{VERSION 3 0 "IBM INTEL NT" "3.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 110 "This Maple worksheet com putes the formal asymptotic expansion for the eigenvalues of the Lame \+ equation " }{XPPEDIT 18 0 "diff(y,`\$`(u,2))+(lambda-kappa^2*sn(u ,k)^2)*y = 0;" "6#/,&-%%diffG6\$%\"yG-%\"\$G6\$%\"uG\"\"#\"\"\"*&,&%'lamb daGF.*&%&kappaG\"\"#-%#snG6\$F,%\"kG\"\"#!\"\"F.F(F.F.\"\"!" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 75 "Written by Hans Volkmer, A pril 27, 2001, send comments to volkmer@uwm.edu ." }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 167 "The computations are based on formulas by Mueller , Asymptotic Expansion of Ellipsoidal Wave Functions and their Charact eristic Numbers, Math. Nachr. 31 (1965), 89-101." }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 34 "The following notations are used: " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 70 "R : we want to determine the asymptotic e xpansion up to and including " }{XPPEDIT 18 0 "h^(-2*R+1);" "6#)%\"hG, &*&\"\"#\"\"\"%\"RGF(!\"\"\"\"\"F(" }{TEXT -1 36 " . We take R=3 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 96 ." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "P[r,t] = " }{XPPEDIT 18 0 "P[r]( t);" "6#-&%\"PG6#%\"rG6#%\"tG" }{TEXT -1 34 ", these are functions of \+ q, k and " }{XPPEDIT 18 0 "Delta;" "6#%&DeltaG" }{TEXT -1 2 " ." }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 7 "M[r] = " }{XPPEDIT 18 0 "M[r];" "6# &%\"MG6#%\"rG" }{TEXT -1 35 " , these are functions of q, k and " } {XPPEDIT 18 0 "Delta;" "6#%&DeltaG" }{TEXT -1 15 ", see page 98 ." }}} {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 35 " , these are fun ctions of q and k." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 11 "g = 1/(2^6 " }{XPPEDIT 18 0 "kappa;" "6#%&kappaG" }{TEXT -1 1 ")" }}}{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 expansion of " }{XPPEDIT 18 0 "Delta;" "6#%&Del taG" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "lambda = q*kappa+Delta/8; " "6#/%'lambdaG,&*&%\"qG\"\"\"%&kappaGF(F(*&%&DeltaGF(\"\")!\"\"F(" } {TEXT -1 52 " , see page 93. The L[r] are functions of q and k . " }}} {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 29 "restart;readlib(powmod):R :=3;" }}}{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 33 "if n=-1 then (q-1)*(q-3)*(1-k^2);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 " elif n=0 then 2*((q^2+1)*(1+k^2)+Delta);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "elif n=1 then (q+1)*(q+3)*(1-k^2)-16*k^2/(1-k^2);" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "else (-1)^n*4*k*((1+k)^n/(1-k)^n-(1 -k)^n/(1+k)^n);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "fi;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end;" }}}{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 "ma keP():" }}}{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 24 "M[1]:=2*(q^2+1)*(1+k^2);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "for r from 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*simplify(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 "for 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 "make M(): " }}}{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 same 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 d o 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;lo cal delta,del,r,n,j,s;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "delta:=ar ray(0..2*R);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "L[0]:=-1/2*M[1];del :=L[0];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "for r from 2 to 2*R do \+ " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "for j from 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 "for 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 "" {MPLTEXT 1 0 66 "conv:=proc(p,n) # make conversion of polynom ials to form in paper " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "local j,w ,pp;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "pp:=p;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 103 "for j from 0 to (n+2)/2 do w[j]:=simplify(subs(k=0,p p));pp:=simplify((pp-w[j]*(1+k^2)^(n-2*j))/k^2);od;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "sum(w[jj]*(k^(2*jj))*(1+k^2)^(n-2*jj),jj=0..(n+2)/ 2);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 50 "Example: Compare with results in paper, page 100. " }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "L[0]/8;" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 16 "conv(-L[1]/8,2);" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 18 "conv(-L[2]/2^5,3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "conv(-L[3]/2^7,4);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "conv(-L[4]/2^11,5); # we have agreement with paper ex cept for one term in L[4] (the one with 19944)" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 19 "conv(-L[5]/2^13,6);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "36" 0 }{VIEWOPTS 1 1 0 1 1 1803 }