{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 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 "Warning" -1 7 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 1 1 3 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Error" -1 8 1 {CSTYLE "" -1 -1 "Courie r" 1 10 255 0 255 1 2 2 2 2 2 1 1 1 3 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 }{PSTYLE "Maple Outpu t" -1 12 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 } 1 3 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 243 "#General d^l_mn rou tines, lensed correlation functions\n#Follow Anthony's D_lm rotation m ethod, generalise to polarization and get full second order result.\n# See ############################## below for start of lensing derivati ons\n#AL Jan 2005" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "assume(l>=0);" }}} {EXCHG {PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 563 "dlmn:=proc(j,m,n,t)\n local s,ss,f,d,prefacto r;\n s:=0; ss:=0:\n for s from 0 to min(j -m,j-n) do\n if j-m-s<0 or j-n-s<0 or m+n+s<0 then\n \+ f:=0\n else f:=1/((j-m-s)!*(j-n-s)!*s!*(m+n +s)!);\n fi;\n ss:=ss + (cos(t/2))**(2*s+m+n)\n \+ *(sin(t/2))**(2*j-2*s-m-n)\n \+ *((-1)**s*f);\n od;\n prefactor:=(j-m)!*(j+m)!*(j-n )!*(j+n)!;\n d:=(-1)**(j-n)*ss*sqrt(prefactor);\n conver t(d,trig);\n factor(%);\nend:\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 136 "eval_d:=proc(inval,ll)\n applyrule(d(x,m::algebraic, n::algebraic)=dlmnp(ll,m,n,t),evalf(subs(l=ll,inval)));\n subs(dlmnp=d lmn,%);\n %;\nend;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%'eval_dGR6$%&i nvalG%#llG6\"F)F)C%-%*applyruleG6$/-%\"dG6%%\"xG'%\"mG%*algebraicG'%\" nGF5-%&dlmnpG6&9%F4F7%\"tG-%&evalfG6#-%%subsG6$/%\"lGF;9$-FA6$/F9%%dlm nG%\"%GFJF)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 107 "rec:=(- m+n*cos(t))/sin(t)*d(x,m,n)=1/2*sqrt((l+n)*(l-n+1))*d(x,m,n-1) + 1/2*s qrt((l-n)*(l+n+1))*d(x,m,n+1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$r ecG/*&*&,&%\"mG!\"\"*&%\"nG\"\"\"-%$cosG6#%\"tGF-F-F--%\"dG6%%\"xGF)F, F-F--%$sinGF0F*,&*&-%%sqrtG6#*&,&%#l|irGF-F,F-F-,(F?F-F,F*F-F-F-F--F36 %F5F),&F,F-F-F*F-#F-\"\"#*(FDF--F;6#*&,&F?F-F,F*F-,(F?F-F,F-F-F-F-F--F 36%F5F),&F,F-F-F-F-F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 109 "r ec := (-m+n*cos(t))*d(x,m,n)/sin(t) = 1/2*sqrt((l+n)*(l-n+1))*d(x,m,n- 1)+1/2*sqrt((l-n)*(l+n+1))*d(x,m,n+1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$recG/*&*&,&%\"mG!\"\"*&%\"nG\"\"\"-%$cosG6#%\"tGF-F-F--%\"dG6 %%\"xGF)F,F-F--%$sinGF0F*,&*&-%%sqrtG6#*&,&%#l|irGF-F,F-F-,(F?F-F,F*F- F-F-F--F36%F5F),&F,F-F-F*F-#F-\"\"#*(FDF--F;6#*&,&F?F-F,F*F-,(F?F-F,F- F-F-F-F--F36%F5F),&F,F-F-F-F-F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "solve(rec,d(x,m,n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&*& -%$sinG6#%\"tG\"\"\",&*&-%%sqrtG6#,$*&,&%#l|irGF*%\"nGF*F*,(F3!\"\"F4F *F*F6F*F6F*-%\"dG6%%\"xG%\"mG,&F4F*F*F6F*F**&-F.6#,$*&,&F3F6F4F*F*,(F3 F*F4F*F*F*F*F6F*-F86%F:F;,&F4F*F*F*F*F*F*F*,&F;F**&F4F*-%$cosGF(F*F6F6 #F6\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "rec_mid:=d(x,m, n)=%;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(rec_midG/-%\"dG6%%\"xG%\"m G%\"nG,$*&*&-%$sinG6#%\"tG\"\"\",&*&-%%sqrtG6#,$*&,&%#l|irGF3F+F3F3,(F F3F>F3-F'6%F)F*,&F+F3F3F>F3F3*&-F76#,$*&,&FF+F3F3,(F< F3F+F3F3F3F3F>F3-F'6%F)F*,&F+F3F3F3F3F3F3F3,&F*F3*&F+F3-%$cosGF1F3F>F> #F>\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "solve(rec,d(x,m ,n+1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&,(*&-%\"dG6%%\"xG%\"mG% \"nG\"\"\"F+F-\"\"#**F.F-F'F-F,F--%$cosG6#%\"tGF-!\"\"*(-%%sqrtG6#,$*& ,&%#l|irGF-F,F-F-,(F " 0 "" {MPLTEXT 1 0 14 "subs(n=n-1,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&,(*&-%\"dG6%%\"xG%\"mG,&%\"nG\"\"\"F.!\"\"F.F+F.\" \"#**F0F.F'F.F,F.-%$cosG6#%\"tGF.F/*(-%%sqrtG6#,$*&,(%#l|irGF.F-F.F.F/ F.,(F=F/F-F.F0F/F.F/F.-F(6%F*F+,&F-F.F0F/F.-%$sinGF4F.F.F.*&-F86#,$*&, &F=F.F-F.F.,(F=F/F-F.F.F/F.F/F.FBF.F/F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "rec_down:=d(x,m,n)=%;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%)rec_downG/-%\"dG6%%\"xG%\"mG%\"nG,$*&,(*&-F'6%F)F*,&F+\"\"\"F 3!\"\"F3F*F3\"\"#**F5F3F0F3F2F3-%$cosG6#%\"tGF3F4*(-%%sqrtG6#,$*&,(%#l |irGF3F+F3F3F4F3,(FBF4F+F3F5F4F3F4F3-F'6%F)F*,&F+F3F5F4F3-%$sinGF9F3F3 F3*&-F=6#,$*&,&FBF3F+F3F3,(FBF4F+F3F3F4F3F4F3FGF3F4F4" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "solve(rec,d(x,m,n-1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&,(*&-%\"dG6%%\"xG%\"mG%\"nG\"\"\"F+F-\"\"#* *F.F-F'F-F,F--%$cosG6#%\"tGF-!\"\"*(-%%sqrtG6#,$*&,&%#l|irGF4F,F-F-,(F " 0 "" {MPLTEXT 1 0 14 "subs(n=n+1,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$* &,(*&-%\"dG6%%\"xG%\"mG,&%\"nG\"\"\"F.F.F.F+F.\"\"#**F/F.F'F.F,F.-%$co sG6#%\"tGF.!\"\"*(-%%sqrtG6#,$*&,(%#l|irGF5F-F.F.F.F.,(F=F.F-F.F/F.F.F 5F.-F(6%F*F+,&F-F.F/F.F.-%$sinGF3F.F.F.*&-F86#,$*&,&F=F5F-F.F.,(F=F.F- F.F.F.F.F5F.FBF.F5F5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "rec _up:=d(x,m,n)=%;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%'rec_upG/-%\"dG6 %%\"xG%\"mG%\"nG,$*&,(*&-F'6%F)F*,&F+\"\"\"F3F3F3F*F3\"\"#**F4F3F0F3F2 F3-%$cosG6#%\"tGF3!\"\"*(-%%sqrtG6#,$*&,(%#l|irGF:F+F3F3F3F3,(FBF3F+F3 F4F3F3F:F3-F'6%F)F*,&F+F3F4F3F3-%$sinGF8F3F3F3*&-F=6#,$*&,&FBF:F+F3F3, (FBF3F+F3F3F3F3F:F3FGF3F:F:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 124 "swapargs:=proc(inval)\n RETURN(subs(dd_tmp=d,applyrule(d(x,a::alg ebraic,b::algebraic)=dd_tmp(x,b,a)*(-1)^(a-b),inval)));\nend;" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#>%)swapargsGR6#%&invalG6\"F(F(-%'RETUR NG6#-%%subsG6$/%'dd_tmpG%\"dG-%*applyruleG6$/-F16%%\"xG'%\"aG%*algebra icG'%\"bGF;*&-F06%F8F=F:\"\"\")!\"\",&F:FAF=FCFA9$F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 12 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 132 "midsimp_all:=proc(invals)\n RETURN(subs(dd_tm p=d,applyrule(d(x,m::algebraic,n::algebraic)=subs(d=dd_tmp,rhs(rec_mid )),invals)));\nend;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%,midsimp_allG R6#%'invalsG6\"F(F(-%'RETURNG6#-%%subsG6$/%'dd_tmpG%\"dG-%*applyruleG6 $/-F16%%\"xG'%\"mG%*algebraicG'%\"nGF;-F-6$/F1F0-%$rhsG6#%(rec_midG9$F (F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 130 "upsimp_all:=proc( invals)\n RETURN(subs(dd_tmp=d,applyrule(d(x,m::algebraic,n::algebraic )=subs(d=dd_tmp,rhs(rec_up)),invals)));\nend;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%+upsimp_allGR6#%'invalsG6\"F(F(-%'RETURNG6#-%%subsG6$ /%'dd_tmpG%\"dG-%*applyruleG6$/-F16%%\"xG'%\"mG%*algebraicG'%\"nGF;-F- 6$/F1F0-%$rhsG6#%'rec_upG9$F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 134 "downsimp_all:=proc(invals)\n RETURN(subs(dd_tmp=d,ap plyrule(d(x,m::algebraic,n::algebraic)=subs(d=dd_tmp,rhs(rec_down)),in vals)));\nend;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%-downsimp_allGR6#% 'invalsG6\"F(F(-%'RETURNG6#-%%subsG6$/%'dd_tmpG%\"dG-%*applyruleG6$/-F 16%%\"xG'%\"mG%*algebraicG'%\"nGF;-F-6$/F1F0-%$rhsG6#%)rec_downG9$F(F( F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 110 "order_simp:=proc(inv als)\n RETURN(applyrule(d(x,m::algebraic,-n::algebraic)=d(x,-m,n)*(-1) ^(m+n),invals));\nend;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%+order_sim pGR6#%'invalsG6\"F(F(-%'RETURNG6#-%*applyruleG6$/-%\"dG6%%\"xG'%\"mG%* algebraicG,$'%\"nGF6!\"\"*&-F16%F3,$F5F:F9\"\"\")F:,&F5F?F9F?F?9$F(F(F (" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 9 "rec_down;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/- %\"dG6%%\"xG%\"mG%\"nG,$*&,(*&-F%6%F'F(,&F)\"\"\"F1!\"\"F1F(F1\"\"#**F 3F1F.F1F0F1-%$cosG6#%\"tGF1F2*(-%%sqrtG6#,$*&,(%#l|irGF1F)F1F1F2F1,(F@ F2F)F1F3F2F1F2F1-F%6%F'F(,&F)F1F3F2F1-%$sinGF7F1F1F1*&-F;6#,$*&,&F@F1F )F1F1,(F@F2F)F1F1F2F1F2F1FEF1F2F2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 211 "midsimp:=proc(inval,i)\n subs(dd_tmp=d,applyrule(d( x,m::algebraic,n::name+i)=subs(d=dd_tmp,rhs(rec_mid)),inval)); \n RETU RN(subs(dd_tmp=d,applyrule(d(x,m::algebraic,i)=subs(\{d=dd_tmp,n=0\},r hs(rec_mid)),%)));\nend;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%(midsimp GR6$%&invalG%\"iG6\"F)F)C$-%%subsG6$/%'dd_tmpG%\"dG-%*applyruleG6$/-F0 6%%\"xG'%\"mG%*algebraicG,&'%\"nG%%nameG\"\"\"9%F?-F,6$/F0F/-%$rhsG6#% (rec_midG9$-%'RETURNG6#-F,6$F.-F26$/-F06%F7F8F@-F,6$<$FC/F=\"\"!FD%\"% GF)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 236 "downsimp:=proc(inval,i)\n local rep;\n rep:=sub s(n=n+i,rhs(rec_down));\nsubs(dd_tmp=d,applyrule(d(x,m::algebraic,n::n ame+i)=subs(d=dd_tmp,rep),inval)); \nRETURN(subs(dd_tmp=d,applyrule(d( x,m::algebraic,i)=subs(\{d=dd_tmp,n=0\},rep),%)));\nend;" }}{PARA 12 " " 1 "" {XPPMATH 20 "6#>%)downsimpGR6$%&invalG%\"iG6#%$repG6\"F+C%>8$-% %subsG6$/%\"nG,&F3\"\"\"9%F5-%$rhsG6#%)rec_downG-F06$/%'dd_tmpG%\"dG-% *applyruleG6$/-F?6%%\"xG'%\"mG%*algebraicG,&'F3%%nameGF5F6F5-F06$/F?F> F.9$-%'RETURNG6#-F06$F=-FA6$/-F?6%FFFGF6-F06$<$FO/F3\"\"!F.%\"%GF+F+F+ " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 233 "upsimp:=proc(inval,i) \n local rep;\n rep:=subs(n=n+i,rhs(rec_up));\n subs(dd_tmp=d,applyrul e(d(x,m::algebraic,n::name+i)=subs(d=dd_tmp,rep),inval));\n RETURN(sub s(dd_tmp=d,applyrule(d(x,m::algebraic,i)=subs(\{d=dd_tmp,n=0\},rep),%) ));\nend;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%'upsimpGR6$%&invalG%\"i G6#%$repG6\"F+C%>8$-%%subsG6$/%\"nG,&F3\"\"\"9%F5-%$rhsG6#%'rec_upG-F0 6$/%'dd_tmpG%\"dG-%*applyruleG6$/-F?6%%\"xG'%\"mG%*algebraicG,&'F3%%na meGF5F6F5-F06$/F?F>F.9$-%'RETURNG6#-F06$F=-FA6$/-F?6%FFFGF6-F06$<$FO/F 3\"\"!F.%\"%GF+F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 256 "ups imp := proc (inval, i) local rep; rep := subs(n = n+i,rhs(rec_up)); su bs(dd_tmp = d,applyrule(d(x,m::algebraic,n::name+i) = subs(d = dd_tmp, rep),inval)); RETURN(subs(dd_tmp = d,applyrule(d(x,m::algebraic,i) = s ubs(\{n = 0, d = dd_tmp\},rep),%))) end proc;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%'upsimpGR6$%&invalG%\"iG6#%$repG6\"F+C%>8$-%%subsG6$/ %\"nG,&F3\"\"\"9%F5-%$rhsG6#%'rec_upG-F06$/%'dd_tmpG%\"dG-%*applyruleG 6$/-F?6%%\"xG'%\"mG%*algebraicG,&'F3%%nameGF5F6F5-F06$/F?F>F.9$-%'RETU RNG6#-F06$F=-FA6$/-F?6%FFFGF6-F06$<$FO/F3\"\"!F.%\"%GF+F+F+" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "d_dlmn:=diff(d(x,m,n),x)=(m* cos(t)-n)*d(x,m,n)/sin(t)^2 + sqrt((l+m)*(l-m+1))*d(x,m-1,n)/sin(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'d_dlmnG/-%%diffG6$-%\"dG6%%\"xG% \"mG%\"nGF,,&*&*&,&*&F-\"\"\"-%$cosG6#%\"tGF4F4F.!\"\"F4F)F4F4*$)-%$si nGF7\"\"#F4F9F4*&*&-%%sqrtG6#*&,&%#l|irGF4F-F4F4,(FFF4F-F9F4F4F4F4-F*6 %F,,&F-F4F4F9F.F4F4F " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 136 "diffsimp:=proc(invals)\n RETURN(subs(dd_tmp=d,appl yrule(diff(d(x,m::algebraic,n::algebraic),x)=subs(d=dd_tmp,rhs(d_dlmn) ),invals)));\nend;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 12 " " 1 "" {XPPMATH 20 "6#>%)diffsimpGR6#%'invalsG6\"F(F(-%'RETURNG6#-%%su bsG6$/%'dd_tmpG%\"dG-%*applyruleG6$/-%%diffG6$-F16%%\"xG'%\"mG%*algebr aicG'%\"nGF>F;-F-6$/F1F0-%$rhsG6#%'d_dlmnG9$F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "#doesn't work??\ndiffsimp(diff(d(x,m,n),x)) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%%diffG6$-%\"dG6%%\"xG%\"mG%\"nG F)" }}}{EXCHG {PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "d1_symm:=\{d(x,1,0)=-d(x,0,1),d(x,1,-1)=d(x,-1,1), d(x,-1,0)=d(x,0,1),d(x,0,-1)=-d(x,0,1)\};" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(d1_symmG<&/-%\"dG6%%\"xG\"\"\"\"\"!,$-F(6%F*F,F+!\" \"/-F(6%F*F+F0-F(6%F*F0F+/-F(6%F*F0F,F./-F(6%F*F,F0F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "trig_subs:=\{cos(t)=x,sin(t)=sqrt(1 -x^2)\};" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "l_term:=[l*(l+1)=llp1]; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*trig_subsG<$/-%$cosG6#%\"tG%\"x G/-%$sinGF)*$-%%sqrtG6#,&\"\"\"F4*$)F+\"\"#F4!\"\"F4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'l_termG7#/*&%#l|irG\"\"\",&F(F)F)F)F)%%llp1G" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "d01_subs:=d(x,0,1)=sin(t)/sq rt(l*(l+1))*diff(P(x),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)d01_su bsG/-%\"dG6%%\"xG\"\"!\"\"\"*&*&-%$sinG6#%\"tGF+-%%diffG6$-%\"PG6#F)F) F+F+*$-%%sqrtG6#*&%#l|irGF+,&F=F+F+F+F+F+!\"\"" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 95 "one_subs:=d(x,0,1)=solve(d(x,0,1)*(1+cos(t))/s qrt(l*(l+1))/sin(t)=d(x,0,0)+d(x,-1,1),d(x,0,1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)one_subsG/-%\"dG6%%\"xG\"\"!\"\"\"*&*(-%%sqrtG6#*&%# l|irGF+,&F2F+F+F+F+F+-%$sinG6#%\"tGF+,&-F'6%F)F*F*F+-F'6%F)!\"\"F+F+F+ F+,&F+F+-%$cosGF6F+F=" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "sw apargs(downsimp(d(x,1,1),1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&, &-%\"dG6%%\"xG\"\"!\"\"\"!\"#*(-%%sqrtG6#,$*&%#l|irGF+,&F3!\"\"F+F5F+F 5F+-F'6%F)F5F+F+-%$sinG6#%\"tGF+F+F+*&-F/6#*&F3F+,&F3F+F+F+F+F+F8F+F5F 5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "subs(d(x,-1,1)=solve(o ne_subs,d(x,-1,1)),%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&,&-%\"dG 6%%\"xG\"\"!\"\"\"!\"#*&*&-%%sqrtG6#,$*&%#l|irGF+,&F4!\"\"F+F6F+F6F+,( F&F6*&F&F+-%$cosG6#%\"tGF+F6*(-F'6%F)F*F*F+-F06#*&F4F+,&F4F+F+F+F+F+-% $sinGF;F+F+F+F+*$-F06#FBF+F6F6F+*&-F06#FBF+FDF+F6F6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "d11_subs:=d(x,1,1)=simplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)d11_subsG/-%\"dG6%%\"xG\"\"\"F**&,(-F'6%F )\"\"!F*F**&F-F*-%$cosG6#%\"tGF*!\"\"*(-F'6%F)F/F/F*-%%sqrtG6#*&%#l|ir GF*,&F=F*F*F*F*F*-%$sinGF3F*F*F**&-F:6#F " 0 "" {MPLTEXT 1 0 9 "s_max:=4;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%&s_maxG\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 564 "reduce_d _max:=proc(aterm,amax)\n local i,tmp;\n tmp:=aterm;\n for i from am ax by -1 to 2 do\n tmp:=downsimp(tmp,i); \n tmp:=swapargs(tmp );\n tmp:=downsimp(tmp,i); \n tmp:=upsimp(tmp,-i); \n t mp:=swapargs(tmp);\n tmp:=upsimp(tmp,-i); \n od;\n tmp:=upsim p(tmp,-1); \n tmp:=swapargs(tmp);\n tmp:=upsimp(tmp,-1); \n t mp:=subs(d11_subs,tmp);\n tmp:=subs(d1_symm,tmp);\n subs(d(x,0,0)=P( x),%);\n subs(d01_subs,%);\n simplify(subs(trig_subs,%));\n simplif y(simplify(%,l_term));\n end;\n\nreduce_d:=proc(aterm)\n reduce_d_max( aterm,s_max);\nend;\n" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%-reduce_d_m axGR6$%&atermG%%amaxG6$%\"iG%$tmpG6\"F,C->8%9$?(8$9%!\"\"\"\"#%%trueGC (>F/-%)downsimpG6$F/F2>F/-%)swapargsG6#F/>F/F9>F/-%'upsimpG6$F/,$F2F4> F/F=>F/FB>F/-FC6$F/F4>F/F=>F/FI>F/-%%subsG6$%)d11_subsGF/>F/-FO6$%(d1_ symmGF/-FO6$/-%\"dG6%%\"xG\"\"!Fgn-%\"PG6#Ffn%\"%G-FO6$%)d01_subsGF[o- %)simplifyG6#-FO6$%*trig_subsGF[o-F`o6#-F`o6$F[o%'l_termGF,F,F," }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%)reduce_dGR6#%&atermG6\"F(F(-%-reduc e_d_maxG6$9$%&s_maxGF(F(F(" }}}{EXCHG {PARA 11 "" 1 "" {TEXT -1 0 "" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "reduce_d(d(x,2,1));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#*&,0-%%diffG6$-%\"PG6#%\"xGF+\"\"#*(\" \"%\"\"\"F%F/F+F/!\"\"*(F,F/F%F/)F+F,F/F/**F,F/%%llp1GF/F+F/F(F/F0*(F, F/F4F/F(F/F/*(F4F/F%F/F2F/F/*&F4F/F%F/F0F/**-%%sqrtG6#,&F/F/*$F2F/F0F/ -F:6#,&%#l|irGF/F,F/F/-F:6#,&FAF/F/F0F/F4F/F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 121 "dP_sub:=proc(aterm,i,j)\n solve(d(x,i,j)=reduc e_d(d(x,i,j)),diff(P(x),x));\n simplify(subs(diff(P(x),x)=%,aterm)); \+ \nend;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%'dP_subGR6%%&atermG%\"iG% \"jG6\"F*F*C$-%&solveG6$/-%\"dG6%%\"xG9%9&-%)reduce_dG6#F0-%%diffG6$-% \"PG6#F3F3-%)simplifyG6#-%%subsG6$/F9%\"%G9$F*F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "reduced_ds:=array(0..s_max,-s_max..s_max);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%+reduced_dsG-%&arrayG6%;\"\"!\"\"%;!\"%F*7\"" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 166 "get_reduced:=proc()\n local i,j;\n global reduced_ds;\nfor i from 0 to s_max do\n for j from -s_m ax to s_max do\n reduced_ds[i,j]:=reduce_d(d(x,i,j));\n od;od;\n null ;\nend;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%,get_reducedGR6\"6$%\"iG% \"jGF&F&C$?(8$\"\"!\"\"\"%&s_maxG%%trueG?(8%,$F/!\"\"F.F/F0>&%+reduced _dsG6$F,F2-%)reduce_dG6#-%\"dG6%%\"xGF,F2%%nullGF&6#F7F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 294 "fast_reduce:=proc(invars)\n local \+ aterm,tmp,i,j;\n tmp:=invars;\n for i from 0 to s_max do\n for j from \+ -s_max to s_max do\n tmp:=subs(d(x,i,j)=0,tmp)+coeff(tmp,d(x,i,j))*re duced_ds[i,j];\n if (i<>0) then tmp:=subs(d(x,-i,j)=0,tmp)+coeff(tmp, d(x,-i,j))*reduced_ds[i,-j]*(-1)^(i-j);fi;\n\n od;od;\nend;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%,fast_reduceGR6#%'invarsG6&%&atermG%$tmpG% \"iG%\"jG6\"F-C$>8%9$?(8&\"\"!\"\"\"%&s_maxG%%trueG?(8',$F6!\"\"F5F6F7 C$>F0,&-%%subsG6$/-%\"dG6%%\"xGF3F9F4F0F5*&-%&coeffG6$F0FCF5&%+reduced _dsG6$F3F9F5F5@$0F3F4>F0,&-F@6$/-FD6%FF,$F3F;F9F4F0F5*(-FI6$F0FUF5&FL6 $F3,$F9F;F5)F;,&F3F5F9F;F5F5F-F-F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 188 "get_dsub:=proc(i,j,k,n)\n local tmp;\n tmp:=solve( \{d(x,i,j)=reduced_ds[i,j],d(x,k,n)=reduced_ds[k,n]\},\{diff(P(x),x),P (x)\});\n if subs(tmp,P(x))=P(x) then\n tmp:=0;\n fi;\n RETURN(tmp) ;\nend;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%)get_dsubGR6&%\"iG%\"jG% \"kG%\"nG6#%$tmpG6\"F-C%>8$-%&solveG6$<$/-%\"dG6%%\"xG9$9%&%+reduced_d sG6$F:F;/-F76%F99&9'&F=6$FBFC<$-%%diffG6$-%\"PG6#F9F9FJ@$/-%%subsG6$F0 FJFJ>F0\"\"!-%'RETURNG6#F0F-F-F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 317 "convert_d:=proc(aterm,i,j,k,n)\nlocal tmp,tmp_reduce ;\n tmp:=get_dsub(i,j,k,n);\n tmp_reduce:=reduce_d(aterm); \n simplify (subs(tmp,tmp_reduce));\n simplify(%,l_term);\n factor(simplify(coeff( %,d(x,i,j))))*d(x,i,j)+factor(simplify(coeff(%,d(x,k,n))))*d(x,k,n)\n \+ +simplify(subs(\{P(x)=0,diff(P(x),x)=0\},tmp_reduce));\n end;" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#>%*convert_dGR6'%&atermG%\"iG%\"jG%\"k G%\"nG6$%$tmpG%+tmp_reduceG6\"F/C'>8$-%)get_dsubG6&9%9&9'9(>8%-%)reduc e_dG6#9$-%)simplifyG6#-%%subsG6$F2F;-FA6$%\"%G%'l_termG,(*&-%'factorG6 #-FA6#-%&coeffG6$FH-%\"dG6%%\"xGF6F7\"\"\"FTFXFX*&-FM6#-FA6#-FR6$FH-FU 6%FWF8F9FXFjnFXFX-FA6#-FD6$<$/-%\"PG6#FW\"\"!/-%%diffG6$FboFWFeoF;FXF/ F/F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "dsubs:=array(0..s_m ax,-s_max..s_max,0..s_max,-s_max..s_max);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&dsubsG-%&arrayG6';\"\"!\"\"%;!\"%F*F(F+7\"" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 627 "get_dsubs:=proc()\n local jmin,nmin,i,j,k,n;\n glo bal dsubs;\n #set to zero elements equivlanet by symmetries\n for i f rom 0 to s_max do\n for j from -s_max to s_max do\n for k from 0 t o s_max do\n for n from -s_max to s_max do\n dsubs[i,j,k,n]:=0 ;\n od;od;od;od; \nfor i from 0 to s_max do\n for j from -s_max to s_max do\n if (i>=1 or j>=0) and (abs(j)>=i) then\n for k from 0 to s_max do\n for n from -s_max to s_max do\n if (k>=1 or n>=0) and (abs(n)>=k) and\n not (i=k and j=n) and (nops(dsubs[k,n,i, j])<2) then\n \n dsubs[i,j,k,n]:=get_dsub(i,j,k,n);\n fi; \+ \n od;od;\n fi;\n od;od;\nend;\n " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%*get_dsubsGR6\"6(%%jminG%%nminG%\"iG%\"jG%\"kG%\"nGF& F&C$?(8&\"\"!\"\"\"%&s_maxG%%trueG?(8',$F3!\"\"F2F3F4?(8(F1F2F3F4?(8)F 7F2F3F4>&%&dsubsG6&F0F6F:FF>-%)get_dsubGF@F&6#F?F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "#read \"dlmn_info.map\":" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "get_reduced();" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%%nullG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "#this takes a while...\n#Skip this as don't use it\n#get_dsubs() ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "save reduced_ds, \"dlm n_info.map\";" }}}{EXCHG {PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "d2m2:=reduce_d(d(x,2,-2));" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#>%%d2m2G*&,4*(%\"xG\"\"\"-%\"PG6#F(F)% %llp1GF)\"\"%*(F(F)F*F))F-\"\"#F)F)*(\"\")F)F*F)F-F)F)**F.F)-%%diffG6$ F*F(F))F(F1F)F-F)!\"\"*&F*F)F0F)F9*(F.F)F5F)F8F)F9*(F.F)F5F)F-F)F)*(\" #7F)F5F)F(F)F9*&F3F)F5F)F9F)*&F-F),*F(!\"#F1F)*&F(F)F-F)F)F-F9F)F9" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "d22:=reduce_d(d(x,2,2));" } }{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$d22G*&,4*&-%%diffG6$-%\"PG6#%\"xGF .\"\"\")F.\"\"#F/!\"%**\"\"%F/F.F/F+F/%%llp1GF/F/*&\"\")F/F(F/!\"\"*( \"#7F/F(F/F.F/F/**F4F/F(F/F0F/F5F/F8*&F+F/)F5F1F/F/*(F.F/F+F/F=F/F/*(F 4F/F(F/F5F/F/*(F7F/F+F/F5F/F8F/*&F5F/,*F.!\"#F1F8*&F.F/F5F/F/F5F/F/F8 " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "d20:=reduce_d(d(x,2,0)) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$d20G,$*&,&*&-%%diffG6$-%\"PG6# %\"xGF/\"\"\"F/F0!\"#*&F,F0%%llp1GF0F0F0**-%%sqrtG6#%#l|irGF0-F66#,&F8 F0F0F0F0-F66#,&F8F0\"\"#F0F0-F66#,&F8F0F0!\"\"F0FCFC" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "P2_sub:=diff(P(x),x,x) = (2*x*diff(P(x),x )-llp1*P(x))/(1-x^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'P2_subG/-% %diffG6$-%\"PG6#%\"xG-%\"$G6$F,\"\"#*&,&*&-F'6$F)F,\"\"\"F,F6F0*&F)F6% %llp1GF6!\"\"F6,&F6F6*$)F,F0F6F9F9" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 149 "reduce_diffs:=proc(invars)\n simplify(subs(P2_sub,i nvars));\n simplify(subs(P2_sub,%));\n simplify(subs(P2_sub,%));\n \+ simplify(subs(P2_sub,%));\n\nend;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#> %-reduce_diffsGR6#%'invarsG6\"F(F(C&-%)simplifyG6#-%%subsG6$%'P2_subG9 $-F+6#-F.6$F0%\"%GF2F2F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 258 "group_ds:=proc(inval)\nlocal i,j,tmp,res;\ntmp:=inval;\nres:=0; \n for i from -s_max to s_max do\n for j from -s_max to s_max do\n r es:=res+ factor(simplify(coeff(tmp,d(x,i,j))))*d(x,i,j);\n tmp:=subs( d(x,i,j)=0,tmp);\n od\n od;\nres:=res+factor(simplify(tmp));\nend;" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#>%)group_dsGR6#%&invalG6&%\"iG%\"jG%$t mpG%$resG6\"F-C&>8&9$>8'\"\"!?(8$,$%&s_maxG!\"\"\"\"\"F8%%trueG?(8%F7F :F8F;C$>F3,&F3F:*&-%'factorG6#-%)simplifyG6#-%&coeffG6$F0-%\"dG6%%\"xG F6F=F:FKF:F:>F0-%%subsG6$/FKF4F0>F3,&F3F:-FC6#-FF6#F0F:F-F-F-" }}} {EXCHG {PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 184 "stupid_maple_subs:=proc(asubs,invars)\n #for when ma ple refuses to substitute in the demoninator\n local tmp;\n tmp:=simpl ify(invars);\n subs(asubs,numer(tmp))/subs(asubs,denom(tmp));\nend;" } }{PARA 12 "" 1 "" {XPPMATH 20 "6#>%2stupid_maple_subsGR6$%&asubsG%'inv arsG6#%$tmpG6\"F+C$>8$-%)simplifyG6#9%*&-%%subsG6$9$-%&numerG6#F.\"\" \"-F56$F7-%&denomGF:!\"\"F+F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 429 "get_P_ser:=proc(ord)\n local i,atone,ser2;\natone:=\{P(1)=1,D (P)(1)=l*(l+1)/2\};\nreduce_diffs(diff(P(x),x,x));\nconvert(%,D);\n\ns ubs(x=1+e,%);\nseries(%,e,ord-2);\nser2:=simplify(subs(atone,%),l_term );\nfor i from 0 to ord-2 do\n subs(atone,ser2);\n solve(`@@`(D,2+i) (P)(1)=factorial(i)*coeff(%,e,i),`@@`(D,2+i)(P)(1));\n atone :=atone \+ union \{`@@`(D,2+i)(P)(1)=%\};\nod;\n\nfactor(simplify(subs(atone,seri es(P(cos(t)),t,ord+1)),l_term));\nend;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "#Get series expansion (in theta) of Legendrea polynomial" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "Pser:=get_P_ser(14);" }}{PARA 12 " " 1 "" {XPPMATH 20 "6#>%*get_P_serGR6#%$ordG6%%\"iG%&atoneG%%ser2G6\"F ,C*>8%<$/-%\"PG6#\"\"\"F5/--%\"DG6#F3F4,$*&%\"lGF5,&F=F5F5F5F5#F5\"\"# -%-reduce_diffsG6#-%%diffG6%-F36#%\"xGFIFI-%(convertG6$%\"%GF9-%%subsG 6$/FI,&F5F5%\"eGF5FM-%'seriesG6%FMFS,&9$F5F@!\"\">8&-%)simplifyG6$-FO6 $F/FM%'l_termG?(8$\"\"!F5FW%%trueGC%-FO6$F/Fen-%&solveG6$/---%#@@G6$F9 ,&F@F5F]oF5F:F4*&-%*factorialG6#F]oF5-%&coeffG6%FMFSF]oF5Fgo>F/-%&unio nG6$F/<#/FgoFM-%'factorG6#-Fgn6$-FO6$F/-FU6%-F36#-%$cosG6#%\"tGFhq,&FX F5F5F5F[oF,F,F," }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%PserG+5%\"tG\"\" \"\"\"!,$%%llp1G#!\"\"\"\"%\"\"#,$*&F*F',&!\"#F'*&\"\"$F'F*F'F'F'#F'\" $#>F-,$*&F*F',(F*!#5*&\"\"&F')F*F.F'F'\"\")F'F'#F,\"&?:\"\"\"',$*&F*F' ,*!$s#F'*&\"$3$F'F*F'F'*&\"$S\"F'F=F'F,*&\"#NF')F*F4F'F'F'#F'\"(g4;&F> ,$*&F*F',,\"%oRF'*&\"%sUF'F*F'F,*&\"%k " 0 "" {MPLTEXT 1 0 30 "##############################" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "### Lensed correlation functions" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "max_ord:=2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(max _ordG\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "TT:=0;TE:=1;C plus:=2;Cminus:=3;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#TTG\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#TEG\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&CplusG\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'Cm inusG\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 150 "cosp1_sub:=(alpha[1]+beta[1])/2/a1 ;\ncosp2_sub:=(alpha[2]+beta[2])/2/a2;\nsinp1_sub:=-I*(alpha[1]-beta[1 ])/2/a1;\nsinp2_sub:=-I*(alpha[2]-beta[2])/2/a2;\n" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%*cosp1_subG,$*&,&&%&alphaG6#\"\"\"F+&%%betaGF*F+F+% #a1G!\"\"#F+\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*cosp2_subG,$*& ,&&%&alphaG6#\"\"#\"\"\"&%%betaGF*F,F,%#a2G!\"\"#F,F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*sinp1_subG*&*&^##!\"\"\"\"#\"\"\",&&%&alphaG6#F +F+&%%betaGF/F)F+F+%#a1GF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*sinp2 _subG*&*&^##!\"\"\"\"#\"\"\",&&%&alphaG6#F*F+&%%betaGF/F)F+F+%#a2GF)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 237 "spin1subs:=proc(invars) \n algsubs(cos(p1)=cosp1_sub,invars):\n algsubs(cos(p2)=cosp2_sub,%): \n algsubs(sin(p1)=sinp1_sub,%):\n algsubs(sin(p2)=sinp2_sub,%):\n alg subs(a1^2=alpha[1]*beta[1],%):\n simplify(algsubs(a2^2=alpha[2]*beta[2 ],%)):\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 12 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "assume(sigma2>0,Cgt>0);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "Cg0:=sigma2+Cgt;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$Cg0G,&%(s igma2|irG\"\"\"%%Cgt|irGF'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "vec:=array([alpha[1],beta[1],alpha[2],beta[2]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$vecG-%'vectorG6#7&&%&alphaG6#\"\"\"&%%betaGF+&F*6 #\"\"#&F.F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg) ;" }}{PARA 7 "" 1 "" {TEXT -1 80 "Warning, the protected names norm an d trace have been redefined and unprotected\n" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7^r%.BlockDiagonalG%,GramSchmidtG%,JordanBlockG%)LUdeco mpG%)QRdecompG%*WronskianG%'addcolG%'addrowG%$adjG%(adjointG%&angleG%( augmentG%(backsubG%%bandG%&basisG%'bezoutG%,blockmatrixG%(charmatG%)ch arpolyG%)choleskyG%$colG%'coldimG%)colspaceG%(colspanG%*companionG%'co ncatG%%condG%)copyintoG%*crossprodG%%curlG%)definiteG%(delcolsG%(delro wsG%$detG%%diagG%(divergeG%(dotprodG%*eigenvalsG%,eigenvaluesG%-eigenv ectorsG%+eigenvectsG%,entermatrixG%&equalG%,exponentialG%'extendG%,ffg ausselimG%*fibonacciG%+forwardsubG%*frobeniusG%*gausselimG%*gaussjordG %(geneqnsG%*genmatrixG%%gradG%)hadamardG%(hermiteG%(hessianG%(hilbertG %+htransposeG%)ihermiteG%*indexfuncG%*innerprodG%)intbasisG%(inverseG% 'ismithG%*issimilarG%'iszeroG%)jacobianG%'jordanG%'kernelG%*laplacianG %*leastsqrsG%)linsolveG%'mataddG%'matrixG%&minorG%(minpolyG%'mulcolG%' mulrowG%)multiplyG%%normG%*normalizeG%*nullspaceG%'orthogG%*permanentG %&pivotG%*potentialG%+randmatrixG%+randvectorG%%rankG%(ratformG%$rowG% 'rowdimG%)rowspaceG%(rowspanG%%rrefG%*scalarmulG%-singularvalsG%&smith G%,stackmatrixG%*submatrixG%*subvectorG%)sumbasisG%(swapcolG%(swaprowG %*sylvesterG%)toeplitzG%&traceG%*transposeG%,vandermondeG%*vecpotentG% (vectdimG%'vectorG%*wronskianG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "corr:=array(1..4,1..4,[[0,Cg0,-Cg2,Cgt],[Cg0,0,Cgt,-Cg2],[-Cg2 ,Cgt,0,Cg0],[Cgt,-Cg2,Cg0,0]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%% corrG-%'matrixG6#7&7&\"\"!,&%(sigma2|irG\"\"\"%%Cgt|irGF-,$%$Cg2G!\"\" F.7&F+F*F.F/7&F/F.F*F+7&F.F/F+F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "evecs:=eigenvectors(corr);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&evecsG6&7%,(%$Cg2G!\"\"*&\"\"#\"\"\"%%Cgt|irGF,F)%(s igma2|irGF)F,<#-%'vectorG6#7&F)F,F)F,7%,&F(F,F.F)F,<#-F16#7&F)F,F,F)7% ,(F(F)*&F+F,F-F,F,F.F,F,<#-F16#7&F,F,F,F,7%,&F(F,F.F,F,<#-F16#7&F,F,F) F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "evecs[1][3];" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#<#-%'vectorG6#7&!\"\"\"\"\"F(F)" }}} {EXCHG {PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 110 "#Get exponent of Gaussian prob distribution function \nfac:=simplify(evalm(transpose(vec)&*inverse(corr)&*vec));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$facG,$*&,\\o**&%&alphaG6#\"\"#\"\"\"&%%be taGF+F-)%%Cgt|irGF,F-%(sigma2|irGF-!\"#**F)F-F.F-)%$Cg2GF,F-F2F-F-**F) F-F.F-F1F-F5F-F-**)F.F,F-F6F-F2F-F1F-F-*(&F*6#F-F-&F/FF-F@*(F9F-F6F-F0F-F-*()F;F,F-F0F-F6F-F-**FKF-F1F-F6F- F2F-F-*,F?F-F;F-F=F-)F2F,F-F1F-F@*,F,F-F;F-F=F-F2F-F0F-F@**F;F-F=F-F5F -F2F-F-**F;F-F=F-F5F-F1F-F-**F;F-F)F-F6F-FNF-F@*,F,F-F;F-F)F-F6F-F0F-F @**F;F-F.F-F1F-FNF-F-*,F,F-F;F-F.F-F0F-F2F-F-**F;F-F.F-F1F-F5F-F-**FDF -F1F-F6F-F2F-F-**F=F-F)F-F1F-FNF-F-*,F,F-F=F-F)F-F0F-F2F-F-**F=F-F)F-F 1F-F5F-F-**F=F-F.F-F6F-FNF-F@*,F,F-F=F-F.F-F6F-F0F-F@**FGF-F6F-F2F-F1F -F-*,F?F-F)F-F.F-F1F-FNF-F@*.F,F-F;F-F)F-F6F-F2F-F1F-F@*.F,F-F=F-F.F-F 6F-F2F-F1F-F@F-,0*$)F2\"\"%F-F-*(F^oF-F>F-F1F-F-*(F^oF-FNF-F0F-F-*(F,F -F5F-FNF-F@**F^oF-F2F-F5F-F1F-F@*(F^oF-F5F-F0F-F@*$)F6F^oF-F-F@F3" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "Prob:=4*a1* a2/(2*Pi)^2*exp(-fac/2)/sqrt(det(corr));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%ProbG*&*(%#a1G\"\"\"%#a2GF(-%$expG6#*&,\\o**&%&alphaG6#\"\"#F (&%%betaGF2F()%%Cgt|irGF3F(%(sigma2|irGF(!\"#**F0F(F4F()%$Cg2GF3F(F8F( F(**F0F(F4F(F7F(F;F(F(**)F4F3F(F " 0 "" {MPLTEXT 1 0 131 "cmplx_subs:=solve( \{cos(p1)=cosp1_sub,cos(p2)=cosp2_sub,sin(p1)=sinp1_sub,sin(p2)=sinp2_ sub\},\{alpha[2],beta[2],alpha[1],beta[1]\});\n\n" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%+cmplx_subsG<&/&%%betaG6#\"\"\"*(^#!\"\"F*%#a1GF*,&-% $sinG6#%#p1GF**&^#F*F*-%$cosGF2F*F*F*/&F(6#\"\"#*(F,F*%#a2GF*,&-F16#%# p2GF**&F5F*-F7F@F*F*F*/&%&alphaGF:,&*&FCF*F=F*F**(F5F*F=F*F?F*F*/&FFF) ,&*&F6F*F.F*F**(F5F*F.F*F0F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "Intfunc:=exp(-(a2^2+a1^2)/sigma2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(IntfuncG-%$expG6#,$*&,&*$)%#a2G\"\"#\"\"\"F/*$)%#a1G F.F/F/F/%(sigma2|irG!\"\"F4" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 187 "#Now series expansion the prob dist function in Cgt (C_gl) and Cg 2 (C_\{gl,2\})\nser:=simplify(subs(cmplx_subs,simplify(series(subs(\{C gt=eps*Cgt,Cg2=eps*Cg2\},Prob)/Intfunc,eps,max_ord+1))));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$serG++%$epsG*&*&%#a1G\"\"\"%#a2GF*F**&)%#PiG \"\"#F*)%(sigma2|irGF/F*!\"\"\"\"!*&*(F)F*F+F*,0*&)F)F/F*%%Cgt|irGF*F* *.F/F*F+F*F9F*-%$cosG6#%#p1GF*F)F*-F<6#%#p2GF*F**.F/F*F+F*F9F*F)F*-%$s inGF=F*-FDF@F*F**.F/F*F)F*F+F*%$Cg2GF*FCF*FEF*F**.F/F*F)F*F+F*FGF*F;F* F?F*F2*&)F+F/F*F9F*F**(F/F*F9F*F1F*F2F*F**&F-F*)F1\"\"%F*F2F*,$*&*(F)F *F+F*,fo*&)FGF/F*F0F*!\"#*(\"\")F*F0F*)F9F/F*F2*&)F)FNF*FXF*F2*&)F+FNF *FXF*F2*,FNF*FJF*FGF*F1F*F9F*F***FNF*F8F*FJF*FTF*F2*,FNF*F8F*F9F*FGF*F 1F*F**2FWF*FJF*FXF*F8F*F;F*FEF*FCF*F?F*F2**FWF*FJF*FXF*F1F*F***F/F*FJF *FTF*F1F*F***FWF*F8F*F1F*FXF*F***F/F*F8F*FTF*F1F*F**.FWF*FTF*)F;F/F*F8 F*)F?F/F*FJF*F2*.FWF*FJF*FGF*F1F*F9F*FaoF*F2*0\"#;F*F)F*F1F*FXF*FCF*F+ F*FEF*F**0FdoF*F)F*F1F*FXF*F;F*F?F*F+F*F**.FWF*F8F*FXF*F`oF*FaoF*FJF*F 2*.FWF*F9F*FGF*F1F*F`oF*F8F*F2*2FdoF*F9F*FGF*F1F*F;F*F)F*F?F*F+F*F2*,F WF*F8F*FJF*FGF*F9F*F2*.FNF*)F)\"\"$F*FXF*F?F*F+F*F;F*F2*2FdoF*F9F*FGF* F1F*F)F*FCF*F+F*FEF*F**2FWF*FTF*F;F*F8F*FCF*F?F*FJF*FEF*F**0FNF*F)F*)F +F\\pF*FGF*F9F*F;F*F?F*F**0FNF*F[pF*F+F*FGF*F9F*F;F*F?F*F**.FNF*F[pF*F XF*FCF*F+F*FEF*F2*0FNF*F)F*F`pF*FGF*F9F*FCF*FEF*F2*0FNF*FGF*F[pF*F9F*F CF*F+F*FEF*F2*.FNF*F`pF*FXF*F;F*F)F*F?F*F2*.FNF*F`pF*FXF*F)F*FCF*FEF*F 2*.FWF*F8F*FJF*FGF*F9F*FaoF*F**.FWF*F8F*FJF*FGF*F9F*F`oF*F**,FNF*F8F*F XF*FJF*FaoF*F***\"\"'F*F8F*FXF*FJF*F2*,FNF*F8F*FXF*F`oF*FJF*F**,FNF*F8 F*FJF*FTF*FaoF*F**,FNF*F8F*FJF*FTF*F`oF*F*F*F**&F-F*)F1F[qF*F2#F2F/F/- %\"OG6#F*F\\p" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 12 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 196 "#Function to do integrals over angles\nmm_int:=proc( invars,m1,m2)\n int(series(invars*exp(I*p1*m1),eps,max_ord+1),p1=0..2* Pi);\n simplify(int(series(%*exp(-I*p2*m2),eps,max_ord+1),p2=0..2*Pi)) ;\n\nend;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%'mm_intGR6%%'invarsG%#m 1G%#m2G6\"F*F*C$-%$intG6$-%'seriesG6%*&9$\"\"\"-%$expG6#*(^#F4F4%#p1GF 49%F4F4%$epsG,&%(max_ordGF4F4F4/F:;\"\"!,$%#PiG\"\"#-%)simplifyG6#-F-6 $-F06%*&%\"%GF4-F66#*(^#!\"\"F4%#p2GF49&F4F4F " 0 "" {MPLTEXT 1 0 357 "#function to return particu lar d_lm related by symmetries to the input\nget_d:=proc(m,n,x)\n loca l tmp,i,j,sgn;\n if (abs(m) > abs(n)) then\n i:=m; j:=n;sgn:=1;\n el se\n i:=n; j:=m;sgn:=(-1)^(m-n);\n end if;\n if i<0 then \n i:=-i; \n j:=-j;\n sgn:=sgn*(-1)^(i+j);\n fi;\n if i=0 and j<0 then\n j :=-j;\n sgn:=sgn*(-1)^j;\n end if; \n return(sgn*d(x,i,j));\nend;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&get_dGR6%%\"mG%\"nG%\"xG6&%$tmpG% \"iG%\"jG%$sgnG6\"F/C&@%2-%$absG6#9%-F46#9$C%>8%F9>8&F6>8'\"\"\"C%>FF>F9>F@)!\"\",&F9FAF6FG@$2F<\"\"!C%>F<,$FF>,$F>FG>F@*&F@FA)FG,&F FAFA@$3/FFKC$>F>FP>F@*&F@FA)FGF>FAO*&F@FA-%\"dG6%9&FFAF/ F/F/" }}}{EXCHG {PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 294 "get_mm_ints:=proc(invars,amax,s1,s2)\n local tm p,ares,res,i,j;\n tmp:=simplify(invars);\n res:=0;\n for i from -amax \+ to amax do\n for j from -amax to amax do\n ares:= mm_int(invars,i,j )*get_d(i,s1,a1)*get_d(j,s2,a2);\n res:=res+factor(simplify(ares))*g et_d(i,j,x); \n od;od; \n return(res);\nend;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%,get_mm_intsGR6&%'invarsG%%amaxG%#s1G%#s2G6'%$tmpG%%a resG%$resG%\"iG%\"jG6\"F1C&>8$-%)simplifyG6#9$>8&\"\"!?(8',$9%!\"\"\" \"\"F?%%trueG?(8(F>FAF?FBC$>8%*(-%'mm_intG6%F8F=FDFA-%&get_dG6%F=9&%#a 1GFA-FM6%FD9'%#a2GFA>F:,&F:FA*&-%'factorG6#-F66#FGFA-FM6%F=FD%\"xGFAFA OF:F1F1F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "s_max:=4;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%&s_maxG\"\"%" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 651 "#Substitute for X_\{imn\}\nint_subs:=proc(inv ars)\n local tmp,alpha,avar,pow,res,i,j;\n res:=invars; \n for alpha f rom 1 to 2 do\n if(alpha=1) then avar:=a1 else avar:=a2; fi;\n tmp :=coeff(res,d(avar,1,0));\n #first relate d_10 to derivative, do by p arts\n res:=subs(d(avar,1,0)=0,res)-simplify(diff(tmp*Intfunc,avar) /Intfunc)/sqrt(llp1)*d(avar,0,0);\n for i from 0 to s_max do\n fo r j from -s_max to s_max do\n tmp:=convert(series(coeff(res,d(ava r,i,j)),avar,9),polynom);\n for pow from 6 to 1 by -1 do\n \+ tmp:=algsubs(avar^pow = sigma2^pow/2*X(pow-1,i,j),tmp);\n end do; \n res:=subs(d(avar,i,j)=0,res)+tmp; \n od;od;\n end;\nend;" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#>%)int_subsGR6#%'invarsG6)%$tmpG%&alph aG%%avarG%$powG%$resG%\"iG%\"jG6\"F0C$>8(9$?(8%\"\"\"F7\"\"#%%trueGC&@ %/F6F7>8&%#a1G>F>%#a2G>8$-%&coeffG6$F3-%\"dG6%F>F7\"\"!>F3,&-%%subsG6$ /FGFJF3F7*&*&-%)simplifyG6#*&-%%diffG6$*&FCF7%(IntfuncGF7F>F7Fen!\"\"F 7-FH6%F>FJFJF7F7-%%sqrtG6#%%llp1GFfnFfn?(8)FJF7%&s_maxGF9?(8*,$F_oFfnF 7F_oF9C%>FC-%(convertG6$-%'seriesG6%-FE6$F3-FH6%F>F^oFaoF>\"\"*%(polyn omG?(8'\"\"'FfnF7F9>FC-%(algsubsG6$/)F>Fbp,$*&)%'sigma2GFbpF7-%\"XG6%, &FbpF7F7FfnF^oFaoF7#F7F8FC>F3,&-FN6$/F]pFJF3F7FCF7F0F0F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "kernel:=2*a1/sigma2*exp(-a1^2/sigma 2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'kernelG,$*&*&%#a1G\"\"\"-%$e xpG6#,$*&*$)F(\"\"#F)F)%(sigma2|irG!\"\"F3F)F)F2F3F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 529 "#Substitutions for some X_\{imn\} in ter ms of X_\{imn\}'\nget_dXsubs:=proc()\n local as1,as2;\n global X4_sub, X2_sub;\nfor as2 from -2 to 2 by 2 do\n as1:=abs(as2);\n diff(kernel,s igma2);\n simplify(%/exp(-a1^2/sigma2)); \n int_subs(%*d(a1,as1,as2)); \n solve(dX(0,as1,as2)=%,X(2,as1,as2));\n X2_sub[as1,as2]:=X(2,as1,as2 )=simplify(%);\n diff(kernel,sigma2,sigma2); \n simplify(%/exp(-a1^2/s igma2));\n int_subs(%*d(a1,as1,as2));\n solve(ddX(0,as1,as2)=%,X(4,as1 ,as2));\n subs(X2_sub[as1,as2],%);\n X4_sub[as1,as2]:=X(4,as1,as2)=sim plify(%);\nod;\n\nend;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%+get_dXsub sGR6\"6$%$as1G%$as2GF&F&?(8%!\"#\"\"#F-%%trueGC.>8$-%$absG6#F+-%%diffG 6$%'kernelG%'sigma2G-%)simplifyG6#*&%\"%G\"\"\"-%$expG6#,$*&*$)%#a1GF- F?F?F9!\"\"FHFH-%)int_subsG6#*&F>F?-%\"dG6%FGF1F+F?-%&solveG6$/-%#dXG6 %\"\"!F1F+F>-%\"XG6%F-F1F+>&%'X2_subG6$F1F+/FX-F;6#F>-F66%F8F9F9F:FI-F Q6$/-%$ddXGFVF>-FY6%\"\"%F1F+-%%subsG6$FfnF>>&%'X4_subGFhn/FcoFjnF&6$F [pFgnF&" }}}{EXCHG {PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "get_dXsubs();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"XG6%\"\"%\"\"#F(*&,(*&-%$ddXG6%\"\"!F(F(\"\"\")%(sigma2|irG F(F0F0*&F(F0-F%F.F0F0*(F'F0-%#dXGF.F0F2F0F0F0*$F1F0!\"\"" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "kernel ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&*&%#a1G\"\"\"-%$expG6#,$*&*$) F&\"\"#F'F'%(sigma2|irG!\"\"F1F'F'F0F1F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "diff(kernel*a1/sigma2,sigma2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&*&)%#a1G\"\"#\"\"\"-%$expG6#,$*&*$F&F)F)%(sigma2|ir G!\"\"F1F)F)*$)F0\"\"$F)F1!\"%*&*(F(F))F'\"\"%F)F*F)F)*$)F0F9F)F1F)" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "simplify(%/exp(-a1^2/sigma 2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&*&)%#a1G\"\"#\"\"\",&%(sig ma2|irG!\"#*$F&F)F)F)F)*$)F+\"\"%F)!\"\"F(" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 22 "int_subs(%*d(a1,2,1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&-%\"XG6%\"\"\"\"\"#F(!\"#*&-F&6%\"\"$F)F(F(%(sigma2 |irGF(F(F(F/!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "X321_s ub:=X(3,2,1)=solve(dX(1,2,1)=%,X(3,2,1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)X321_subG/-%\"XG6%\"\"$\"\"#\"\"\"*&,&*&-%#dXG6%F+F* F+F+%(sigma2|irGF+F+*&F*F+-F'F1F+F+F+F2!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "simplify(diff(kernel*a1/sigma2,sigma2)/exp(-a1^2 /sigma2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&*&)%#a1G\"\"#\"\"\", &%(sigma2|irG!\"#*$F&F)F)F)F)*$)F+\"\"%F)!\"\"F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "int_subs(%*d(a1,3,2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&-%\"XG6%\"\"\"\"\"$\"\"#!\"#*&-F&6%F)F)F*F(%(sigma2 |irGF(F(F(F/!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "X332_s ub:=X(3,3,2)=solve(dX(1,3,2)=%,X(3,3,2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)X332_subG/-%\"XG6%\"\"$F)\"\"#*&,&*&-%#dXG6%\"\"\"F) F*F1%(sigma2|irGF1F1*&F*F1-F'F0F1F1F1F2!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "simplify(diff(kernel*a1/sigma2,sigma2)/exp(-a1^2 /sigma2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&*&)%#a1G\"\"#\"\"\", &%(sigma2|irG!\"#*$F&F)F)F)F)*$)F+\"\"%F)!\"\"F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "int_subs(%*d(a1,3,-2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&-%\"XG6%\"\"\"\"\"$!\"#F**&-F&6%F)F)F*F(%(sigma2|ir GF(F(F(F.!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "X33m2_sub :=X(3,3,-2)=solve(dX(1,3,-2)=%,X(3,3,-2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*X33m2_subG/-%\"XG6%\"\"$F)!\"#*&,&*&-%#dXG6%\"\"\"F) F*F1%(sigma2|irGF1F1*&\"\"#F1-F'F0F1F1F1F2!\"\"" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 58 "simplify(diff(kernel*a1/sigma2,sigma2)/exp(-a1 ^2/sigma2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&*&)%#a1G\"\"#\"\" \",&%(sigma2|irG!\"#*$F&F)F)F)F)*$)F+\"\"%F)!\"\"F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "int_subs(%*d(a1,1,-2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&-%\"XG6%\"\"\"F(!\"#F)*&-F&6%\"\"$F(F)F(%(sigma2 |irGF(F(F(F.!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "X31m2_ sub:=X(3,1,-2)=solve(dX(1,1,-2)=%,X(3,1,-2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*X31m2_subG/-%\"XG6%\"\"$\"\"\"!\"#*&,&*&-%#dXG6%F*F* F+F*%(sigma2|irGF*F**&\"\"#F*-F'F1F*F*F*F2!\"\"" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 117 "#####Get each correlation function kernel\n#N eed to repeat code from here several times with different \"what\" set tings" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "what:=Cplus;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%whatG\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 146 "if what=TT then\n\n s1:=0;\n s2:=0;\nelif what \+ = Cplus then\n s1:=2;\n s2:=2;\nelif what = Cminus then\n s1:=2;\n s2: =-2;\nelse\n s1:=0;\n s2:=-2; \nend if; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#s1G\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#s2G\" \"#" }}}{EXCHG {PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "#Make some coffee while this runs...\nres[what] :=get_mm_ints(convert(ser,polynom)*exp(-I*(s1*p1-s2*p2)),4,s1,s2);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#>&%$resG6#\"\"#,<**+%%$epsG,$*&*()%#a1 G\"\"$\"\"\")%#a2GF1F2)%%Cgt|irGF'F2F2*$)%(sigma2|irG\"\"'F2!\"\"F'F'F 2-%\"dG6%F0F'\"\"!F2-F=6%F4F'F?F2-F=6%%\"xGF?F?F2F2**+%F+,$*&*,F/F2F4F 2%$Cg2GF2F6F2,&*$)F4F'F2F2F9F;F2F2*$F8F2F;!\"%F'F2F " 0 "" {MPLTEXT 1 0 20 "int_subs(res[what]):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "res_ser:=subs(eps=1,group_ds (%));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%(res_serG,4*()%%Cgt|irG\"\" #\"\"\")-%\"XG6%F)F)\"\"!F)F*-%\"dG6%%\"xGF/F/F*#F*F)*&**F(F*-F-6%F*F) F*F*,(*&%(sigma2|irGF*F7F*!\"\"*(\"\"%F*F(F*F7F*F***F)F*F(F*-F-6%\"\"$ F)F*F*F;F*FF*FTF*F'F*F***FIF*FUF*F'F*-F-6%F>F)F)F*F**&F TF*)FGF)F*F**,F)F*FIF*F;F*FLF*FfnF*F<*,\"\")F*FIF*F;F*FLF*F'F*F<*,F)F* FIF*FUF*FLF*F(F*F**(FUF*)FLF)F*FfnF*F***F)F*FUF*F\\oF*F'F*F*F*-F16%F3F )F)F*F**$FUF*FF*F7F*FeoF *F(F*F<**F7F*F(F*-F-6%FBFBF)F*F;F*F***F(F*F@F*F;F*FeoF*F*F*-F16%F3FBF* F*F*F;FF*F(F*FeoF*F<**F)F*F(F*FioF* F;F*F*F*-F16%F3FBFBF*F*F;FF/F*-F-6%F)F>F)F*F,F*F**& *.F)F*FGF*F(F*FipF*FHF*-F16%F3F>F)F*F*F;FF>F*)Fi pF)F*F*" }}}{EXCHG {PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 148 "res_ser2:=simplify(subs(\{X31m2_sub,X33m2_su b,X321_sub,X332_sub,X2_sub[2,-2],X4_sub[2,-2],X2_sub[2,2],X4_sub[2,2], X2_sub[0,0],X4_sub[0,0]\},res_ser));" }}{PARA 12 "" 1 "" {XPPMATH 20 " 6#>%)res_ser2G,D*()%%Cgt|irG\"\"#\"\"\")-%\"XG6%F)F)\"\"!F)F*-%\"dG6%% \"xGF/F/F*#F*F)*(F(F*)-F-6%F*F)F*F)F*-F16%F3F*F*F*F**,F)F*F'F*F7F*F9F* -%#dXGF8F*F**.F)F*%$Cg2GF*F(F*F,F*-F=6%F/F)F)F*-F16%F3F)F/F*!\"\"*&-F1 6%F3F)F)F*)-F-FAF)F*F***FFF*FIF*F'F*-%$ddXGFAF*F**,F)F*FFF*FIF*F(F*F@F *F**(FFF*)F?F)F*)F@F)F*F***F)F*FFF*F'F*FPF*F**,F)F*F?F*-F16%F3\"\"$F*F *F7F*-F-6%F*FUF)F*F**.F)F*F?F*FSF*F(F*F7F*-F=FWF*F**.F)F*F?F*FSF*FVF*F (F*F " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "res_nonpert:=array(TT..Cminus);\nread \"results_nonpert.map\":" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,res_nonpertG-%&arrayG6$;\"\"!\"\" $7\"" }}{PARA 8 "" 1 "" {TEXT -1 44 "Error, unable to read `results_no npert.map`\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "res_nonpert [what]:=group_ds(res_ser2);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>&%,res _nonpertG6#\"\"#,4*()%%Cgt|irGF'\"\"\")-%\"XG6%F'F'\"\"!F'F,-%\"dG6%% \"xGF1F1F,#F,F'**-F/6%F,F'F,F,F+F,,&*&-%#dXGF9F,F+F,F'F8F,F,-F36%F5F,F ,F,F,*.F'F,%$Cg2GF,F+F,F.F,-F=6%F1F'F'F,-F36%F5F'F1F,!\"\"*&,,*$)-F/FC F'F,F,*(FKF,F*F,-%$ddXGFCF,F,**F'F,FKF,F+F,FBF,F,*&)FAF'F,)FBF'F,F,*(F 'F,F*F,FRF,F,F,-F36%F5F'F'F,F,**F'F,FAF,,(*&F8F,-F/6%F,\"\"$F'F,F,*(F+ F,F8F,-F=FZF,F,*(FYF,F+F,F " 0 "" {MPLTEXT 1 0 40 "save res_nonpert , \"results_nonpert.map\";" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "########That's it! (just repeat for the other \"what\" spectra\n\n " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "#Now find forms for the X_\{imn\} integrals as series in sigma^2" }}}{EXCHG {PARA 12 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 330 "X_ser:=pro c(w,i,j)\n global X_ser_sub; \n reduce_d(d(x,i,j));\n simplify(series( subs(\{diff(P(x),x)=-1/sin(t)*diff(Pser,t),P(x)=Pser,x=cos(t)\},%),t,1 5)); \n int(2*(t/sigma2)^(1+w)*exp(-t^2/sigma2)*(convert(%,polynom)),t =0..infinity);\n factor(convert(simplify(series(%,sigma2,5)),polynom)) ;\nseries(%,sigma2,3);\nfactor(simplify(%));\nend;" }}{PARA 12 "" 1 " " {XPPMATH 20 "6#>%&X_serGR6%%\"wG%\"iG%\"jG6\"F*F*C(-%)reduce_dG6#-% \"dG6%%\"xG9%9&-%)simplifyG6#-%'seriesG6%-%%subsG6$<%/F2-%$cosG6#%\"tG /-%\"PG6#F2%%PserG/-%%diffG6$FEF2,$*&-FK6$FHFC\"\"\"-%$sinGFB!\"\"FT% \"%GFC\"#:-%$intG6$,$*()*&FCFQ%'sigma2GFT,&9$FQFQFQFQ-%$expG6#,$*&*$)F C\"\"#FQFQFhnFTFTFQ-%(convertG6$FU%(polynomGFQFbo/FC;\"\"!%)infinityG- %'factorG6#-Fdo6$-F66#-F96%FUFhn\"\"&Ffo-F96%FUFhn\"\"$-F\\p6#-F66#FUF *6#%*X_ser_subGF*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "X_ser( 0,0,0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#++%(sigma2|irG\"\"\"\"\"!,$ %%llp1G#!\"\"\"\"%F%,$*&F(F%,&!\"#F%*&\"\"$F%F(F%F%F%#F%\"#'*\"\"#-%\" OG6#F%F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "X_ser(2,2,0);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#++%(sigma2|irG,$*&*&%%llp1G\"\"\",&! \"#F)F(F)F)F)**-%%sqrtG6#%#l|irGF)-F.6#,&F0F)F)F)F)-F.6#,&F0F)\"\"#F)F )-F.6#,&F0F)F)!\"\"F)F;#F)\"\"%\"\"!,$*&*&F(F))F*F7F)F)**-F.6#F0F)-F.6 #F3F)-F.6#F6F)-F.6#F:F)F;#F;\"#;F),$*&*(F(F)F*F),(*$)F(F7F)\"#:*&\"#qF )F(F)F;\"$O\"F)F)F)**-F.6#F0F)-F.6#F3F)-F.6#F6F)-F.6#F:F)F;#F)\"%?>F7- %\"OG6#F)\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "X_ser(0,2 ,2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#++%(sigma2|irG\"\"\"\"\"!,&%%l lp1G#!\"\"\"\"%F%F%F%,(F(#!#8\"#[*&#F%\"#KF%)F(\"\"#F%F%#\"#<\"#CF%F4- %\"OG6#F%\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "X_ser(1,2 ,1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#++%(sigma2|irG,$*&,&!\"#\"\"\" %%llp1GF)F)*&-%%sqrtG6#,&%#l|irGF)\"\"#F)F)-F-6#,&F0F)F)!\"\"F)F5#F5F1 \"\"!,$*&*&F'F),&F*\"\"$\"\")F5F)F)*&-F-6#F/F)-F-6#F4F)F5#F)\"#CF),$*& *&F'F),(*$)F*F1F)\"\"&*&\"#IF)F*F)F5\"#oF)F)F)*&-F-6#F/F)-F-6#F4F)F5#F 5\"$?$F1-%\"OG6#F)F<" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "X_s er(1,3,2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#++%(sigma2|irG,$*&,&%%ll p1G\"\"\"\"\"'!\"\"F)*&-%%sqrtG6#,&%#l|irGF)\"\"#F+F)-F.6#,&F1F)\"\"$F )F)F+#F+F2\"\"!,$*&*&F'F),&F(F6\"#?F+F)F)*&-F.6#F0F)-F.6#F5F)F+#F)\"#C F),$*&*&F'F),(*$)F(F2F)\"\"&*&\"#qF)F(F)F+\"$3$F)F)F)*&-F.6#F0F)-F.6#F 5F)F+#F+\"$?$F2-%\"OG6#F)F6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "X_ser(2,4,2);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#++%(sigma2|irG,$* &*&,&%%llp1G\"\"\"\"\"'!\"\"F*,&F)F*\"#7F,F*F***-%%sqrtG6#,&%#l|irGF* \"\"%F*F*-F16#,&F4F*\"\"$F,F*-F16#,&F4F*F9F*F*-F16#,&F4F*\"\"#F,F*F,#F *F5\"\"!,$*&*(F(F*F-F*,&F)F*\"#5F,F*F***-F16#F3F*-F16#F8F*-F16#FF@-%\"OG6#F*F9" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "0 0 0" 239 } {VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }