{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 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 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT 256 70 " \+ Eigenvalues for Lame Polynomials" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 93 " A Maple worksheet writt en by Hans Volkmer, February 10, 2001. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 83 " send c omments to volkmer@uwm.edu" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 93 "In \+ this worksheet we compute the eigenvalues for Lame polynomials as eige nvalues of matrices." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "res tart;with(linalg):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "Digit s:=10; # select the required precison " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "matrixu:=proc(nu,k2) # nu even EMO (18)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "local d,A,r;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "d:=nu/2+1;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "A:=matrix(d,d,0 );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "for r from 0 to d-1 do A[r+1, r+1]:=4*r^2*(2-k2) od;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "if nu>0 t hen A[1,2]:=(nu-1)*(nu+2)*k2 fi;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "for r from 1 to d-2 do A[r+1,r+2]:=0.5*(nu-2*r-1)*(nu+2*r+2)*k2 od; \+ " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "for r from 1 to d-1 do A[r+1,r] :=0.5*(nu-2*r+2)*(nu+2*r-1)*k2 od;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "A;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end;" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 38 "matrixd:=proc(nu,k2) # nu odd EMO (19)" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "local d,A,r;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "d:=(nu+1)/2;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "A: =matrix(d,d,0);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "for r from 0 to \+ d-1 do A[r+1,r+1]:=4*r^2*(2-k2) od;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "if nu>1 then A[1,2]:=nu*(nu+1)*k2 fi;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "for r from 1 to d-2 do A[r+1,r+2]:=0.5*(nu-2*r)*(nu+2 *r+1)*k2 od; " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "for r from 1 to d- 1 do A[r+1,r]:=0.5*(nu-2*r+1)*(nu+2*r)*k2 od;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "A;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "matrixs:=proc(nu,k2) # nu od d EMO (20) " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "local d,A,r;" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "d:=(nu+1)/2;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "A:=matrix(d,d,0);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "A[1,1]:=2-k2+0.5*nu*(nu+1)*k2;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "for r from 1 to d-1 do A[r+1,r+1]:=(2*r+1)^2*(2-k2) od;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "for r from 0 to d-2 do A[r+1,r+2]:=0.5*(n u-2*r-2)*(nu+2*r+3)*k2 od; " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "for \+ r from 1 to d-1 do A[r+1,r]:=0.5*(nu-2*r+1)*(nu+2*r)*k2 od;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "A;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "en d;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "matrixsd:=proc(nu,k2) # nu even EMO (21) " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "local d,A, r;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "d:=nu/2;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "A:=matrix(d,d,0);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "A[1,1]:=2-k2+0.5*nu*(nu+1)*k2;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "for r from 1 to d-1 do A[r+1,r+1]:=(2*r+1)^2*(2-k2) od;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "for r from 0 to d-2 do A[r+1,r+2]:=0.5*(n u-2*r-1)*(nu+2*r+2)*k2 od; " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "for \+ r from 1 to d-1 do A[r+1,r]:=0.5*(nu-2*r)*(nu+2*r+1)*k2 od;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "A;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "en d;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "matrixsc:=proc(nu,k2) # nu even EMO (22) " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "local d,A, r;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "d:=nu/2;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "A:=matrix(d,d,0);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "for r from 0 to d-1 do A[r+1,r+1]:=(2*r+2)^2*(2-k2) od;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "for r from 0 to d-2 do A[r+1,r+2]:=0.5*(n u-2*r-3)*(nu+2*r+4)*k2 od; " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "for \+ r from 1 to d-1 do A[r+1,r]:=0.5*(nu-2*r)*(nu+2*r+1)*k2 od;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "A;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "en d;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "matrixscd:=proc(nu,k2 ) # nu odd EMO (23) " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "local d,A, r;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "d:=(nu-1)/2;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "A:=matrix(d,d,0);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "for r from 0 to d-1 do A[r+1,r+1]:=(2*r+2)^2*(2-k2) od;" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "for r from 0 to d-2 do A[r+1,r+2]:= 0.5*(nu-2*r-2)*(nu+2*r+3)*k2 od; " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "for r from 1 to d-1 do A[r+1,r]:=0.5*(nu-2*r-1)*(nu+2*r+2)*k2 od; " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "A;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "mat rixc:=proc(nu,k2) # nu odd EMO (24) " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "local d,A,r;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "d:=(nu+1)/2; " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "A:=matrix(d,d,0);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "A[1,1]:=2-k2-0.5*nu*(nu+1)*k2;" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 55 "for r from 1 to d-1 do A[r+1,r+1]:=(2*r+1)^2*( 2-k2) od;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "for r from 0 to d-2 do A[r+1,r+2]:=0.5*(nu-2*r-2)*(nu+2*r+3)*k2 od; " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "for r from 1 to d-1 do A[r+1,r]:=0.5*(nu-2*r+1)*(nu+2 *r)*k2 od;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "A;" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 4 "end;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "m atrixcd:=proc(nu,k2) # nu even EMO (25) " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "local d,A,r;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "d:= nu/2;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "A:=matrix(d,d,0);" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "A[1,1]:=2-k2-0.5*nu*(nu+1)*k2;" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "for r from 1 to d-1 do A[r+1,r+1]:= (2*r+1)^2*(2-k2) od;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "for r from \+ 0 to d-2 do A[r+1,r+2]:=0.5*(nu-2*r-1)*(nu+2*r+2)*k2 od; " }}{PARA 0 " > " 0 "" {MPLTEXT 1 0 63 "for r from 1 to d-1 do A[r+1,r]:=0.5*(nu-2*r )*(nu+2*r+1)*k2 od;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "A;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "a:=proc(m,nu,k2) # eigenvalue a for m=0..nu" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "local h,A,L;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "if type(nu,even) and type(m,even) then" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 18 "A:=matrixu(nu,k2);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "L:=[eigenvalues(A)];L:=sort(L);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "h:=0.5*(L[m/2+1]+nu*(nu+1)*k2);return(h);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "fi;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "if type(nu,even) and type(m,odd) then" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "A:=matrixsd(nu,k2);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "L:=[eige nvalues(A)];L:=sort(L);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "h:=0.5*( L[(m+1)/2]+nu*(nu+1)*k2);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "fi;" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "if type(nu,odd) and type(m,even) th en" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "A:=matrixd(nu,k2);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "L:=[eigenvalues(A)];L:=sort(L);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "h:=0.5*(L[m/2+1]+nu*(nu+1)*k2);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "fi;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 " if type(nu,odd) and type(m,odd) then" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "A:=matrixs(nu,k2);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "L:=[eig envalues(A)];L:=sort(L);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "h:=0.5* (L[(m+1)/2]+nu*(nu+1)*k2);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "fi;" } }{PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "h;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "b:=proc(m,nu,k 2) # eigenvalue b for m=1..nu" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "lo cal h,A,L;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "if type(nu,even) and \+ type(m,even) then" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "A:=matrixsc(nu ,k2);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "L:=[eigenvalues(A)];L:=sor t(L);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "h:=0.5*(L[m/2]+nu*(nu+1)*k 2);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "fi;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "if type(nu,even) and type(m,odd) then" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 19 "A:=matrixcd(nu,k2);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "L:=[eigenvalues(A)];L:=sort(L);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "h:=0.5*(L[(m+1)/2]+nu*(nu+1)*k2);" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 3 "fi;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "if type(nu ,odd) and type(m,even) then" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "A:=m atrixscd(nu,k2);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "L:=[eigenvalues (A)];L:=sort(L);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "h:=0.5*(L[m/2]+ nu*(nu+1)*k2);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "fi;" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 36 "if type(nu,odd) and type(m,odd) then" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "A:=matrixc(nu,k2);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "L:=[eigenvalues(A)];L:=sort(L);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "h:=0.5*(L[(m+1)/2]+nu*(nu+1)*k2);" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 3 "fi;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "h;" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 149 "list_lame:=proc(nu,k2) # list 2*nu+1 eigenvalues for Lame polynomials of degree nu in format (Arscott's type,m=number of z eros in (0,K),eigenvalue h)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "loca l m,d,A,L;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "if type(nu,even) then " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "d:=nu/2+1;" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 18 "A:=matrixu(nu,k2);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "L:=[eigenvalues(A)];L:=sort(L);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "for m from 1 to d do print('u',m-1,0.5*(L[m]+nu*(nu+1)*k2)) od; " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "d:=nu/2;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "A:=matrixsc(nu,k2);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "L:=[eigenvalues(A)];L:=sort(L);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "for m from 1 to d do print('sc',m-1,0.5*(L[m]+nu*(nu+1)*k2)) od; " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "A:=matrixsd(nu,k2);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "L:=[eigenvalues(A)];L:=sort(L);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "for m from 1 to d do print('sd',m-1,0.5*(L[ m]+nu*(nu+1)*k2)) od;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "A:=matrixc d(nu,k2);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "L:=[eigenvalues(A)];L: =sort(L);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "for m from 1 to d do p rint('cd',m-1,0.5*(L[m]+nu*(nu+1)*k2)) od;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "else" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "d:=(nu+1)/2 ;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "A:=matrixs(nu,k2);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "L:=[eigenvalues(A)];L:=sort(L);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "for m from 1 to d do print('s',m-1,0.5*(L[m ]+nu*(nu+1)*k2)) od;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "A:=matrixc( nu,k2);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "L:=[eigenvalues(A)];L:=s ort(L);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "for m from 1 to d do pri nt('c',m-1,0.5*(L[m]+nu*(nu+1)*k2)) od;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "A:=matrixd(nu,k2);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "L:=[e igenvalues(A)];L:=sort(L);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "for m from 1 to d do print('d',m-1,0.5*(L[m]+nu*(nu+1)*k2)) od;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "d:=(nu-1)/2;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "A:=matrixscd(nu,k2);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "L:= [eigenvalues(A)];L:=sort(L);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "for m from 1 to d do print('scd',m-1,0.5*(L[m]+nu*(nu+1)*k2)) od;" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "fi;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "Example 1: List all \+ " }{XPPEDIT 18 0 "2*nu+1;" "6#,&*&\"\"#\"\"\"%#nuGF&F&F&F&" }{TEXT -1 46 " eigenvalues for Lame polynomials for a given " }{XPPEDIT 18 0 "nu ;" "6#%#nuG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "k^2;" "6#*\$%\"kG\"\"# " }{TEXT -1 41 " including type and oscillation number m." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "list_lame(4,0.5);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "list_lame(5,0.1);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "Example 2: Plot eigenvalues a,b as functions of k. \+ " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "nu:=2;" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 42 "a0:=plot(k->a(0,nu,k*k),0..1,color=black):" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "a1:=plot(k->a(1,nu,k*k),0.. 1,color=red):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "a2:=plot(k ->a(2,nu,k*k),0..1,color=blue):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "b1:=plot(k->b(1,nu,k*k),0..1,color=green):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "b2:=plot(k->b(2,nu,k*k),0..1,color=brown): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "display(a0,a1,a2,b1,b2, labels=[k,h],title=\"five eigenvalues for Lame polynomials if nu=2\"); " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "Example: Test Erdelyi identit y: " }{XPPEDIT 18 0 "a[nu]^m;" "6#)&%\"aG6#%#nuG%\"mG" }{TEXT -1 1 "( " }{XPPEDIT 18 0 "k^2;" "6#*\$)%\"kG\"\"#\"\"\"" }{TEXT -1 4 ") + " } {XPPEDIT 18 0 "a[nu]^(nu-m);" "6#)&%\"aG6#%#nuG,&F'\"\"\"%\"mG!\"\"" } {TEXT -1 1 "(" }{XPPEDIT 18 0 "1-k^2;" "6#,&\"\"\"F\$*\$)%\"kG\"\"#F\$!\" \"" }{TEXT -1 4 ") = " }{XPPEDIT 18 0 "b[nu]^m;" "6#)&%\"bG6#%#nuG%\"m G" }{TEXT -1 1 "(" }{XPPEDIT 18 0 "k^2;" "6#*\$)%\"kG\"\"#\"\"\"" } {TEXT -1 4 ") + " }{XPPEDIT 18 0 "b[nu]^(nu-m+1);" "6#)&%\"bG6#%#nuG,( F'\"\"\"%\"mG!\"\"F)F)" }{TEXT -1 1 "(" }{XPPEDIT 18 0 "1-k^2;" "6#,& \"\"\"F\$*\$)%\"kG\"\"#F\$!\"\"" }{TEXT -1 3 ") =" }{XPPEDIT 18 0 "nu*(nu +1);" "6#*&%#nuG\"\"\",&F\$F%F%F%F%" }{TEXT -1 4 " . " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "nu:=5;k2:=0.6;" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 61 "for m from 0 to nu do print(m,a(m,nu,k2)+a(n u-m,nu,1-k2)) od;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "for m \+ from 1 to nu do print(m,b(m,nu,k2)+b(nu-m+1,nu,1-k2)) od;" }}{PARA 0 " > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "# looks good" }}}}{MARK "34" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }