{VERSION 5 0 "IBM INTEL NT" "5.0" }
{USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 
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0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 }
{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 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 "Maple Output" -1 
11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 
0 0 0 1 0 1 0 2 2 0 1 }}
{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 53 "We express (Laplacian)^2 of psi in polar \+
coordinates." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 178 "Lpsi := di
ff(psi(r,theta),r$2)+1/r*diff(psi(r,theta),r) + 1/r^2*diff(psi(r,theta
),theta$2);\nL2psi := diff(Lpsi,r$2)+1/r*diff(Lpsi,r) + 1/r^2*diff(Lps
i,theta$2):\ncollect(L2psi,r);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%L
psiG,(-%%diffG6$-%$psiG6$%\"rG%&thetaG-%\"$G6$F,\"\"#\"\"\"*&F,!\"\"-F
'6$F)F,F2F2*&F,!\"#-F'6$F)-F/6$F-F1F2F2" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#,,-%%diffG6$-%$psiG6$%\"rG%&thetaG-%\"$G6$F*\"\"%\"\"\"*(\"\"#F0
F*!\"\"-F%6$F'-F-6$F*\"\"$F0F0*&,&-F%6$F'-F-6$F*F2F3*&F2F0-F%6%F'F=-F-
6$F+F2F0F0F0F*!\"#F0*&,&-F%6$F'F*F0*&F2F0-F%6%F'F*FBF0F3F0F*!\"$F0*&,&
*&F/F0-F%6$F'FBF0F0-F%6$F'-F-6$F+F/F0F0F*!\"%F0" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 81 "We compute (Laplacian)^2 of the nth term of the Four
ier series (in theta) of psi." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 257 "nterm := psi[n](r)*exp(I*n*theta);\nLnterm := diff(nterm,r$2)+1
/r*diff(nterm,r) + 1/r^2*diff(nterm,theta$2);\nL2nterm := collect(diff
(Lnterm,r$2)+1/r*diff(Lnterm,r) + 1/r^2*diff(Lnterm,theta$2),r):\ncoll
ect(simplify(L2nterm*exp(-I*n*theta)),r)*exp(I*n*theta); " }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#>%&ntermG*&-&%$psiG6#%\"nG6#%\"rG\"\"\"-%$expG6
#*(F*F-%&thetaGF-^#F-F-F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'Lnterm
G,(*&-%%diffG6$-&%$psiG6#%\"nG6#%\"rG-%\"$G6$F0\"\"#\"\"\"-%$expG6#*(F
.F5%&thetaGF5^#F5F5F5F5*(F0!\"\"-F(6$F*F0F5F6F5F5**F0!\"#F*F5F.F4F6F5F
=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,,-%%diffG6$-&%$psiG6#%\"nG6#%
\"rG-%\"$G6$F.\"\"%\"\"\"*(\"\"#F3-F&6$F(-F06$F.\"\"$F3F.!\"\"F3*&,&-F
&6$F(-F06$F.F5F;*(F5F3F>F3)F,F5F3F;F3F.!\"#F3*&,&-F&6$F(F.F3*(F5F3FGF3
FCF3F3F3F.!\"$F3*&,&*(F2F3F(F3FCF3F;*&F(F3)F,F2F3F3F3F.!\"%F3F3-%$expG
6#*(F,F3%&thetaGF3^#F3F3F3" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 109 "We
 solve the ODE that results from setting (Laplacian)^2 of the nth term
 of the Fourier series equal to zero." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 182 "ode1 := diff(psi[n](r),r$4) + 2/r*diff(psi[n](r),r$3
) - (2*n^2+1)/r^2*diff(psi[n](r),r$2) \n         + (2*n^2+1)/r^3*diff(
psi[n](r),r) + (n^4 - 4*n^2)/r^4*psi[n](r) = 0;\ndsolve(ode1);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%ode1G/,,-%%diffG6$-&%$psiG6#%\"nG6#
%\"rG-%\"$G6$F0\"\"%\"\"\"*(\"\"#F5-F(6$F*-F26$F0\"\"$F5F0!\"\"F5*(,&F
5F5*&F7F5)F.F7F5F5F5F0!\"#-F(6$F*-F26$F0F7F5F=*(F?F5F0!\"$-F(6$F*F0F5F
5*(,&*$)F.F4F5F5*&F4F5FAF5F=F5F0!\"%F*F5F5\"\"!" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-&%$psiG6#%\"nG6#%\"rG,**&%$_C1G\"\"\")F*,&F(F.\"\"#F.
F.F.*&%$_C2GF.)F*,$F(!\"\"F.F.*&%$_C3GF.)F*F(F.F.*&%$_C4GF.)F*,&F(F6F1
F.F.F." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 59 "But the solution has a \+
different form when n = 0, 1 and -1." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 237 "for j from 0 to 1 do\nsprintf(\"for n = %d:\",j);\no
de1 := diff(psi[j](r),r$4) + 2/r*diff(psi[j](r),r$3) - (2*j^2+1)/r^2*d
iff(psi[j](r),r$2) \n         + (2*j^2+1)/r^3*diff(psi[j](r),r) + (j^4
 - 4*j^2)/r^4*psi[j](r) = 0;\ndsolve(ode1);\nend do;" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#Q+for~n~=~0:6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>
%%ode1G/,*-%%diffG6$-&%$psiG6#\"\"!6#%\"rG-%\"$G6$F0\"\"%\"\"\"*(\"\"#
F5F0!\"\"-F(6$F*-F26$F0\"\"$F5F5*&F0!\"#-F(6$F*-F26$F0F7F5F8*&F0!\"$-F
(6$F*F0F5F5F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-&%$psiG6#\"\"!6#%\"
rG,**&%$_C1G\"\"\")F*\"\"#F.F.*(%$_C2GF.F/F.-%#lnGF)F.F.%$_C3GF.*&%$_C
4GF.F3F.F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#Q+for~n~=~1:6\"" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%ode1G/,,-%%diffG6$-&%$psiG6#\"\"\"6
#%\"rG-%\"$G6$F0\"\"%F.*(\"\"#F.F0!\"\"-F(6$F*-F26$F0\"\"$F.F.*(F<F.F0
!\"#-F(6$F*-F26$F0F6F.F7*(F<F.F0!\"$-F(6$F*F0F.F.*(F<F.F0!\"%F*F.F7\"
\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-&%$psiG6#\"\"\"6#%\"rG,**&%$_
C1GF(F*!\"\"F(*&%$_C2GF()F*\"\"$F(F(*&%$_C3GF(F*F(F(*(%$_C4GF(F*F(-%#l
nGF)F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "So these are the gene
ral solutions:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 159 "psi[n] :
= r -> _C1*r^(-n+2)+_C2*r^(n+2)+_C3*r^(-n)+_C4*r^n;\npsi[0] := r -> _C
1*r^2+_C2*r^2*ln(r)+_C3+_C4*ln(r);\npsi[1] := r -> _C1*r^3+_C2/r+_C3*r
+_C4*r*ln(r);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%$psiG6#%\"nGf*6#%
\"rG6\"6$%)operatorG%&arrowGF+,**&%$_C1G\"\"\")9$,&F'!\"\"\"\"#F2F2F2*
&%$_C2GF2)F4,&F'F2F7F2F2F2*&%$_C3GF2)F4,$F'F6F2F2*&%$_C4GF2)F4F'F2F2F+
F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%$psiG6#\"\"!f*6#%\"rG6\"6$%
)operatorG%&arrowGF+,**&%$_C1G\"\"\")9$\"\"#F2F2*(%$_C2GF2F3F2-%#lnG6#
F4F2F2%$_C3GF2*&%$_C4GF2F8F2F2F+F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#>&%$psiG6#\"\"\"f*6#%\"rG6\"6$%)operatorG%&arrowGF+,**&%$_C1GF')9$\"
\"$F'F'*&%$_C2GF'F3!\"\"F'*&%$_C3GF'F3F'F'*(%$_C4GF'F3F'-%#lnG6#F3F'F'
F+F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "11 0 \+
0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }