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2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 262 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Normal" -1 263 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 38 "Numerical evalua tion of Heun functions" }}{PARA 256 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 0 "" }{TEXT 256 16 "E.S. Cheb-Terrab" }}{PARA 256 "" 0 "" {TEXT 257 29 "CECM, Simon Fraser University" }}{PARA 256 "" 0 "" {TEXT 258 26 "Research Fellow, Maplesoft" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 14 "" 0 "" {TEXT -1 290 "The five multiparameter Heun equ ations have been popping up with surprising frequency in applications \+ during the last 10 years. Heun equations include as particular cases t he Lame, Mathieu, spheroidal wave, hypergeometric, and with them most \+ of the known equations of mathematical physics.\n" }}{PARA 14 "" 0 "" {TEXT -1 316 "Five Heun functions are defined as the solutions to each of these five Heun equations. In this talk, the difficulties for nume rically evaluating these functions are summarized and an a hybrid appr oach resolving the problem, exploring exact Heun function identities a nd numerical evaluation techniques, is presented. " }}{PARA 14 "" 0 " " {TEXT -1 333 "\nFor those more familiar with linear ODE topics, in m ore technical words, this presentation is about a hybrid symbolic & nu meric approach for tackling the \"two point connection problem\" (TPCP ), for a function with four singularities and depending on 7 complex p arameters, in a case where the exact solution to the TPCP is not known ." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 14 "Heun equations" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "restart;with(DEtools):with(PDEtools):\ndeclare(y(x),prime=x);\ng a := convert(gamma,`local`):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(-%\" yG6#%\"xG\"\"\"%9will~now~be~displayed~asGF(F%F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(% " 0 "" {MPLTEXT 1 0 108 "GHE := diff(y(x),x,x)+(ga/x+delta/(x-1)+epsilon/(x-a ))*diff(y(x),x)+(alpha*beta*x-q)/x/(x-1)/(x-a)*y(x) = 0;" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%$GHEG/,(%$y''G\"\"\"*&,(*&%&gammaGF(%\"xG!\"\" F(*&%&deltaGF(,&F-F(F(F.F.F(*&%(epsilonGF(,&F-F(%\"aGF.F.F(F(%#y'GF(F( *,,&*(%&alphaGF(%%betaGF(F-F(F(%\"qGF.F(F-F.F1F.F4F.%\"yGF(F(\"\"!" }} }{PARA 0 "" 0 "" {TEXT -1 34 "with four regular singular points," }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "singularities( GHE );" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$/%(regularG<&\"\"!\"\"\"%)infinityG%\" aG/%*irregularG<\"" }}}{PARA 0 "" 0 "" {TEXT -1 260 " and four other e quations derived from the GHE by \"coalescing singularities\"; these a re the Confluent (CHE), Biconfluent (BHE), Doubleconfluent (DHE) and T riconfluent (THE) equations. The five Heun equations respectively dep end on 6, 5, 4, 4 and 3 parameters." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 52 "Example: coalescing singularities to obt ain the CHE." }}{PARA 0 "" 0 "" {TEXT -1 28 "Redefine the parameters u sin" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "beta=beta*a, epsilon= epsilon*a, q=q*a;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/%%betaG*&F$\"\" \"%\"aGF&/%(epsilonG*&F)F&F'F&/%\"qG*&F,F&F'F&" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 14 "eval(GHE,[%]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(%$y''G\"\"\"*&,(*&%&gammaGF&%\"xG!\"\"F&*&%&deltaGF&,&F+F&F&F ,F,F&*(%(epsilonGF&%\"aGF&,&F+F&F2F,F,F&F&%#y'GF&F&*,,&**%&alphaGF&%%b etaGF&F2F&F+F&F&*&%\"qGF&F2F&F,F&F+F,F/F,F3F,%\"yGF&F&\"\"!" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 10 "Now take " }{XPPEDIT 18 0 "a -> i nfinity" "6#j+6#%\"aG7\"6$%)operatorG%&arrowG6\"%)infinityGF*F*F*6#\"+ 'yu.3#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "Limit(%,a=infinit y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&LimitG6$/,(%$y''G\"\"\"*&,(* &%&gammaGF)%\"xG!\"\"F)*&%&deltaGF),&F.F)F)F/F/F)*(%\"aGF)%(epsilonGF) ,&F.F)F4F/F/F)F)%#y'GF)F)*,,&**%&alphaGF)%%betaGF)F4F)F.F)F)*&%\"qGF)F 4F)F/F)F.F/F2F/F6F/%\"yGF)F)\"\"!/F4%)infinityG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "If we evaluate this limit, the singularity at " } {TEXT 314 1 "a" }{TEXT -1 37 " \"coalesces\" with the singularity at \+ " }{XPPEDIT 18 0 "infinity" "6#%)infinityG" }{TEXT -1 2 ". " }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 114 "CHE := (lhs-rhs)(collect(is olate( value(%), diff(y(x),x,x)), [y(x),diff(y(x),x)], u -> convert(u, parfrac,x))) = 0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$CHEG/,(%$y''G \"\"\"*&,(%(epsilonGF(*&%&deltaGF(,&%\"xGF(F(!\"\"F0F0*&%&gammaGF(F/F0 F0F(%#y'GF(F0*&,&*&,&*&%&alphaGF(%%betaGF(F(%\"qGF0F(F.F0F(*&F;F(F/F0F (F(%\"yGF(F0\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 93 "Since the tw o singularities being coalesced are regular, the resulting single sing ularity at " }{XPPEDIT 18 0 "infinity" "6#%)infinityG" }{TEXT -1 67 " \+ will be irregular (this is typical; there are exceptions though..)" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "singularities(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/%(regularG<$\"\"!\"\"\"/%*irregularG<#%)inf inityG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 127 "The remaining three He un Bi, Double and Tri confluent equations are obtained in essentially \+ the same way; these equations are: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 117 "BHE := diff(y(x),x,x)+(-2*x-beta+(1+alpha)/x)*diff(y (x),x)+1/2*((2*ga-2*alpha-4)*x-alpha*beta-delta-beta)/x*y(x) = 0;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$BHEG/,(%$y''G\"\"\"*&,(*&\"\"#F(%\" xGF(!\"\"%%betaGF.*&,&F(F(%&alphaGF(F(F-F.F(F(%#y'GF(F(**F,F.,**&,(*&F ,F(%&gammaGF(F(*&F,F(F2F(F.\"\"%F.F(F-F(F(*&F2F(F/F(F.%&deltaGF.F/F.F( F-F.%\"yGF(F(\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "singu larities(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/%(regularG<#\"\"!/%*i rregularG<#%)infinityG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 134 " DHE := diff(y(x),x,x)+(alpha+alpha/x^2+1/x)*diff(y(x),x)+1/4*((4*ga+2* alpha)*x^2+(4*delta+2*alpha^2-1)*x+4*beta-2*alpha)/x^3*y(x) = 0;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$DHEG/,(%$y''G\"\"\"*&,(%&alphaGF(*& F+F(%\"xG!\"#F(*&F(F(F-!\"\"F(F(%#y'GF(F(**\"\"%F0,**&,&*&F3F(%&gammaG F(F(*&\"\"#F(F+F(F(F()F-F:F(F(*&,(*&F3F(%&deltaGF(F(*&F:F()F+F:F(F(F(F 0F(F-F(F(*&F3F(%%betaGF(F(*&F:F(F+F(F0F(F-!\"$%\"yGF(F(\"\"!" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "singularities(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/%(regularG<\"/%*irregularG<$\"\"!%)infinity G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "THE := diff(y(x),x,x)- (ga+3*x^2)*diff(y(x),x)+(alpha+(beta-3)*x)*y(x) = 0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$THEG/,(%$y''G\"\"\"*&,&%&gammaGF(*&\"\"$F()%\"xG \"\"#F(F(F(%#y'GF(!\"\"*&,&%&alphaGF(*&,&%%betaGF(F-F2F(F/F(F(F(%\"yGF (F(\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "singularities(% );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/%(regularG<\"/%*irregularG<#%)i nfinityG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 316 5 "Note:" }{TEXT -1 89 " despite the fact that, for instance , the THE has a single irregular point at infinity, " }{TEXT 315 196 "none of the solutions to these five equations can be expressed the fu nctions of the mathematical language unless we particularize the param eters or we introduce the Heun functions in the language." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 256 "" 0 "" {TEXT -1 74 " ______________________________________________________________________ ____" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 275 32 " Heun equations in the literature" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 15 "" 0 "" {OLE 1 25097 1 "[xm]Br=WfoRrB:::wk;nyyI;G:;:j::>:B>N: F:nyyyyy]::yyyyyy::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::::fyyyyyA=nYnyyyYE:G:I:K:M:O:Q:S:UJ:n;v;;JBB:]:_:a:c:e :g:i:wAwAo:q:s:u:w:y:;[:F>N>V>^>f>n>v>>?F?N?V?^?f?n?v?nYvY:::::::::::: :::::::::::::::::::::::::::::::::::::::::_lqvGcMJ:::::::JE f:yyyxIN::<::::::JDJ:j:VBYmp>HYLkNG>::::::::N:<::::::C:wdC;j``pkDqqHqqTPtZF::n_mBwmcf]]:>::::yayA:<::j]NHEmlVG>:::::::::^fpC:>=N>I:;KxyyI>>yI:TmJk:^>\\:[R<Z:::yW:<^:vxYIff C<>Z:^v;Jr@Z:^:f?>=>::[V:vYDt;Dj:@m[vGe=MmlNH;KajuB:cldNDYt@QeW^ZoGlmwyvioYAwYAop@obougWyLYsGohNIrsjFnqw_yXs;Qd]GhsYsPFeb?kj@m\\Gk\\HtQqobvu Yquxgv@lhp_EojGQmExj]@kLiuNyySa\\A`d dnrA`g_eQHywyyBwdT?ixpe`WgmGv 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:::::::J:Zy=ZIO\\PY\\[a[Gwyyuy::::::Zj>`jF?j@vx^uyxI::::::Zx[cb:L;p:LKDN:bYry:::::::::>:ry:^_LFv@wyyuy :::::::Jy:DJD^\\NZ:ry:^xqIdymy;:::::::tA[SDJ;:MkNb Yry:::::::::J:^Z:>Z:v^;vyyY:;B:;:::Ja@Na`^:>Z::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::4:" }{TEXT -1 15 " searching \+ for " }{TEXT 259 16 "\"Heun equations\"" }{TEXT -1 99 " returns 3,650 \+ links, from which, more or less, 1 in 3 is about the topic and around \+ 1/2 repeated.\n" }}{PARA 256 "" 0 "" {TEXT -1 74 "____________________ ______________________________________________________" }}{PARA 15 "" 0 "" {OLE 1 28682 1 "[xm]Br=WfoRrB:::wk;nyyI;G:;:j::>:B>N:F:nyyyyy]::y yyyyy::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::fyyyyyq@nYnyyyYE:G:I:K:M:O:Q:S:UJ:n;v;;JBB:]:_:a:c:e:g:i:k:m:o:q :s:u:w:y:;[:F>N>V>^>f>n>v>>?F?N?V?^?f?n?v?>`:B:];_;a;c;e;wAwAyA::::::: :::::::::::::::::::::::::::::::::::::::::_lqvGcMJ:::::::JE f:yyyxIN::;`:Z@[::JZLWW`JY@V\\Aj;J:@:<:=j[vGUMrvC?MoJ::::::::JCN 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a[d=LBs\\V;::xAsgKqtyvYJoCVZV;:\\C:yay=::::::::JoCVZV;JrBeFRjy;::ReZV; ::jMckvYxI:a>^E:::lDS_ymy;:::::::::g`VKh::k>NWxIyA^JJh::S_;=xI:::::_WN cURSUvs\\V;:::::tB;idawyvY::::::::::TGBQ:Z>pBS_d=LBs\\V;:Jm:Yaymy;OajK hBQ:ZRHlD=Pnjy;::ReZV;::h@Sw@VdqbHUrBV::Z PikT\\y=:::::::::jtctNW:jq;yA:nkuBVsDm[=:nTjy;::::::g`WPDoUNT@s\\V;:h@ MVJTVufx]uyvY:::::::::::TGs\\V;:LBg`o=tBBV:Z>pbThvr@RvY:JosY>HVJP;RjnTxIyAZ;=hwGYD:TG=`ymqCeVUxIya Z=dIS_pV@VdnP@uNFRVxI;a;MUSfZJ=dHS_ymyCrFPlDS_@=xIBn^OhFryZ;M esyvY::jtSfMcJs_p@aHWuUxnymyCnZf<:<:::::::::::::::::::vYxI:;Z::::::::::3:" }{TEXT -1 15 " searching for " }{TEXT 267 6 "\"Heun\"" }{TEXT -1 63 " in the tit le: 90 publications, mostly during the last years.\n\n" }{TEXT 270 28 "Some relate to new solutions" }{TEXT -1 31 "\n* Belmehdi, S.; Chehab, J.-P. " }{TEXT 260 73 "\"Integral representation of the solutions to \+ Heun's biconfluent equation\"" }{TEXT -1 68 ". Appl. Anal. 4 (2004).\n \n* Ishkhanyan, Artur; Suominen, Kalle-Antti " }{TEXT 269 42 "\"New so lutions of Heun's general equation\"" }{TEXT -1 28 ". J. Phys. A 5, 36 (2003).\n\n" }{TEXT 271 52 "Some link Heun equations to other relevan t problems\n" }{TEXT -1 63 "* Dorey, Patrick; Suzuki, Junji; Tateo, Ro berto Finite lattice " }{TEXT 261 44 "\"Bethe ansatz systems and the H eun equation\"" }{TEXT -1 48 ". J. Phys. A 6, 37 (2004).\n\n* Takemura , Kouichi " }{TEXT 262 60 "\"The Heun equation and the Calogero-Moser- Sutherland system\"" }{TEXT -1 40 ". J. Differential Equations 15 (200 4).\n\n" }{TEXT 272 46 "Some are related the \"special function\" aspe ct" }{TEXT -1 16 "\n* Ronveaux, A. " }{TEXT 263 51 "\"Factorization of the Heun's differential operator\"" }{TEXT -1 150 ". Advanced special functions and related topics in differential equations (Melfi, 2001). Appl. Math. Comput. 141, 1 (2003). \n\n* Smirnov, Alexander O. " } {TEXT 264 39 "\"Elliptic solitons and Heun's equation\"" }{TEXT -1 89 ". The Kowalevski property (Leeds, 2000), 287--305, CRM Proc. Lecture \+ Notes, 32, (2002).\n\n" }{TEXT 273 85 "Some are related to group theor y and an important connection with Painleve equations\n" }{TEXT -1 18 "* Kazakov, A. Ya. " }{TEXT 265 43 "\"Symmetries of the confluent Heun equation\"" }{TEXT -1 122 ", Mat. Vopr. Teor. Rasprostr. Voln. 30, 55 --71, 311; translation in J. Math. Sci. (N. Y.) 117 (2003).\n\n* S. Yu . Slavyanov " }{TEXT 266 70 "\"Isomonodromic deformations of Heun equa tions, and Painlev\351 equations\"" }{TEXT -1 41 " Theoret. and Math. \+ Phys. 1, 125 (2000).\n" }}{PARA 256 "" 0 "" {TEXT -1 74 "_____________ _____________________________________________________________" }} {PARA 15 "" 0 "" {OLE 1 26633 1 "[xm]Br=WfoRrB:::wk;nyyI;G:;:j::>:B>N: F:nyyyyy]::yyyyyy::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::::fyyyyyI=nYnyyyYE:G:I:K:M:O:Q:S:UJ:n;v;;JBB:]:_:a:c:e :g:i:k:wAwAq:s:u:w:y:;[:F>N>V>^>f>n>v>>?F?N?V?^?f?n?v?>`:B:];_;wAyA::: :::::::::::::::::::::::::::::::::::::::::::::::JcvGYMt>^:fBWMtNHm=;::: ::::nbs^lKnupvsKflxBjKGJMStB::>j[iaCna;ar;V:>Z;Z:j:VBYmp>H YLkNG>::::::::N:VBYLa^DELajcJ::::::::JCj:j;jysy; Z::::::J<>:WYDB:VB:E`:B:::::::JFNZ;f:vYxI>:<::::::?:c:;:=J:nYnY :yA::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::=Jyyy;d:yayAbs^L::A;UTTAeVYuVYeScEB[DTYEWYEBEDSMEWYeV;s FWCF;CNMuRcUW_USt:VdJ?gik\\LpnJ:l::JQH]nIalHH>::::yayA:<:::::::::[CBsd R[CB=Z`VWh[o`j^ejGfagf\\Ofj>^VOg]_gcWhn_h[Ofu_ed^fpC <:C:=Z:vYxY;J:J[aJ:Nx@Zj>:;:;]:>F>;B:@K;LQJ`NZF:_B: ;jDj>B:;::::FffCh;B:=Z:V=^;>::[V:B:Dj:Dj:DkcFi:fG?mq^G>:::::::: :alv::;::::?B:gMd>:;J\\uZ:>FwtQs?Flv?fEA=kWx_jbPjBmQvXJDfg?Flv?kEA=Ku lkJPlWD>i;^<]:[TlkB:;EVYNRmSpn;l;Glv?esQ<;J:>s?Jl`YCJ:>S>\\Va<[cx_jbJfYJD:SCqcJFlWD:;JVJ b`F;J:^L^>=j<>Z:^yrQ=>h?FlvwYxypPjBOfSFlvWlSFlv?j:ylv?hSFlv ?K>A^_V]Jiyy_ssQs:I:IJ: Z:>:cex_:>HvYxypjg:IEv;FYwHocxW>B=J:>ZJm:>:;:kS:>]VMb` F;J:^btM:>:[wy_jsas;mQrZ:>^ZJRumkMZ:ar;>r yWnEF`:aR[Cj;rx[CjCX:;J:^;ZJAZy;kHx_ZtQ ^<;c`;FYw@;J:<::J\\dW\\FZt_VNZ:>:;:::::?PsUjIxkIx;:::::::::::::::::::::::: ::::::::::::::::::::j:@j:@jIxkIx;::=:=rAyjI:::::::::::::R:\\ZEV[EV[J> \\EV;;BAB>[B>;B:=B<;JBAr:<[EV[:>[Yv]Yv=:::::::::::=j?<;wbAIbAw r;XJItKItk;tKI\\k;]RVZ=VZ=::>\\=V[E>[;v=x;::::::::::=B:\\KFXj=Xj=n ]AvZXn=w:j;bAr:AbAAr:Ar::ZJ><;B[ENZHvZAvZAbAIbAI:n]AJI t;V:w:Z=V:::[:;B<@J>\\j:jI:::::::B<]TvZTvZAvZA:wZAvZXn]AbAwbAj; t[Xn]=V:A:::JBB>QB>;rAyrAy:::::::Z:>]SN]TN]Tj=d;X:JIr;Ij=t;t[Rn]An=wr: V::::B>Ar:@jIx[Y:::::::B?mb=Qj?hJBtKItk;t;HZ= :::Z:v]YrA::::::ZB>[BF]DN]TN=ZAN=v:bAIj=n]AbA;BZJ>\\=n]XV:H::::r< =rAyjI::::::Z:V;Wb:v=::::::B:AJ;HKGdJGdZTN=j=XZTvZTr;XZX>]EFZYjIj:x[;>[YB>[bAwZ=r::::Q: yrA::::::B:kr\\Bv =x;:::::ZBFZBVZE^]SN[DN[TN;JGd;v:X:F]J>[YjI::hJIHJIbAt[=VZ=::>\\Jv]YrA ::::::=r`K?dZDN]D:dKG:XJGdK?[:n]XVZXn=HJIV:A::B>K:y ::::::F:[B:QJ>hkCdJ?dJ?b@Ob\\=vZXn]Xn]=JIV:A::ZX>[>V\\DV\\DN[TN[DZDN]TN=r;Or<=rAv=::>ZEF:kr:kZX:wZ= V::>\\BZY::::::=B:krV[An]X:wj ;V::B:rA:::::Z:>\\JB@QBObZRn]XV:t[=r:tKIV :A:\\R>[EVZBn[MV\\D:OZTN[DN[Tb@N]R>ZYv=y::::=B:t;wbAHJIHJ IHj;Hj;:Z:B:?R;Or>Or>dZD:JGdJ?`K;xkIxkI::::KB:[r;wbAt[ =n=wj;t;Ar:V;rA:::::R:;r;QBabH[DN[D:oJGdkFtJ>xkIv=::::Kr@kb Aw:VZXn=Aj;JB\\:x;:::::>:[j;F:v=::::@ZRn=tKIt KIHj;tk;VZ=ZEv]YrA:::::J>\\KFOJ?dZDZMF]Ev]YrA::::Z;j=tKI bAr:wZ=V:tjajCdjCHkCb\\BZY:::::j:hjFhJ>d;E:r>:V[YjI:::::Z;V=w:wj;t[XJIHj;<:x;:::::\\KBLkFPjCH[M:ObZYv=y::::::\\J:\\k=ZXZ=j;t[JF:v=:::::]B n[?fZMVx;y::::::r:v=:::::\\jGdK>Lj=\\J;Pj< H[?V\\MV\\M:`K>xkIv=:::::J>Ej< jCV<`K;V[Bv=x;::::>ZE>ZYjI\\JFXJIXJIt[X:AJF<[YjI:::::^ZR>]BNZR>ZSfZ?V< PZMr>R@;:x;::::J>hJ;[YjI:::::^\\RN]:NZTR@EjJAL jI@j:xk:jI::::J>>Z:>ZRv:v:wJIV:wJI\\k?xkIv=:::::=r@orER;Ej\\:ZY:::::FZ]TvZAr;wj=t[XJIJFZY:::::raR;ajCjFtjIxkI\\J;hjIx[Y::::FZ:>ZE>]Tv:XZXZAr;tKIt[UFZYjI:::::hKF@jF@kEb:=B]TvZTv:X:bAv:wJI\\K:x;y::: ::BPj[:>ZBFZ:F]Dr;X:JIr;bAtk?\\j IrA::::ZB>]DV;OrmrAyrAy::@JD@j:xkIv=Kr`KFZ<^]YFZYrar>mB?y rAyBN\\KF<]jP:ER;PJ; n[?NZHrAyB?V\\Bv=x;::::>[HV\\<^\\ MNZ>F\\KF\\K:P:]j= =:y:::::=b=mR@cr>qB;_R>FfZ?:H;ER;V\\MjFLkIrA>[<>]SV\\MV\\MN[D b[YjI::::ZPjCPj? @jIrAj:_JC@KCF<]:EZKR;R; r>aR;Eb=Qj:x;y::\\Bv=x;::::\\jFPj?lk<ib >]ZKf:@kBPZ?f:fZ?:?R:yrA@jI^=cB:KR:KB<=B<;b=Ob]R>@;PZKF=:y::::Z;NZ?n[;n[H^]QN\\KN<_R>DkB@;@kBj>[Bn [?ZMr>ajH[DN[Db@Obi:]ZLv\\KF\\KFdJGdZ TN;OJGdk=N=XJG>]>v[Gf[Of\\Pn\\@nZFv]Yv]N^\\V^]>F[>v[If[GR=Ur=YB;?R;EB; ;R;QB;uR>_b>XKCJCF\\LF<]R>f:@[?R;VZSfZ>>[KNZVN\\LNV\\?R;aR@qb:;B?KR:=R:=ZB^\\:^Z]jBR>@kFZYFZ;>[:N:ObOZTN;obx:PkDxjA`J<@kB@k:xjFLjEpKCDKCD;ZKF\\KNZEn[SfZ?JHxkIx K>\\j?\\E^]If[Of\\Pn:ljIFZN^\\N J_b>_:r?DkBDKC@[H^ZNv]YFZ;B]TbA[B>[j?hj?L[:>Z:>[BFZ;^]Cf\\G:G:yrA::Z: F\\?N:Eb:Mb>Gb>gb>_b>_:R>XK:@jI\\j?DJF`kFdjF`KFhk?hJ>x;yR:[R@ob@qr<;R: =rAyjIZ;FZ:B@wr;HJB?B ?Er=sB=Sb;_b>Gb>_b>_jBF\\LNZ;v=?R@ar>mR@kR@kR@ob>[;FZ;v=yJ>hJBHj;HJFHJB><[B>QJ:\\JH`j@PkDTK=^[Fj:L[VJmB?ib=sb?_b>Gb;_b>_b>jELkI\\J;PjFDJ:\\J>j:\\J>V;Ir>Ob<@ZB> ZEB>kB@HJFHJF\\J>xkI^]CR?gb?n:yrA^\\V^=v[Gb?eb?e R?YZ>R>]B?CR;cR=_B=Gb>Gb>f]Lv\\>FZYR@hKDxkIxkI@j:@j:v=xK>hjGh[YFZ:B@ob @XjF\\kFXJF\\kF\\k=X:rZ;v]VV[[_B;=R>?BAu RAub>Sb?MBAyrA;r?ER@QrA=jFHkQr\\ZE>]R>]Xv:tKIHj;Hj;><=:y::=JDl[VF;YR=nxj>`JHL kI@J:tj<`K;@jI^Z>^:MRF[CB; CRmb=CBACJHLZ>^=;r=mR@]b=CB;MR<`ZIv[Cv[Cv[I j>v[IF[CRV[E^ZE^ZE>Z>^ZCR=R?gb?Gb;Sb;SB=yZ;B?CJ<`jApZPnZPnLJF[>F[CF[CB;LJ<`j>Lj>xj>jAj>xj@xj @xjApjAv[IZCRR=eb?n\\OjAxj>pJCDK< LKCD[Lf]?v\\<^];F:?R>]R;mb=CB;Cj>`j>xj>v;xZCF[CF[Iv[If;UR=YR=Yr=Yr=UR? 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" }{TEXT -1 12 "That means: " }{TEXT 317 217 "all but for of them can be solved at the cost of a single fac torization of a fourth degree polynomial. Also essentially all the app lications behind Kamke's linear examples are in fact formulated using \+ Heun equations. 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Sleeman and V. Kuznetsov." }} {PARA 256 "" 0 "" {TEXT -1 74 "_______________________________________ ___________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 14 "Heun functions" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "Five Heun functions, " }{XPPEDIT 18 0 "HeunG, HeunC, HeunB, HeunD" "6&%&HeunGG% &HeunCG%&HeunBG%&HeunDG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "HeunT;" " 6#%&HeunTG" }{TEXT -1 0 "" }{TEXT -1 67 " are defined as the solutions to each of these five Heun equations " }{XPPEDIT 18 0 "GHE, CHE, BHE, DHE, THE" "6'%$GHEG%$CHEG%$BHEG%$DHEG%$THEG" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 15 "" 0 "" {TEXT -1 139 "Definin g a function from the differential equation it satisfies is a process, which, computationally speaking, involves mainly three steps " }} {PARA 0 "" 0 "" {TEXT -1 89 "\n 1. Define the function's different iation rule to satisfy the differential equation\n" }}{PARA 0 "" 0 "" {TEXT -1 149 " 2. Define the function itself in terms of known obj ects, e.g. a power series solution of the ODE; set a convention for th e initial conditions. \n" }}{PARA 0 "" 0 "" {TEXT -1 117 " 3. Defi ne the numerical evaluation rules (requires defining the so-called \"a nalytic extensions\" of the function)" }}{PARA 256 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 74 "_______________________ ___________________________________________________" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 277 8 "Example:" }{TEXT -1 24 " The Heun Triconfluent (" }{XPPEDIT 18 0 "HeunT" "6#%&HeunTG" }{TEXT -1 10 ") function" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 " " 0 "" {TEXT -1 33 "1. Differentiation rule for HeunT" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "OFF; THE; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(-%%diffG6$-%\"yG6#%\"xG-%\"$G6$ F+\"\"#\"\"\"*&,&%&gammaGF0*&\"\"$F0)F+F/F0F0F0-F&6$F(F+F0!\"\"*&,&%&a lphaGF0*&,&%%betaGF0F5F9F0F+F0F0F0F(F0F0\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 83 "Let's call our new function \"HT\", and state that HT \+ is a solution to this equation " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "y(x) = HT(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"x G-%#HTGF&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "Substitute into the \+ equation" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "eval(THE, %);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(-%%diffG6$-%#HTG6#%\"xG-%\"$G6$F+ \"\"#\"\"\"*&,&%&gammaGF0*&\"\"$F0)F+F/F0F0F0-F&6$F(F+F0!\"\"*&,&%&alp haGF0*&,&%%betaGF0F5F9F0F+F0F0F0F(F0F0\"\"!" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 94 "Introduce now the function \"derivative of HT = HTPrime \", so our first differentiation rule is:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 36 "eq(1) := diff(HT(x),x) = HTPrime(x);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>-%#eqG6#\"\"\"/-%%diffG6$-%#HTG6#%\"xGF/-%(HTPr imeGF." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "eval(%%,%); " }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/,(-%%diffG6$-%(HTPrimeG6#%\"xGF+\"\" \"*&,&%&gammaGF,*&\"\"$F,)F+\"\"#F,F,F,F(F,!\"\"*&,&%&alphaGF,*&,&%%be taGF,F1F4F,F+F,F,F,-%#HTGF*F,F,\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 65 "So the differentiation rule for HTPrime, completing this step i s:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "eq(2) := isolate(%, d iff(HTPrime(x),x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%#eqG6#\"\"#/ -%%diffG6$-%(HTPrimeG6#%\"xGF/,&*&,&%&gammaG\"\"\"*&\"\"$F4)F/F'F4F4F4 F,F4F4*&,&%&alphaGF4*&,&%%betaGF4F6!\"\"F4F/F4F4F4-%#HTGF.F4F>" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "Our function depends also on " } {XPPEDIT 18 0 "alpha,beta,gamma" "6%%&alphaG%%betaG%&gammaG" }{TEXT -1 19 ", so it is called " }{XPPEDIT 18 0 "HeunT(alpha,beta,gamma,x) " "6#-%&HeunTG6&%&alphaG%%betaG%&gammaG%\"xG" }{TEXT -1 12 ", so we ha ve" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "sol_THE := y(x) = Heu nT(alpha,beta,ga,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(sol_THEG/-% \"yG6#%\"xG-%&HeunTG6&%&alphaG%%betaG%&gammaGF)" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 10 "diff(%,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /-%%diffG6$-%\"yG6#%\"xGF*-%+HeunTPrimeG6&%&alphaG%%betaG%&gammaGF*" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "diff(%,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F*\"\"#,&*&,&%&gam maG\"\"\"*&\"\"$F3)F*F.F3F3F3-%+HeunTPrimeG6&%&alphaG%%betaGF2F*F3F3*& ,&F:F3*&,&F;F3F5!\"\"F3F*F3F3F3-%&HeunTGF9F3F@" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 126 "After defining these differentiation rules using th e differential equation, the function automatically satisfies this equ ation" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "ON; THE; sol_THE; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(%$y''G\"\"\"*&,&%&gammaGF&*&\" \"$F&)%\"xG\"\"#F&F&F&%#y'GF&!\"\"*&,&%&alphaGF&*&,&%%betaGF&F+F0F&F-F &F&F&%\"yGF&F&\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"yG-%&HeunTG 6&%&alphaG%%betaG%&gammaG%\"xG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "odetest( %, %% );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&-%+He unTPrimeG6&%&alphaG%%betaG%&gammaG%\"xG\"\"\"F*F,F,*&F%F,%&gammaGF,!\" \"" }}}{PARA 256 "" 0 "" {TEXT -1 74 "________________________________ __________________________________________" }{TEXT 278 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 279 78 "2. Define HeunT as the power series solution of the equation a round the origin" }{TEXT -1 11 " such that " }{XPPEDIT 286 0 "y(0)=1, \+ Eval(diff(y(x),x),x=0) = 0" "6$/-%\"yG6#\"\"!\"\"\"/-%%EvalG6$-%%diffG 6$-F%6#%\"xGF2/F2F'F'" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 283 11 " The basics:" }}{PARA 15 "" 0 "" {TEXT -1 42 "Power series solutions ar e of the form " }{XPPEDIT 18 0 "y = (x-x[0])^rho*Sum(c[n]*(x-x[0])^ n,n = 0 .. infinity);" "6#/%\"yG*&),&%\"xG\"\"\"&F(6#\"\"!!\"\"%$rhoGF )-%$SumG6$*&&%\"cG6#%\"nGF)),&F(F)&F(6#F,F-F6F)/F6;F,%)infinityGF)" } {TEXT -1 58 " . Since we are defining the function around the origin, " }{XPPEDIT 18 0 "x[0] = 0;" "6#/&%\"xG6#\"\"!F'" }{TEXT -1 1 "\n" }} {PARA 15 "" 0 "" {TEXT -1 42 "Power series expansions converge wheneve r " }{XPPEDIT 18 0 "x[0]" "6#&%\"xG6#\"\"!" }{TEXT -1 88 " is an ordin ary point (Taylor series), or a regular singular point (Frobenius seri es).\n\n" }}{PARA 15 "" 0 "" {TEXT -1 5 "When " }{XPPEDIT 18 0 "x[0]; " "6#&%\"xG6#\"\"!" }{TEXT -1 30 " is a regular singular point, " } {XPPEDIT 18 0 "rho;" "6#%$rhoG" }{TEXT -1 71 " is any of the two solut ions of the so-called \"indicial equation\" (see " }{HYPERLNK 17 "DEto ols[indicialeq]" 2 "DEtools[indicialeq]" "" }{TEXT -1 2 ")\n" }}{PARA 15 "" 0 "" {TEXT -1 58 "In practice, inside Maple, we directly \"see\" the value of " }{XPPEDIT 18 0 "rho;" "6#%$rhoG" }{TEXT -1 7 " using \+ " }{HYPERLNK 17 "DEtools[formal_sol]" 2 "DEtools[formal_sol]" "" }} {PARA 259 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 74 "_____ _____________________________________________________________________ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 284 26 "The \+ power series solution:" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 46 "In the THE case, the origin is a regu lar point" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "THE;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(%$y''G\"\"\"*&,&%&gammaGF&*&\"\"$F&)%\"xG \"\"#F&F&F&%#y'GF&!\"\"*&,&%&alphaGF&*&,&%%betaGF&F+F0F&F-F&F&F&%\"yGF &F&\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "singularities(% );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/%(regularG<\"/%*irregularG<#%)i nfinityG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "Hence a standard " } {TEXT 280 22 "Taylor series solution" }{TEXT -1 39 " of the equation i s sufficient and the " }{TEXT 281 21 "radius of convergence" }{TEXT -1 4 " is " }{XPPEDIT 18 0 "infinity" "6#%)infinityG" }{TEXT -1 34 " ( the function is then said to be " }{TEXT 282 6 "entire" }{TEXT -1 95 " ). We can see the first terms of the two independent series solutions, as well as the value of " }{XPPEDIT 18 0 "rho" "6#%$rhoG" }{TEXT -1 15 " (for the THE, " }{XPPEDIT 18 0 "rho=0" "6#/%$rhoG\"\"!" }{TEXT -1 6 "), via" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "DEtools[for mal_sol](THE, y(x), x=0, order=3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# 7$+)%\"xG\"\"\"F&,$*&\"\"#!\"\"%&gammaGF&F&F)-%\"OG6#F&\"\"$++F%F*\"\" !,$*&F)F*F+F&F*F&,&*&F)F*%&alphaGF&F&*&\"\"%F*F+F)F*F)F,F/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 73 "To numerically evaluate the function, how ever, we cannot rely on calling " }{HYPERLNK 17 "formal_sol" 2 "DEtool s[formal_sol]" "" }{TEXT -1 67 " at every evaluation point: we need a \+ formula for the coefficients " }{XPPEDIT 18 0 "c[n];" "6#&%\"cG6#%\"nG " }{TEXT -1 42 " entering this series expansion solution " }{XPPEDIT 18 0 "y(x) = Sum(c[n]*x^n,n = 0 .. infinity);" "6#/-%\"yG6#%\"xG-%$Sum G6$*&&%\"cG6#%\"nG\"\"\")F'F/F0/F/;\"\"!%)infinityG" }{TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 56 "Start int roducing the series expansion into the equation" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 45 "sol := y(x) = Sum(c[nu]*x^nu,nu=0..infinity); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$solG/%\"yG-%$SumG6$*&&%\"cG6#%# nuG\"\"\")%\"xGF.F//F.;\"\"!%)infinityG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "eval(THE,sol);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,( -%$SumG6$,&**&%\"cG6#%#nuG\"\"\")%\"xGF-F.F-\"\"#F0!\"#F.**F*F.F/F.F-F .F0F2!\"\"/F-;\"\"!%)infinityGF.*&,&%&gammaGF.*&\"\"$F.)F0F1F.F.F.-F&6 $**F*F.F/F.F-F.F0F4F5F.F4*&,&%&alphaGF.*&,&%%betaGF.F=F4F.F0F.F.F.-F&6 $*&F*F.F/F.F5F.F.F7" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 167 "Terms of \+ each sum will combine with terms of the other sums, and the summation \+ indices are redefined in order to become a single sum of the form sol \+ above, arriving at" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 137 "e7 : = Sum(x^(nu+1)*((beta-3*nu-3)*c[nu]+c[nu+1]*alpha+(-c[nu+2]*ga+c[nu+3] *(nu+3))*(nu+2)),nu = 0 .. infinity)+2*c[2]-c[1]*ga+c[0]*alpha;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#e7G,*-%$SumG6$*&)%\"xG,&\"\"\"F-%#n uGF-F-,(*&,(%%betaGF-*&\"\"$F-F.F-!\"\"F4F5F-&%\"cG6#F.F-F-*&&F76#F,F- %&alphaGF-F-*&,&*&&F76#,&F.F-\"\"#F-F-%&gammaGF-F5*&&F76#,&F.F-F4F-F-F HF-F-F-FBF-F-F-/F.;\"\"!%)infinityGF-*&FCF-&F76#FCF-F-*&&F76#F-F-FDF-F 5*&&F76#FKF-F " 0 "" {MPLTEXT 1 0 15 "eval(e7,S um=0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&\"\"#\"\"\"&%\"cG6#F%F&F &*&&F(6#F&F&%&gammaGF&!\"\"*&&F(6#\"\"!F&%&alphaGF&F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 7 "we get " }{XPPEDIT 18 0 "c[2]" "6#&%\"cG6# \"\"#" }{TEXT -1 25 " in terms of (arbitrary) " }{XPPEDIT 18 0 "c[0]" "6#&%\"cG6#\"\"!" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "c[1]" "6#&%\"cG6 #\"\"\"" }{TEXT -1 2 ", " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "isolate(%, c[2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"#,& *&#\"\"\"F'F+*&&F%6#F+F+%&gammaGF+F+F+*&#F+F'F+*&&F%6#\"\"!F+%&alphaGF +F+!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "and with these three \+ " }{XPPEDIT 18 0 "c[n]" "6#&%\"cG6#%\"nG" }{TEXT -1 109 " at hands we \+ compute the other coefficients in the series imposing that the whole e xpression is zero, that is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "e8 := e7-%%=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#e8G/-%$SumG6$ *&)%\"xG,&\"\"\"F-%#nuGF-F-,(*&,(%%betaGF-*&\"\"$F-F.F-!\"\"F4F5F-&%\" cG6#F.F-F-*&&F76#F,F-%&alphaGF-F-*&,&*&&F76#,&F.F-\"\"#F-F-%&gammaGF-F 5*&&F76#,&F.F-F4F-F-FHF-F-F-FBF-F-F-/F.;\"\"!%)infinityGFK" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "remove(has, op(1,lhs(e8)), x )= 0; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*&,(%%betaG\"\"\"*&\"\"$F(%#nuG F(!\"\"F*F,F(&%\"cG6#F+F(F(*&&F.6#,&F(F(F+F(F(%&alphaGF(F(*&,&*&&F.6#, &F+F(\"\"#F(F(%&gammaGF(F,*&&F.6#,&F+F(F*F(F(F@F(F(F(F:F(F(\"\"!" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "isolate(%,c[nu+3]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#,&%#nuG\"\"\"\"\"$F)*&,&*&,&* &,(%%betaGF)*&F*F)F(F)!\"\"F*F3F)&F%6#F(F)F3*&&F%6#,&F)F)F(F)F)%&alpha GF)F3F),&F(F)\"\"#F)F3F)*&&F%6#F;F)%&gammaGF)F)F)F'F3" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 285 7 "Example" }{TEXT -1 23 ": compute the valu e of " }{XPPEDIT 18 0 "c[3]" "6#&%\"cG6#\"\"$" }{TEXT -1 13 " in terms of " }{XPPEDIT 18 0 "c[2],c[1],c[0];" "6%&%\"cG6#\"\"#&F$6#\"\"\"&F$6 #\"\"!" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "ev al(%,nu=0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"$,(*&#\"\" \"\"\"'F+*&,&%%betaGF+F'!\"\"F+&F%6#\"\"!F+F+F0*&#F+F,F+*&&F%6#F+F+%&a lphaGF+F+F0*&#F+F'F+*&&F%6#\"\"#F+%&gammaGF+F+F+" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 105 "This recurrence formula for the coefficients can \+ be obtained directly (when the expansion is of the form " }{XPPEDIT 18 0 "y(x) = Sum(c[n]*x^n,n = 0 .. infinity)" "6#/-%\"yG6#%\"xG-%$SumG 6$*&&%\"cG6#%\"nG\"\"\")F'F/F0/F/;\"\"!%)infinityG" }{TEXT -1 12 ") us ing the " }{HYPERLNK 17 "gfun[diffeqtorec]" 2 "gfun[diffeqtorec]" "" } {TEXT -1 8 " command" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 12 "We see th at " }{XPPEDIT 18 0 "c[0]" "6#&%\"cG6#\"\"!" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "c[1]" "6#&%\"cG6#\"\"\"" }{TEXT -1 161 " remained arbit rary. This is the typical case when we expand around an ordinary (not \+ singular) point. We choose the values of these first two coefficients \+ to be " }{XPPEDIT 18 0 "c[0] = 1,c[1] = 0;" "6$/&%\"cG6#\"\"!\"\"\"/&F %6#F(F'" }{TEXT -1 30 ", so that HeunT is of the form" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "HeunT(alpha,beta,ga,x) = 1 + c[2]*x ^2 + `...`*O(x^3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&HeunTG6&%&al phaG%%betaG%&gammaG%\"xG,(\"\"\"F,*&&%\"cG6#\"\"#F,)F*F1F,F,*&%$...GF, -%\"OG6#*$)F*\"\"$F,F,F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "In th is way we have defined " }{XPPEDIT 18 0 "HeunT(alpha,beta,gamma,x)" "6 #-%&HeunTG6&%&alphaG%%betaG%&gammaG%\"xG" }{TEXT -1 51 " as the soluti on to the Triconfluent Heun equation " }{XPPEDIT 18 0 "`y''`-(gamma+3* x^2)*`y'`+(alpha+(beta-3)*x)*y = 0" "6#/,(%$y''G\"\"\"*&,&%&gammaGF&*& \"\"$F&*$%\"xG\"\"#F&F&F&%#y'GF&!\"\"*&,&%&alphaGF&*&,&%%betaGF&F+F0F& F-F&F&F&%\"yGF&F&\"\"!" }{TEXT -1 36 " satisfying the initial conditio ns " }{XPPEDIT 18 0 "y(0) = 1, Eval(diff(y(x),x),x = 0) = 0" "6$/-%\" yG6#\"\"!\"\"\"/-%%EvalG6$-%%diffG6$-F%6#%\"xGF2/F2F'F'" }{TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {SECT 1 {PARA 5 "" 0 "" {TEXT -1 32 "3. Numerical evaluation of HeunT " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 73 "Since the function is entire (the radius of convergence of the series is " }{XPPEDIT 18 0 "infinity" "6#%)infinityG" }{TEXT -1 157 "), we can use this series expansion as a way to evaluate the function numerically i n the whole complex plane. For that, first implement the series expans ion:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "HeunT(alpha,beta,ga, x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&HeunTG6&%&alphaG%%betaG%&gam maG%\"xG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "series(%, x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+/%\"xG\"\"\"\"\"!,$*&\"\"#!\"\"%&al phaGF%F*F),(*&\"\"'F*%%betaGF%F*#F%F)F%*(F.F*%&gammaGF%F+F%F*\"\"$,**& #F%\"#CF%*$)F+F)F%F%F%*&#F%F7F%*&F2F%F/F%F%F**&#F%\"\")F%F2F%F%*&#F%F7 F%*&)F2F)F%F+F%F%F*\"\"%,.*&#F%\"#IF%*&F+F%F/F%F%F%*&#F%FDF%F+F%F**&#F %\"#gF%*&F2F%F9F%F%F%*&#F%\"$?\"F%*&FCF%F/F%F%F**&#F%\"#SF%*$FCF%F%F%* &#F%FRF%*&)F2F3F%F+F%F%F*\"\"&-%\"OG6#F%F." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 121 "A numerical example and \+ the first images of the real and imaginary parts of HeunT, as a functi on of the complex argument " }{TEXT 287 1 "z" }{TEXT -1 22 ", in the u nit circle " }{XPPEDIT 18 0 "abs(z) = 1" "6#/-%$absG6#%\"zG\"\"\"" } {TEXT -1 10 ". Consider" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 " HT := HeunT(4/5,-2/3,1/7,z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#HTG -%&HeunTG6&#\"\"%\"\"&#!\"#\"\"$#\"\"\"\"\"(%\"zG" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 101 "First assuming z \+ is real: HeunT will be real (expected, from the power series expansion definition..)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "PlotFunct ion( HT ) assuming real;" }}{PARA 13 "" 1 "" {GLPLOT2D 921 921 921 {PLOTDATA 2 "6@-%'CURVESG6'7$7$$\"\"!F)\"#57$F(!#57$7$$!+++++5!\")F,7$ F*F,7$7$$\"+++++IF1F*7$F5F/7$7$$\"+++++?F1F(7$\"#SF(-%'LEGENDG6#QB__ne ver_display_this_legend_entry6\"-%%TEXTG6&F2Q\"xFB%+ALIGNRIGHTG%+ALIGN ABOVEG-FD6&F-FD6%7$FTFT%$-1.G%+ALIGNBELOWG-FD6%7$$!++++]_FZFT%$-.5GFdq-FD6%7$ $!#DFRFT%#0.GFdq-FD6%7$$\"*+++v%F1FT%#.5GFdq-FD6%7$$\"*+++v*F1FT%$1.0G Fdq-FD6%7$Fjo$!+g+\\x(*FZ%#.4G%*ALIGNLEFTG-FD6%7$Fjo$!+ACxNaFZ%#.6GFcs -FD6%7$Fjo$!+%yaS4\"FZ%#.8GFcs-FD6%7$Fjo$\"*&GmZKF1F\\sFcs-FD6%7$Fjo$ \"*\\!Q*e(F1%$1.2GFcs-F$6-7$7$$\"#?F)Feo7$F_uFjo7$7$$\"+++++DF1Feo7$Fd uFjo7$7$$\"#IF)Feo7$FiuFjo7$7$$\"+++++NF1Feo7$F^vFjo7$7$$\"+++++SF1Feo 7$FcvFjo7$7$$\"++++DIF1FO7$$\"++++vHF1FO7$7$FhvFX7$F[wFX7$7$FhvF(7$F[w F(7$7$FhvF[o7$F[wF[o7$7$FhvF`o7$F[wF`oF>-FD6%7$$\"%v>FRFjoFcqFdq-FD6%7 $$\"++++vCF1FjoFjqFdq-FD6%7$$\"%vHFRFjoF`rFdq-FD6%7$$\"++++vMF1FjoFfrF dq-FD6%7$$\"++++vRF1FjoF\\sFdq-FD6%7$F[wFTFcqFcs-FD6%7$F[wFhqFjqFcs-FD 6%7$F[wF^rF`rFcs-FD6%7$F[wFdrFfrFcs-FD6%7$F[wFjrF\\sFcs-F$6$7S7$F/$!+, +++5F17$$!++-$F17$$!+(y$pZi FZ$!*+2\\=#F17$$!+$yaE\"eFZ$!*&[1r8F17$$!+\">s%HaFZ$!)[Q%)pF17$$!+]$*4 )*\\FZ$\"'3\\MF17$$!+]_&\\c%FZ$\")b\\XkF17$$!+]1aZTFZ$\"*z]))>\"F17$$! +/#)[oPFZ$\"*htnk\"F17$$!+$=exJ$FZ$\"*QA.6#F17$$!+L2$f$HFZ$\"*g**\\W#F 17$$!+PYx\"\\#FZ$\"*ti$pFF17$$!+L7i)4#FZ$\"*(Gy+IF17$$!+P'psm\"FZ$\"*o G#)>$F17$$!+74_c7FZ$\"*[#\\NLF17$$!+!3x%z#)F,$\"*374V$F17$$!++s$QM%F,$ \"*8H-[$F17$$!++5zr)*!#7$\"*Paw\\$F17$$\"++!o2J%F,$\"*=fD[$F17$$\"++%Q #\\\")F,$\"*X#)pW$F17$$\"+g\"*[H7FZ$\"*.7/R$F17$$\"++dxd;FZ$\"*X))yJ$F 17$$\"+I0xw?FZ$\"*%R#)QKF17$$\"+g&p@[#FZ$\"*P.+;$F17$$\"+!3'HKHFZ$\"*s >g2$F17$$\"+qZvOLFZ$\"*`b#4IF17$$\"+]2goPFZ$\"*?XI&HF17$$\"++u\"*fTFZ$ \"*T-3#HF17$$\"+])Hxe%FZ$\"**G)>\"HF17$$\"+I!o-*\\FZ$\"*/f\\$HF17$$\"+ 5k.6aFZ$\"*d1%)*HF17$$\"+?WTAeFZ$\"*zEq5$F17$$\"+g!*3`iFZ$\"*W$\\zKF17 $$\"+I*zym'FZ$\"*H4H^$F17$$\"+5N1#4(FZ$\"**4jKQF17$$\"+IYt7vFZ$\"*_adC %F17$$\"++xG**yFZ$\"*NYas%F17$$\"+S6KU$)FZ$\"*v#H;aF17$$\"+IbdQ()FZ$\" *Tqt='F17$$\"+g`1h\"*FZ$\"*g+J?(F17$$\"+S?Wl&*FZ$\"*z\"=.%)F17$F`oF`o- %'COLOURG6&%$RGBG$F*!\"\"F(F(-FD6$7$F)\"#8%#ReG-F$6$7SF97$$\"+3VfV?F1F (7$$\"+$[D:3#F1F(7$$\"+cI=C@F1F(7$$\"+\"RBr;#F1F(7$$\"+Q'f)4AF1F(7$$\" +5;[\\AF1F(7$$\"+my]!H#F1F(7$$\"+!GPHL#F1F(7$$\"+@1BvBF1F(7$$\"+AXt=CF 1F(7$$\"+\"y_qX#F1F(7$$\"+l+>+DF1F(7$$\"+vW]VDF1F(7$$\"+NfC&e#F1F(7$$ \"+!=^Ji#F1F(7$$\"+#=C#oEF1F(7$$\"+FpS1FF1F(7$$\"+OD#3v#F1F(7$$\"+xy8! z#F1F(7$$\"+OIFLGF1F(7$$\"+4zMuGF1F(7$$\"+H_? " 0 "" {MPLTEXT 1 0 38 "P lotFunction( HT ) assuming imaginary;" }}{PARA 13 "" 1 "" {GLPLOT2D 921 921 921 {PLOTDATA 2 "6G-%'CURVESG6$7eo7$$!+++++5!\")$\")]))eQ!\"(7 $$!+g%G?y*!\"*$\").(Rj&F-7$$!+s%HaF1$\")t(4P\"F-7$$!+]$*4)*\\F1$!(\\F^#F-7$$!+]_&\\c%F1$!)% *=.=F-7$$!+]1aZTF1$!)L)3?$F-7$$!+/#)[oPF1$!)^R')F1$\")-D?**F-7$$\"+IbdQ() F1$\")[J?)*F-7$$\"+X/#)\\*)F1$\")ncf%*F-7$$\"+g`1h\"*F1$\")0\\x))F-7$$ \"++PDj$*F1$\")\\F*3)F-7$$\"+S?Wl&*F1$\")54^qF-7$$\"+?5s#y*F1$\")7!*Gc F-7$$\"+++++5F*F+-%'COLOURG6&%$RGBG$\"#5!\"\"$\"\"!FgblFfbl-%%TEXTG6$7 $Fgbl\"#8%#ReG-F$6'7$7$FfblFdbl7$FfblFgu7$7$F(Fgu7$FdblFgu7$7$$\"+++++ IF*Fdbl7$FhclF(7$7$$\"+++++?F*Ffbl7$\"#SFfbl-%'LEGENDG6#QB__never_disp lay_this_legend_entry6\"-Fibl6&FeclQ\"yFedl%+ALIGNRIGHTG%+ALIGNABOVEG- Fibl6&F_dlFhdlFidlFjdl-F$627$7$$FguFgbl$!$v*!\"#7$Fael$!%D5Fdel7$7$$!+ ++++]F1Fbel7$FjelFfel7$7$FfblFbel7$FfblFfel7$7$$\"*++++&F*Fbel7$FbflFf el7$7$F]blFbel7$F]blFfel7$7$$\"#DFdel$!+Y/*o-)F17$$!)+++DF*F\\gl7$7$Fj 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ask for double number of points per axis" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "PlotFun ction( HT, 2 ); " }{TEXT -1 16 " (approx 5 sec.)" }}{PARA 13 "" 1 "" {GLPLOT3D 921 921 921 {PLOTDATA 3 "6K-%%MESHG6$X.6\"6\"F'[gl'!%\"!!$[f o\"5\"5\"$C01651355E73356DC023FFFFFFFD11F7C0181F4D2133449CC013B749B713 AC3BC0226938FD2353EBC002FF38D874DE6EC012C30B935DF75BC020D271FA8DC16BBF D9171642BDBCD1C012A8027F1F28B2C01E7755EFF05DD63FE750AF7873AFA4C012FF7E DD1A1F21C01B49C7EA9C51EB3FF650FDAF084E33C0139582F988A6D8C0181C39E555E8 4D3FFD01408675F5FDC0144F2226ECB728C014EEABE001DC5F4000B7620C54F816C015 1E14168DFDA8C011C11DDABB72C2400240D79CA97D1FC015FAAF041E49A8C00D271FAA EA124740035B488C9A2E5BC016E0F74EECFE75C006CC03A041FA6D400427138AEC7105 C017CF37DE1FD087C00070E795B527324004B218F343EE69C018C570B1D03183BFF42B 97161A1EAF4004FC58C5BC5ACFC019C556E2BEFB4CBFDDD57C0401E0DF4004F7F2A65E F017C01AD2E6119011C13FD503645064B8FD40048487E6B965D6C01BF5C802AB173B3F 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2 0 1 10 0 2 1 1 1 1 1.000000 90.000000 0.000000 1 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve \+ 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Curv e 12" "Curve 13" "Curve 14" "Curve 15" "Curve 16" "Curve 17" "Curve 18 " "Curve 19" "Curve 20" "Curve 21" "Curve 22" "Curve 23" "Curve 24" "C urve 25" "Curve 26" "Curve 27" "Curve 28" "Curve 29" "Curve 30" "Curve 31" "Curve 32" "Curve 33" "Curve 34" "Curve 35" "Curve 36" "Curve 37 " "Curve 38" "Curve 39" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 39 "Numerical evaluation of HeunC and HeunG" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "The numerical evaluation \+ of " }{XPPEDIT 18 0 "HeunT, HeunD" "6$%&HeunTG%&HeunDG" }{TEXT -1 5 " \+ and " }{XPPEDIT 18 0 "HeunB" "6#%&HeunBG" }{TEXT -1 29 " presents no d ifficulty. For " }{XPPEDIT 18 0 "HeunC" "6#%&HeunCG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "HeunG" "6#%&HeunGG" }{TEXT -1 73 ", the two more ge neral Heun functions, the situation is more problematic:" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 15 "" 0 "" {XPPEDIT 18 0 "HeunC" "6#%&Heu nCG" }{TEXT -1 25 " solves an equation with " }{TEXT 289 25 "two regul ar singularities" }{TEXT -1 4 " at " }{XPPEDIT 18 0 "\{0,1\}" "6#<$\" \"!\"\"\"" }{TEXT -1 5 " and " }{TEXT 288 13 "one irregular" }{TEXT -1 16 " singularity at " }{XPPEDIT 18 0 "infinity" "6#%)infinityG" } {TEXT -1 56 ". The series expansion around the origin converges till \+ " }{XPPEDIT 18 0 "abs(z) < 1" "6#2-%$absG6#%\"zG\"\"\"" }{TEXT -1 113 ". An analytic extension is necessary to compute the function outside \+ the unit circle and we cannot expand around " }{XPPEDIT 18 0 "infinity " "6#%)infinityG" }{TEXT -1 81 " because, being an irregular point, th e series does not automatically converges.\n" }}{PARA 15 "" 0 "" {XPPEDIT 18 0 "HeunG" "6#%&HeunGG" }{TEXT -1 55 " solves an equation w ith four regular singularities at " }{XPPEDIT 18 0 "\{0,1,a,infinity\} " "6#<&\"\"!\"\"\"%\"aG%)infinityG" }{TEXT -1 104 ". The series expans ion around the origin converges till \"the next singularity\", which c an be at 1 or at " }{TEXT 295 1 "a" }{TEXT -1 114 ". Here too an anal ytic extension is necessary to compute the function outside the circle centered at the origin.\n" }}{PARA 15 "" 0 "" {TEXT -1 772 "Analytic \+ extensions are constructed expanding the function around a different p oint (typically another singularity) and \"connecting this new series \" with the series expansion around the origin. This is called the \"t wo point connection problem\". That this connection is always possible follows from the fact that any solution of the equation (e.g. the Heu n function, defined around the origin) can always be expressed as a li near combination of any pair of independent solutions (the two series \+ solutions around any point which is not the origin). The problem is th at nobody knows how to compute exactly these constants connecting ser ies expansions around different singularities for the Heun equations ( people know how to do that in the case of hypergeometric equations)." }}{PARA 259 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 74 "___ ______________________________________________________________________ _" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 291 34 "The numerical evaluation of HeunG " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 29 "Following the steps used for " }{XPPEDIT 18 0 "HeunT" "6#%&HeunTG" }{TEXT -1 77 " we obtained the recurrence f or the series expansion around the origin to be" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 39 "HG := HeunG(a,q,alpha,beta,ga,delta,x);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#HGG-%&HeunGG6)%\"aG%\"qG%&alphaG%%b etaG%&gammaG%&deltaG%\"xG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "series(HG,x,3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#++%\"xG\"\"\"\" \"!*(%\"qGF%%&gammaG!\"\"%\"aGF*F%,$*,\"\"#F*,2F(F%*(F(F%%&deltaGF%F+F %F%*(F(F%F)F%F+F%F%*&F(F%F1F%F**&F(F%%&alphaGF%F%*&F(F%%%betaGF%F%*$)F (F.F%F%**F5F%F7F%F)F%F+F%F*F%F)F*F+!\"#,&F)F%F%F%F*F%F.-%\"OG6#F%\"\"$ " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "If " }{XPPEDIT 18 0 "abs(a) > 1" "6#2\"\"\"-%$absG6#%\"aG" }{TEXT -1 30 ", this series converges ti ll " }{XPPEDIT 18 0 "abs(z) < 1" "6#2-%$absG6#%\"zG\"\"\"" }{TEXT -1 18 ", otherwise till " }{XPPEDIT 18 0 "abs(z) < abs(a)" "6#2-%$absG6# %\"zG-F%6#%\"aG" }{TEXT -1 58 ". Picturing the situation in the simpl er situation where " }{XPPEDIT 18 0 "abs(a) > 1" "6#2\"\"\"-%$absG6#% \"aG" }{TEXT -1 9 ", we have" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "with(plots):with(plottools):" }}{PARA 7 "" 1 "" {TEXT -1 50 "War ning, the name changecoords has been redefined\n" }}{PARA 7 "" 1 "" {TEXT -1 74 "Warning, the assigned names arrow and translate now have \+ a global binding\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 118 "disp lay([circle([0,0], 1, color=red),circle([1,0], 1, color=blue),circle([ 0,0], 2, color=white)],scaling=constrained);" }}{PARA 13 "" 1 "" {GLPLOT2D 479 479 479 {PLOTDATA 2 "6&-%'CURVESG6$7U7$$\"\"\"\"\"!$F*F* 7$$\"+8q9@**!#5$\"+OBL`7F/7$$\"+6;$eo*F/$\"+t))*o[#F/7$$\"+f[w(H*F/$\" +FbC\"o$F/7$$\"++o1j()F/$\"+Tn`<[F/7$$\"+V*p,4)F/$\"+CD&y(eF/7$$\"+tio *G(F/$\"+g5ZXoF/7$$\"+'*)RUP'F/$\"+HC80xF/7$$\"+^zEe`F/$\"+a#zKW)F/7$$ \"+=HzdUF/$\"+B0F[!*F/7$$\"+Q*p,4$F/$\"+l^c5&*F/7$$\"+UJ\"Q(=F/$\"+3D( G#)*F/7$$\"+D>0zi!#6$\"+%Gn-)**F/7$$!+m>0ziFaoFbo7$$!+YJ\"Q(=F/$\"+2D( G#)*F/7$$!+U*p,4$F/$\"+j^c5&*F/7$$!+8HzdUF/$\"+E0F[!*F/7$$!+YzEe`F/$\" +d#zKW)F/7$$!+$*)RUP'F/$\"+JC80xF/7$$!+$G'o*G(F/$\"+]5ZXoF/7$$!+]*p,4) F/$\"+9D&y(eF/7$$!+0o1j()F/$\"+Kn`<[F/7$$!+i[w(H*F/$\"+>bC\"o$F/7$$!+8 ;$eo*F/$\"+l))*o[#F/7$$!+9q9@**F/$\"+IBL`7F/7$$!\"\"F*$!+:w1-T!#>7$$!+ 8q9@**F/$!+QBL`7F/7$$!+6;$eo*F/$!+t))*o[#F/7$$!+f[w(H*F/$!+FbC\"o$F/7$ $!+,o1j()F/$!+Sn`<[F/7$$!+Y*p,4)F/$!+?D&y(eF/7$$!+xio*G(F/$!+c5ZXoF/7$ $!+())RUP'F/$!+PC80xF/7$$!+RzEe`F/$!+i#zKW)F/7$$!+1HzdUF/$!+H0F[!*F/7$ $!+M*p,4$F/$!+m^c5&*F/7$$!+QJ\"Q(=F/$!+4D(G#)*F/7$$!+%)=0ziFao$!+&Gn-) **F/7$$\"+2?0ziFao$!+%Gn-)**F/7$$\"+]J\"Q(=F/$!+2D(G#)*F/7$$\"+Y*p,4$F /$!+i^c5&*F/7$$\"+!\"*F07$$\"+hJeo>F]\\lF57$$\"+'[w(H>F]\\lF :7$$\"+!o1j(=F]\\lF?7$$\"+%*p,4=F]\\lFD7$$\"+F'o*G0zi5F]\\lFbo7$$\"+.[4s$*F/Fbo7$$\"+ao=E\")F/ Fjo7$$\"+e+$)4pF/F_p7$$\"+(32Au&F/Fdp7$$\"+a?tTYF/Fip7$$\"+2,wDOF/F^q7 $$\"+F/Fhq7$$\"+&>LpB\"F/F]r7$$\"*Q^B-(F/Fbr7$$ \"*(QoTJF/Fgr7$$\")')H&)yF/F\\s7$F+Fas7$$\")()H&)yF/Fgs7$$\"**QoTJF/F \\t7$$\"*T^B-(F/Fat7$$\"+*>LpB\"F/Fft7$$\"+a+$)4>F/F[u7$$\"+BPJ5FF/F`u 7$$\"+8,wDOF/Feu7$$\"+h?tTYF/Fju7$$\"+%42Au&F/F_v7$$\"+m+$)4pF/Fdv7$$ \"+io=E\")F/Fiv7$$\"+7[4s$*F/F^w7$$\"+?0zi5F]\\lFcw7$$\"+:8Q(=\"F]\\lF hw7$$\"+&*p,48F]\\lF]x7$Fd]lFbx7$Fa]lFgx7$F^]lFjx7$$\"+H'o*GF]\\l$\"+sYm1DF/7$ $\"+Aj;P>F]\\l$\"+Yxzt\\F/7$$\"+sHbf=F]\\l$\"+a5\\itF/7$$\"+gLh_F]\\l7$$\"+%GEwu $F/$\"+-Xdk>F]\\l7$$\"+&Q5eD\"F/$\"+dM0'*>F]\\l7$$!+$R5eD\"F/F`hl7$$!+ #HEwu$F/$\"+,Xdk>F]\\l7$$!+%))R.='F/Ffgl7$$!+Eee:&)F/Fagl7$$!+*e`;2\"F ]\\lF\\gl7$$!+zz%[F\"F]\\lFgfl7$$!+ds$zX\"F]\\l$\"+5U4p8F]\\l7$$!+!*R. =;F]\\l$\"+.0dv6F]\\l7$$!+hLh_F]\\l$\"+Ixzt\\F/7$$!+.%HU)>F]\\l$\"+gYm1DF/7$$!\"#F*$ !+I_8/#)Fcs7$F`[m$!+wYm1DF/7$$!+Aj;P>F]\\l$!+Yxzt\\F/7$Ffjl$!+a5\\itF/ 7$$!+gLh_F]\\l7$$!+wii ZPF/$!+-Xdk>F]\\l7$$!+x.\"eD\"F/$!+dM0'*>F]\\l7$$\"+,/\"eD\"F/F__m7$$ \"++jiZPF/$!+,Xdk>F]\\l7$$\"+#*)R.='F/$!+KI6->F]\\l7$$\"+Mee:&)F/$!+0T l4=F]\\l7$Fjfl$!+^el)o\"F]\\l7$Fefl$!+'[E5a\"F]\\l7$$\"+ds$zX\"F]\\l$! +4U4p8F]\\l7$$\"+\"*R.=;F]\\l$!+-0dv6F]\\l7$$\"+hLh_F]\\l$!+Axzt\\F/7$Fgdl$!+_Ym1DF/7 $Fh[l$\"+Yq#3k\"!#=-F^[l6&F`[lF)F)F)-%(SCALINGG6#%,CONSTRAINEDG" 1 2 0 1 10 0 2 9 1 4 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2 " "Curve 3" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "The ODE solution a round " }{XPPEDIT 18 0 "z = 0" "6#/%\"zG\"\"!" }{TEXT -1 11 ", that is , " }{XPPEDIT 18 0 "HeunG(`...`,z)" "6#-%&HeunGG6$%$...G%\"zG" }{TEXT -1 5 " for " }{XPPEDIT 18 0 "abs(z) < 1" "6#2-%$absG6#%\"zG\"\"\"" } {TEXT -1 121 ", inside the red circle, can always be expressed as a li near combination of the two independent series expansions around " } {XPPEDIT 18 0 "z=1" "6#/%\"zG\"\"\"" }{TEXT -1 5 ". So " }{TEXT 296 21 "inside the red circle" }{TEXT -1 30 " we have something of the for m" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 " HeunG(p[0],z) = c[1]*HeunG(p[1],z-1)+HeunG(p[2],z-1);" "6#/-%&HeunGG6$ &%\"pG6#\"\"!%\"zG,&*&&%\"cG6#\"\"\"F1-F%6$&F(6#F1,&F+F1F1!\"\"F1F1-F% 6$&F(6#\"\"#,&F+F1F1F7F1" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "" 0 "" {TEXT -1 29 "This formula is taken as the " }{TEXT 290 99 "ana lytic extension of HeunG for z outside the red circle and inside the b lue circle centered at one" }{TEXT -1 51 ", but as said: there is no k nown exact formula for " }{XPPEDIT 18 0 "c[1]" "6#&%\"cG6#\"\"\"" } {TEXT -1 5 " and " }{XPPEDIT 18 0 "c[2]" "6#&%\"cG6#\"\"#" }{TEXT -1 33 " in terms of the Heun parameters " }{XPPEDIT 18 0 "p[0]" "6#&%\"pG 6#\"\"!" }{TEXT -1 1 "." }}{PARA 259 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 286 "So our first idea: we know that between 0 and 1 t he two series should return the same numerical value for the function. So compute the lhs and rhs at two points between 0 and 1, so that the formula above transforms into two equations with numerical coefficien ts, linear in the unknowns " }{XPPEDIT 18 0 "\{c[1],c[2]\}" "6#<$&%\"c G6#\"\"\"&F%6#\"\"#" }{TEXT -1 6 ". The " }{XPPEDIT 18 0 "c[i]" "6#&% \"cG6#%\"iG" }{TEXT -1 86 " we compute solving this system are valid o nly for the given values of the parameters " }{XPPEDIT 18 0 "a,q,alpha ,beta,gamma,delta" "6(%\"aG%\"qG%&alphaG%%betaG%&gammaG%&deltaG" } {TEXT -1 20 "). But once we have " }{XPPEDIT 18 0 "c[1]" "6#&%\"cG6#\" \"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "c[2]" "6#&%\"cG6#\"\"#" } {TEXT -1 124 ", we can use them to compute the value of the function o utside the red circle, provided the point is inside the blue circle." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 292 7 "Example " }{TEXT -1 18 ": evaluating with " }{XPPEDIT 18 0 "abs(a) > 1" "6#2\" \"\"-%$absG6#%\"aG" }{TEXT -1 31 " and inside the red circle, so " } {XPPEDIT 18 0 "abs(z) < 1" "6#2-%$absG6#%\"zG\"\"\"" }{TEXT -1 0 "" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "HeunG(4+1/2,1,1/3,-2/3,1/7, -3/2, 3/4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&HeunGG6)#\"\"*\"\"# \"\"\"#!\"#\"\"$#F)F,#F)\"\"(#!\"$F(#F,\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+ a$47*=!\"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 "Evaluating now for " }{XPPEDIT 18 0 "abs(a) > 1" "6#2\"\" \"-%$absG6#%\"aG" }{TEXT -1 84 ", but outside the red circle and insid e the blue circle using the approach described" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 38 "HeunG(4+1/2,1,1/3,-2/3,1/7,-3/2, 5/4);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%&HeunGG6)#\"\"*\"\"#\"\"\"#!\"#\"\"$ #F)F,#F)\"\"(#!\"$F(#\"\"&\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^#$!+$\\JMM#!#6" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 74 "__________________________________________________________________ ________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 15 "" 0 "" {TEXT -1 5 "When " }{XPPEDIT 18 0 "abs(a) < 1" "6#2-%$absG6#%\"aG\"\"\"" } {TEXT -1 129 ", the problem is more complicated because the radius of \+ convergence of the series around 0 and the series around 1 may be smal ler" }}{PARA 259 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "" 0 "" {TEXT 297 9 "Example: " }{TEXT -1 18 "the situation for " }{XPPEDIT 18 0 "a = 1/ 2;" "6#/%\"aG*&\"\"\"F&\"\"#!\"\"" }}{PARA 259 "" 0 "" {TEXT -1 0 "" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 156 "display([circle([0,0], 1/ 2, color=red),circle([1,0], 1/2, color=blue),circle([1/2,0], 1/2, colo r=black),circle([0,0], 1, color=white)],scaling=constrained);" }} {PARA 13 "" 1 "" {GLPLOT2D 931 931 931 {PLOTDATA 2 "6'-%'CURVESG6$7U7$ $\"+++++]!#5$\"\"!F,7$$\"+1Ndg\\F*$\"+!ohmE'!#67$$\"+1e\"H%[F*$\"+O%\\ MC\"F*7$$\"+IC))[YF*$\"+kFiS=F*7$$\"++M`\"Q%F*$\"+q$o(3CF*7$$\"+s\\3XS F*$\"+ii#*QHF*7$$\"+OJ%[k$F*$\"+IbtAMF*7$$\"+[*>r=$F*$\"+9ic_QF*7$$\"+ wR8zEF*$\"+F'R;A%F*7$$\"+fk*)G@F*$\"+i_8CXF*7$$\"+p\\3X:F*$\"+#e#GbZF* 7$$\"+5d1p$*F2$\"+aiV6\\F*7$$\"+if_RJF2$\"+UO8!*\\F*7$$!+$)f_RJF2Fbo7$ $!+Id1p$*F2F]o7$$!+r\\3X:F*Fhn7$$!+ck*)G@F*$\"+j_8CXF*7$$!+tR8zEF*$\"+ G'R;A%F*7$$!+Y*>r=$F*$\"+;ic_QF*7$$!+UJ%[k$F*$\"+DbtAMF*7$$!+v\\3XSF*$ \"+di#*QHF*7$$!+-M`\"Q%F*$\"+m$o(3CF*7$$!+JC))[YF*$\"+gFiS=F*7$$!+1e\" H%[F*$\"+K%\\MC\"F*7$$!+2Ndg\\F*$\"+];mmiF27$$!+++++]F*$!+3Q.^?!#>7$$! 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" }}{PARA 259 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 79 "We propose th e a strategy which can be represented graphically by this picture:" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 240 "display([circle([2,0], 2, \+ color=black),circle([0,-2], 2, color=black),circle([-2,0], 2, color=bl ack),circle([0,2], 2, color=black),circle([0,0], 2, color=green),circl e([0,0], 1, color=red),circle([1,0], 1, color=blue)],scaling=constrain ed);" }}{PARA 13 "" 1 "" {GLPLOT2D 931 931 931 {PLOTDATA 2 "6*-%'CURVE SG6$7U7$$\"\"%\"\"!$F*F*7$$\"+.%HU)R!\"*$\"+sYm1D!#57$$\"+Aj;PRF/$\"+Y xzt\\F27$$\"+sHbfQF/$\"+a5\\itF27$$\"+gLh_PF/$\"+#[t]j*F27$$\"+*)R.=OF /$\"+00dv6F/7$$\"+bs$zX$F/$\"+7U4p8F/7$$\"+zz%[F$F/$\"+'[E5a\"F/7$$\"+ !f`;2$F/$\"+^el)o\"F/7$$\"+%ee:&GF/$\"+0Tl4=F/7$$\"+))R.=EF/$\"+LI6->F /7$$\"+GEwuBF/$\"+-Xdk>F/7$$\"+Q5eD@F/$\"+dM0'*>F/7$$\"+h*=W(=F/Fbo7$$ \"+rtBD;F/$\"+,Xdk>F/7$$\"+7g'>Q\"F/Fhn7$$\"+<9W[6F/FY7$$\"*6kMG*F/FT7 $$\"*@?:D(F/FO7$$\"*VF1U&F/$\"+5U4p8F/7$$\"*5g'>QF/$\"+.0dv6F/7$$\"*Rm QZ#F/$\"+kM2N'*F27$$\"*GqWS\"F/$\"+Q5\\itF27$$\")xO$G'F/$\"+Ixzt\\F27$ $\")(fqd\"F/$\"+gYm1DF27$F+$!+I_8/#)!#>7$Fbr$!+wYm1DF27$$\")yO$G'F/$!+ Yxzt\\F27$Fhq$!+a5\\itF27$$\"*SmQZ#F/$!+![t]j*F27$$\"*6g'>QF/$!+/0dv6F /7$$\"*XF1U&F/$!+6U4p8F/7$$\"*B?:D(F/$!+([E5a\"F/7$$\"*7kMG*F/$!+_el)o \"F/7$$\"+>9W[6F/$!+1Tl4=F/7$$\"+8g'>Q\"F/$!+LI6->F/7$$\"+stBD;F/$!+-X dk>F/7$$\"+i*=W(=F/$!+dM0'*>F/7$$\"+S5eD@F/F`v7$$\"+IEwuBF/$!+,Xdk>F/7 $$\"+*)R.=EF/$!+KI6->F/7$$\"+$ee:&GF/$!+0Tl4=F/7$FR$!+^el)o\"F/7$FM$!+ '[E5a\"F/7$$\"+ds$zX$F/$!+4U4p8F/7$$\"+\"*R.=OF/$!+-0dv6F/7$$\"+hLh_PF /$!+eM2N'*F27$$\"+tHbfQF/$!+I5\\itF27$$\"+Bj;PRF/$!+Axzt\\F27$F-$!+_Ym 1DF27$F($\"+Yq#3k\"!#=-%'COLOURG6&%$RGBGF*F*F*-F$6$7U7$$\"\"#F*$!\"#F* 7$$\"+.%HU)>F/$!+LNL\\F/$!+D-i-:F/7$$\"+sHbf=F/$!+&*3vj7 F/7$$\"+gLh_F/7$$\"+w)R.='F2$!)np)y*F/7$$\"+%GEwu$F2$!))\\Da$F/7$$\"+& Q5eD\"F2$!(Vl%RF/7$$!+$R5eD\"F2F`^l7$$!+#HEwu$F2$!)*\\Da$F/7$$!+%))R.= 'F2Ff]l7$$!+Eee:&)F2Fa]l7$$!+*e`;2\"F/F\\]l7$$!+zz%[F\"F/Fg\\l7$$!+ds$ zX\"F/$!*!z04jF/7$$!+!*R.=;F/$!*(\\HW#)F/7$$!+hLh_F/$!+F-i-:F/7$$!+.%HU)>F/$!+MNL\\F/$!+v(zt\\#F/7$Ff`l$!+0\"\\it#F/7$$ !+gLh_F/$!+s(zt\\#F/7$Fgz$!+lkm]AF/7$Fbz$!+)*** ****>F/Fjy-F$6$7U7$F+F+7$$!)(fqd\"F/F07$$!)yO$G'F/F67$$!*GqWS\"F/F;7$$ !*SmQZ#F/F@7$$!*6g'>QF/FE7$$!*XF1U&F/FJ7$$!*@?:D(F/FO7$$!*5kMG*F/FT7$$ !+;9W[6F/FY7$$!+7g'>Q\"F/Fhn7$$!+stBD;F/F]o7$$!+i*=W(=F/Fbo7$$!+R5eD@F /Fbo7$$!+HEwuBF/Fjo7$$!+))R.=EF/Fhn7$$!+$ee:&GF/FY7$$!+*e`;2$F/FT7$$!+ zz%[F$F/FO7$$!+ds$zX$F/F[q7$$!+!*R.=OF/F`q7$$!+hLh_PF/Feq7$$!+sHbfQF/F jq7$$!+Bj;PRF/F_r7$$!+.%HU)RF/Fdr7$$!\"%F*Fgr7$F`]mF[s7$$!+Aj;PRF/F`s7 $Fj\\mFcs7$$!+gLh_PF/Fhs7$$!+*)R.=OF/F]t7$$!+bs$zX$F/Fbt7$$!+xz%[F$F/F gt7$$!+)e`;2$F/F\\u7$$!+\"ee:&GF/Fau7$$!+()R.=EF/Ffu7$$!+GEwuBF/F[v7$$ !+Q5eD@F/F`v7$$!+g*=W(=F/F`v7$$!+qtBD;F/Fhv7$$!+6g'>Q\"F/F]w7$$!+<9W[6 F/Fbw7$F`jlFew7$F]jlFhw7$$!*VF1U&F/F]x7$$!*4g'>QF/Fbx7$$!*RmQZ#F/Fgx7$ $!*FqWS\"F/F\\y7$$!)xO$G'F/Fay7$F[ilFdy7$F+FgyFjy-F$6$7U7$FbzFbz7$Fgz$ \"+nkm]AF/7$F\\[l$\"+v(zt\\#F/7$Fa[l$\"+0\"\\it#F/7$Ff[l$\"+[t]jHF/7$F [\\l$\"+00dvJF/7$F`\\l$\"+7U4pLF/7$Fe\\l$\"+'[E5a$F/7$Fj\\l$\"+^el)o$F /7$F_]l$\"+0Tl4QF/7$Fd]l$\"+LI6-RF/7$Fi]l$\"+-XdkRF/7$F^^l$\"+dM0'*RF/ 7$Fc^lFjcm7$Ff^l$\"+,XdkRF/7$F[_lFdcm7$F^_lFacm7$Fa_lF^cm7$Fd_lF[cm7$F g_l$\"+5U4pLF/7$F\\`l$\"+.0dvJF/7$Fa`l$\"+Yt]jHF/7$Ff`l$\"+/\"\\it#F/7 $F[al$\"+t(zt\\#F/7$F`al$\"+mkm]AF/7$Fdz$\"+********>F/7$F`al$\"+KNL\\ F/7$Fadl$\")np)y*F/7$Ffdl$\"))\\Da$F/7$F[el$\"(Vl%RF/7$F`elF[hm7$F cel$\")*\\Da$F/7$Fhel$\")op)y*F/7$F]fl$\"*&*eM!>F/7$Fj\\l$\"*\\TM6$F/7 $Fe\\l$\"*9N(*e%F/7$Fhfl$\"*\"z04jF/7$F]gl$\"*)\\HW#)F/7$Fbgl$\"+aE\\O 5F/7$Fggl$\"+(*3vj7F/7$F\\hl$\"+G-i-:F/7$Fgz$\"+NNL\\0zi!#6$\"+%Gn-)**F27$$!+m>0ziF]bnF^ bn7$$!+YJ\"Q(=F2$\"+2D(G#)*F27$$!+U*p,4$F2$\"+j^c5&*F27$$!+8HzdUF2$\"+ E0F[!*F27$$!+YzEe`F2$\"+d#zKW)F27$$!+$*)RUP'F2$\"+JC80xF27$$!+$G'o*G(F 2$\"+]5ZXoF27$$!+]*p,4)F2$\"+9D&y(eF27$$!+0o1j()F2$\"+Kn`<[F27$$!+i[w( H*F2$\"+>bC\"o$F27$$!+8;$eo*F2$\"+l))*o[#F27$$!+9q9@**F2$\"+IBL`7F27$$ !\"\"F*$!+:w1-TFir7$$!+8q9@**F2$!+QBL`7F27$$!+6;$eo*F2$!+t))*o[#F27$$! +f[w(H*F2$!+FbC\"o$F27$$!+,o1j()F2$!+Sn`<[F27$$!+Y*p,4)F2$!+?D&y(eF27$ $!+xio*G(F2$!+c5ZXoF27$$!+())RUP'F2$!+PC80xF27$$!+RzEe`F2$!+i#zKW)F27$ $!+1HzdUF2$!+H0F[!*F27$$!+M*p,4$F2$!+m^c5&*F27$$!+QJ\"Q(=F2$!+4D(G#)*F 27$$!+%)=0ziF]bn$!+&Gn-)**F27$$\"+2?0ziF]bn$!+%Gn-)**F27$$\"+]J\"Q(=F2 $!+2D(G#)*F27$$\"+Y*p,4$F2$!+i^c5&*F27$$\"+F/Ff^n7$$\"+hJeo>F/F[_n7$$\"+'[w(H>F/F`_ n7$$\"+!o1j(=F/Fe_n7$$\"+%*p,4=F/Fj_n7$$\"+F'o*G0zi5F/F^bn7$$\"+.[4s$*F2F^bn7$$\"+ao=E\")F2Ffbn7$$\"+e+ $)4pF2F[cn7$$\"+(32Au&F2F`cn7$$\"+a?tTYF2Fecn7$$\"+2,wDOF2Fjcn7$$\"+

F2Fddn7$$\"+&>LpB\"F2Fidn7$$\"*Q^B-(F2F^en7$$\"*( QoTJF2Fcen7$$\")')H&)yF2Fhen7$F+F]fn7$$\")()H&)yF2Fbfn7$$\"**QoTJF2Fgf n7$$\"*T^B-(F2F\\gn7$$\"+*>LpB\"F2Fagn7$$\"+a+$)4>F2Ffgn7$$\"+BPJ5FF2F [hn7$$\"+8,wDOF2F`hn7$$\"+h?tTYF2Fehn7$$\"+%42Au&F2Fjhn7$$\"+m+$)4pF2F _in7$$\"+io=E\")F2Fdin7$$\"+7[4s$*F2Fiin7$$\"+?0zi5F/F^jn7$$\"+:8Q(=\" F/Fcjn7$$\"+&*p,48F/Fhjn7$Ff_oF][o7$Fc_oFb[o7$F`_oFe[o7$$\"+H'o*G 0" "6#0&%\"zG6#\"\"!F'" } {TEXT -1 42 " gets connected with the expansion around " }{XPPEDIT 18 0 "z[0] = 0" "6#/&%\"zG6#\"\"!F'" }{TEXT -1 168 ". In fact, depending \+ on the region, problems involving two connections can be mapped into p roblems involving only one connection recursively using the identities above." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 311 6 "Note: " }{TEXT -1 197 "these two identities can be derived by playi ng with Mobius transformations and the HeunG equation: we essentially \+ search for transformations which leave three of the four singularities \"unchanged\"." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 312 8 "Example:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "nHG := HeunG(4/5,1,1/3,-2/3,1/7,-3/ 2,z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$nHGG-%&HeunGG6)#\"\"%\"\"& \"\"\"#!\"#\"\"$#F+F.#F+\"\"(#!\"$\"\"#%\"zG" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 5 "Here " }{XPPEDIT 18 0 "abs(a) < 1" "6#2-%$absG6#%\"aG\" \"\"" }{TEXT -1 57 ", so we will use two connections in order to compu te for " }{XPPEDIT 18 0 "abs(z) > 1 + 1/5" "6#2,&\"\"\"F%*&F%F%\"\"&! \"\"F%-%$absG6#%\"zG" }{TEXT -1 62 ". 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