 do\nsimplify(subs(lambda=k*(k+1), y(x)=p[k], q));\nod;
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 43 "Alternatively, in Legendre's equation, set " }
{XPPEDIT 18 0 "lambda" "6#%'lambdaG" }{TEXT -1 4 " to " }{XPPEDIT 18 
0 "k*(k+1)" "6#*&%\"kG\"\"\",&F$F%F%F%F%" }{TEXT -1 84 " and solve, an
ticipating the solution will be the corresponding Legendre polynomial.
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 83 "for k from 0 to 4 do\ns||k := dsolve(subs(lambda=k*(k
+1),q),y(x), output=basis);\nod;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 158 "Notice that \+
in each case there are two solutions, one a polynomial, and one contai
ning logarithmic terms which are unbounded at the endpoints of the int
erval " }{XPPEDIT 18 0 "[-1,1]" "6#7$,$\"\"\"!\"\"F%" }{TEXT -1 88 ". \+
 Also notice that the polynomials are scaled differently.  The \"Legen
dre polynomials\" " }{XPPEDIT 18 0 "P[k](x)" "6#-&%\"PG6#%\"kG6#%\"xG
" }{TEXT -1 24 " are normalized so that " }{XPPEDIT 18 0 "P[k](1)=1" "
6#/-&%\"PG6#%\"kG6#\"\"\"F*" }{TEXT -1 16 ", as verified by" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "f
or k from 0 to 5 do\nsubs(x=1,p[k]);\nod;" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "but the pol
ynomials generated by " }{TEXT 261 6 "dsolve" }{TEXT -1 9 ", namely," 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 33 "for k from 0 to 4 do\ns||k[1];\nod;" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 63 "are clearl
y scaled differently, since their endpoint values are" }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "for k fr
om 0 to 4 do\nsubs(x=1,s||k[1]);\nod;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 363 "The orthogon
ality of the Legendre polynomials has already been discussed in an ear
lier lesson, so we conclude this study of Legendre's equation and the \+
resulting eigenvalue problem.  Legendre polynomials and the Fourier-Le
gendre series will appear again in Sections 29.4 and 29.5 in the conte
xt of boundary value problems involving partial differential equations
." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}}}{MARK "1" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }
{PAGENUMBERS 0 1 2 33 1 1 }