{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 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 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{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 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 22 " " } {TEXT 256 69 "Asymptotic Expansion of Eigenvalues for Prolate Spheroid al Functions" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 83 " \+ A Maple worksheet written by Hans Volkmer, April 2, 2001" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 74 " \+ send comments to volkmer@uwm.edu " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 136 "This Maple program computes the formal asymptotic expa nsion for the eigenvalues of the spheroidal equation in the prolate c ase " }{XPPEDIT 18 0 "diff((1-x^2)*diff(y,x),x)+(lambda+4*h^2*(1 -x^2)-m^2/(1-x^2))*y = 0;" "6#/,&-%%diffG6\$*&,&\"\"\"F**\$%\"xG\"\"#!\" \"F*-F&6\$%\"yGF,F*F,F**&,(%'lambdaGF**(\"\"%F**\$%\"hGF-F*,&F*F**\$F,F-F .F*F**&%\"mGF-,&F*F**\$F,F-F.F.F.F*F1F*F*\"\"!" }{TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 183 "The computations are based on for mulas by Mueller, Asymptotic Expansion of Prolate Spheroidal Wave Func tions and their Characteristic Numbers, J. reine angew. Math. 212 (196 3), 26-48." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "The following notat ions are used: " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 70 "R : we want to determine the asymptotic expansion up to and including " }{XPPEDIT 18 0 "h^(-2*R+1);" "6#)%\"hG,&*&\"\"#\"\"\"%\"RGF(!\"\"F(F(" }{TEXT -1 62 ", in the paper we have R=3 . We take R=4 but R can be changed. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 11 "f(q,qq) =( " }{XPPEDIT 18 0 " q,qq;" "6\$%\"qG%#qqG" }{TEXT -1 16 "), see page 30 ." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "P[r,t] = " }{XPPEDIT 18 0 "P[r](t);" "6#-&%\"PG6 #%\"rG6#%\"tG" }{TEXT -1 36 ", these are polynomials in q, m and " } {XPPEDIT 18 0 "Delta;" "6#%&DeltaG" }{TEXT -1 15 ", see page 32 ." }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 7 "M[r] = " }{XPPEDIT 18 0 "M[r];" "6# &%\"MG6#%\"rG" }{TEXT -1 37 " , these are polynomials in q, m and " } {XPPEDIT 18 0 "Delta;" "6#%&DeltaG" }{TEXT -1 15 ", see page 32 ." }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "N[r,j] = coefficient of " } {XPPEDIT 18 0 "Delta^j;" "6#)%&DeltaG%\"jG" }{TEXT -1 4 " in " } {XPPEDIT 18 0 "M[r];" "6#&%\"MG6#%\"rG" }{TEXT -1 37 " , these are pol ynomials in q and m." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "g = 1/(2 ^7 h) , " }{XPPEDIT 18 0 "nu = m^2-1/4;" "6#/%#nuG,&*\$%\"mG\"\"#\"\"\" *&F)F)\"\"%!\"\"F," }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 18 0 "Delta;" "6#%&DeltaG" }{TEXT -1 24 " = L[0] + L[1] g+ L[2] " } {XPPEDIT 18 0 "g^2;" "6#*\$%\"gG\"\"#" }{TEXT -1 29 "+... asymptotic ex pansion of " }{XPPEDIT 18 0 "Delta;" "6#%&DeltaG" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "lambda+2*h^2+1/4 = -2*h^2+2*h*q+Delta/8;" "6#/,(%'l ambdaG\"\"\"*&\"\"#F&*\$%\"hGF(F&F&*&F&F&\"\"%!\"\"F&,(*&F(F&*\$F*F(F&F- *(F(F&F*F&%\"qGF&F&*&%&DeltaGF&\"\")F-F&" }{TEXT -1 78 " , see page 29 . The L[r] are polynomials in q and m . " }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 101 "Usage: press enter until you rea ch the end of the worksheet where the polynomials L[r] are listed . " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "restart;R:=4;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "f:=proc(q,qq)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "local n;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "n:=(qq- q)/4;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "if n=-1 then (q-1)*(q-3); " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "elif n=0 then 2*(q^2+1+Delta)-1 6*nu;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "elif n=1 then (q+1)*(q+3)- 64*nu;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "else -64*n*nu;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "fi;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "e nd;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "P:=array(0..R,-R..2* R);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "makeP:=proc() # find P[r,t] for r=0..R and t=-r..2*R-r" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "global P,R;local r,s,t,u;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "P[ 0,0]:=1;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "for t from 1 to 2*R do \+ P[0,t]:=0 od;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "for r from 1 to R \+ do P[r,0]:=0 od;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "for r from 1 to R do for t from -r to 2*R-r do if t<>0 then " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "u:=0;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "for s from -r+1 to t+1 do u:=u+f(q+4*s,q+4*t)*P[r-1,s] od;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "P[r,t]:=simplify(u/t) ;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "fi;od;od;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end;" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "makeP():" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "M:=array(0..2*R); " }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 13 "makeM:=proc()" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "global P,M,R;local r,t,s;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 " M[1]:=2*(q^2+1)-16*nu;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "for r fro m 1 to R do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "s:=0;" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 57 "for t from 1 to r do s:=s+t*P[r,t]*subs(q=-q,P [r,-t]) od;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "M[2*r]:=(-1)^r*simpl ify(s-subs(q=-q,s));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od;" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "for r from 1 to R-1 do" }}{PARA 0 " > " 0 "" {MPLTEXT 1 0 5 "s:=0;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "f or t from 1 to r do s:=s+t*P[r,t]*subs(q=-q,P[r+1,-t])-t*P[r,-t]*subs( q=-q,P[r+1,t]) od;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "s:=s+(r+1)*P[ r,r+1]*subs(q=-q,P[r+1,-r-1]);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "M [2*r+1]:=(-1)^(r+1)*simplify(s):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 " od;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "makeM(): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "N:=array(0..2*R,0..2*R); " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "makeN:=proc() # N is the sam e as M but as an array" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "global M, N;local r,n;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "for r from 2 to 2*R do for n from 0 to r-2 do N[r,n]:=coeff(M[r],Delta,n):" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 6 "od od;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "makeN():" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "L:=array(0..2*R); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "makeL:=proc() " }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 35 "global N,L;local delta,del,r,n,j,s;" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 21 "delta:=array(0..2*R);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "del:=-1/2*M[1];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "for r from 2 to 2*R do " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "for j f rom 0 to r-2 do delta[j]:=powmod(del,j,g^(r-j-1),g) od; " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "s:=0;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "fo r j from 2 to r do for n from 0 to j-2 do s:=s+N[j,n]*coeff(delta[n],g ,r-j); od;od;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "L[r-1]:=simplify(- s/2);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "del:=del+L[r-1]*g^(r-1);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "makeL():" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "Example: We compare with the paper , page 33." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "collect(subs( nu=m^2-1/4,-L[1]/8),m);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 " collect(subs(nu=m^2-1/4,-2^(-2)*L[2]/8),m);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "collect(subs(nu=m^2-1/4,-2^(-4)*L[3]/8),m);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "collect(subs(nu=m^2-1/4,-2^( -8)*L[4]/8),m);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "collect( subs(nu=m^2-1/4,-2^(-10)*L[5]/8),m);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 110 "# we have agreement with the paper except for a typo in the paper in the last term: it should be 32 not 1024/3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "collect(subs(nu=m^2-1/4,-2^(-11)*L[ 6]/8),m);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "collect(subs(n u=m^2-1/4,-2^(-12)*L[7]/8),m);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "0 0 0" 22 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }