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Painlev\351 4 equation." } {TEXT 205 1 "\n" }{TEXT 205 114 "For convenience the irregular times a nd monodromies are denoted t_\{i,j\} at lambda=infinity and s_\{i,j\} \+ at lambda=t" }{TEXT 205 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 205 118 " We first use the expression of the coefficients of the spectral curve \+ in terms of the irregular times and monodromies." }{TEXT 205 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "restart:\n" }{MPLTEXT 1 0 140 "P2:=unapply(s10*s20/(lambda-t)^2+P012/(lambda-t)-t12*s10-t12*s20- t12*t10+t10*t22+t11*t21+(t11*t22+t12*t21)*lambda+t12*t22*lambda^2,lamb da);\n" }{MPLTEXT 1 0 126 "tdP2:=unapply(s10*s20/(lambda-t)^2-t12*s10- t12*s20-t12*t10+t10*t22+t11*t21+(t11*t22+t12*t21)*lambda+t12*t22*lambd a^2,lambda);\n" }{MPLTEXT 1 0 67 "P1:=unapply((s10+s20)/(lambda-t)-t11 -t21+(-t12-t22)*lambda,lambda);" }{MPLTEXT 1 0 1 "\n" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I#P2G6\"f*6#I'lambdaGF$F$6$I)operatorGF$I&arrowGF$F$, 4*(I$s10GF$\"\"\"I$s20GF$F.,&9$F.I\"tGF$!\"\"!\"#F.*&I%P012GF$F.F0F3F. *&I$t12GF$F.F-F.F3*&F8F.F/F.F3*&F8F.I$t10GF$F.F3*&F;F.I$t22GF$F.F.*&I$ t11GF$F.I$t21GF$F.F.*&,&*&F?F.F=F.F.*&F8F.F@F.F.F.F1F.F.*(F8F.F=F.F1\" \"#F.F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I%tdP2G6\"f*6#I'lambdaGF $F$6$I)operatorGF$I&arrowGF$F$,2*(I$s10GF$\"\"\"I$s20GF$F.,&9$F.I\"tGF $!\"\"!\"#F.*&I$t12GF$F.F-F.F3*&F6F.F/F.F3*&F6F.I$t10GF$F.F3*&F9F.I$t2 2GF$F.F.*&I$t11GF$F.I$t21GF$F.F.*&,&*&F=F.F;F.F.*&F6F.F>F.F.F.F1F.F.*( F6F.F;F.F1\"\"#F.F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I#P1G6\"f*6# I'lambdaGF$F$6$I)operatorGF$I&arrowGF$F$,**&,&I$s10GF$\"\"\"I$s20GF$F/ F/,&9$F/I\"tGF$!\"\"F4F/I$t11GF$F4I$t21GF$F4*&,&I$t12GF$F4I$t22GF$F4F/ F2F/F/F$F$F$" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 249 30 "Expression of t he Lax matrix L" }}{EXCHG {PARA 210 "" 0 "" {TEXT 250 17 "Study at inf inity" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 207 "logPsi1Infty:=-t1 2/2/h*lambda^2-t11/h*lambda-t10/h*ln(lambda)+A10-A12/(2-1)/lambda^(2-1 )-A13/(3-1)/lambda^(3-1)-A14/(4-1)/lambda^(4-1)-A15/(5-1)/lambda^(5-1) -A16/(6-1)/lambda^(6-1)-A17/(7-1)/lambda^(7-1) ;\n" }{MPLTEXT 1 0 220 "logPsi2Infty:=-t22/2/h*lambda^2-t21*lambda/h-t20/h*ln(lambda)-1*l n(lambda)+A20-A22/(2-1)/lambda^(2-1)-A23/(3-1)/lambda^(3-1)-A24/(4-1)/ lambda^(4-1)-A25/(5-1)/lambda^(5-1)-A26/(6-1)/lambda^(6-1)-A27/(7-1)/l ambda^(7-1) ;\n" }{MPLTEXT 1 0 218 "Llogpsi1Infty:=-Lt12/2/h*lambda^2 -Lt11/h*lambda-Lt10/h*ln(lambda)+LA10-LA12/(2-1)/lambda^(2-1)-LA13/(3- 1)/lambda^(3-1)-LA14/(4-1)/lambda^(4-1)-LA15/(5-1)/lambda^(5-1)-LA16/( 6-1)/lambda^(6-1)-LA17/(7-1)/lambda^(7-1) ;\n" }{MPLTEXT 1 0 218 "Llo gpsi2Infty:=-Lt22/2/h*lambda^2-Lt21*lambda/h-Lt20/h*ln(lambda)+LA20-LA 22/(2-1)/lambda^(2-1)-LA23/(3-1)/lambda^(3-1)-LA24/(4-1)/lambda^(4-1)- LA25/(5-1)/lambda^(5-1)-LA26/(6-1)/lambda^(6-1)-LA27/(7-1)/lambda^(7-1 ) ;\n" }{MPLTEXT 1 0 358 "Lpsi1Infty := exp(1/h*(-1/2*t12*lambda^2-t1 1*lambda-t10*ln(lambda)+h*A10-h*A12/lambda-1/2*h*A13/lambda^2-1/3*h*A1 4/lambda^3-1/4*h*A15/lambda^4-1/5*h*A16/lambda^5-1/6*h*A17/lambda^6))* 1/h*(-1/2*Lt12*lambda^2-Lt11*lambda-Lt10*ln(lambda)+h*LA10-h*LA12/lamb da-1/2*h*LA13/lambda^2-1/3*h*LA14/lambda^3-1/4*h*LA15/lambda^4-1/5*h*L A16/lambda^5-1/6*h*LA17/lambda^6);\n" }{MPLTEXT 1 0 371 "Lpsi2Infty := exp(1/h*(-1/2*t22*lambda^2-t21*lambda-t20*ln(lambda)-h*ln(lambda)+h*A 20-h*A22/lambda-1/2*h*A23/lambda^2-1/3*h*A24/lambda^3-1/4*h*A25/lambda ^4-1/5*h*A26/lambda^5-1/6*h*A27/lambda^6))*1/h*(-1/2*Lt22*lambda^2-Lt2 1*lambda-Lt20*ln(lambda)+h*LA20-h*LA22/lambda-1/2*h*LA23/lambda^2-1/3* h*LA24/lambda^3-1/4*h*LA25/lambda^4-1/5*h*LA26/lambda^5-1/6*h*LA27/lam bda^6);\n" }{MPLTEXT 1 0 30 "psi1Infty:=exp(logPsi1Infty);\n" } {MPLTEXT 1 0 30 "psi2Infty:=exp(logPsi2Infty);\n" }{MPLTEXT 1 0 43 "dp si1dlambdaInfty:=diff(psi1Infty,lambda):\n" }{MPLTEXT 1 0 43 "dpsi2dla mbdaInfty:=diff(psi2Infty,lambda):\n" }{MPLTEXT 1 0 47 "d2psi1dlambda2 Infty:=diff(psi1Infty,lambda$2):\n" }{MPLTEXT 1 0 47 "d2psi2dlambda2In fty:=diff(psi2Infty,lambda$2):\n" }{MPLTEXT 1 0 52 "Vinfty1:=-t12/2/la mbda^2-t11/lambda+t10*ln(lambda);\n" }{MPLTEXT 1 0 52 "Vinfty2:=-t22/2 /lambda^2-t21/lambda+t20*ln(lambda);\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 89 "WronskianLambdaInfty:=h*factor(psi1Infty *dpsi2dlambdaInfty-psi2Infty*dpsi1dlambdaInfty):\n" }{MPLTEXT 1 0 132 "WronskianLambdabisInfty:=h*simplify(factor( (diff(logPsi2Infty,lambda )-diff(logPsi1Infty,lambda))*exp(logPsi1Infty+logPsi2Infty))):\n" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 116 "WronskianTildeLambdaInfty:=h^3* factor(dpsi2dlambdaInfty*d2psi1dlambda2Infty-dpsi1dlambdaInfty*d2psi2d lambda2Infty):\n" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I-logPsi1InftyG6\", 6*(I$t12GF$\"\"\"I\"hGF$!\"\"I'lambdaGF$\"\"##F*F,*(I$t11GF$F(F)F*F+F( F**(I$t10GF$F(F)F*-I#lnG6$%*protectedGI(_syslibGF$6#F+F(F*I$A10GF$F(*& I$A12GF$F(F+F*F**&I$A13GF$F(F+!\"#F-*&I$A14GF$F(F+!\"$#F*\"\"$*&I$A15G F$F(F+!\"%#F*\"\"%*&I$A16GF$F(F+!\"&#F*\"\"&*&I$A17GF$F(F+!\"'#F*\"\"' " }}{PARA 11 "" 1 "" {XPPMATH 20 ">I-logPsi2InftyG6\",8*(I$t22GF$\"\" \"I\"hGF$!\"\"I'lambdaGF$\"\"##F*F,*(I$t21GF$F(F+F(F)F*F**(I$t20GF$F(F )F*-I#lnG6$%*protectedGI(_syslibGF$6#F+F(F*F2F*I$A20GF$F(*&I$A22GF$F(F +F*F**&I$A23GF$F(F+!\"#F-*&I$A24GF$F(F+!\"$#F*\"\"$*&I$A25GF$F(F+!\"%# F*\"\"%*&I$A26GF$F(F+!\"&#F*\"\"&*&I$A27GF$F(F+!\"'#F*\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I.Llogpsi1InftyG6\",6*(I%Lt12GF$\"\"\"I\"hGF$ !\"\"I'lambdaGF$\"\"##F*F,*(I%Lt11GF$F(F)F*F+F(F**(I%Lt10GF$F(F)F*-I#l nG6$%*protectedGI(_syslibGF$6#F+F(F*I%LA10GF$F(*&I%LA12GF$F(F+F*F**&I% LA13GF$F(F+!\"#F-*&I%LA14GF$F(F+!\"$#F*\"\"$*&I%LA15GF$F(F+!\"%#F*\"\" %*&I%LA16GF$F(F+!\"&#F*\"\"&*&I%LA17GF$F(F+!\"'#F*\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I.Llogpsi2InftyG6\",6*(I%Lt22GF$\"\"\"I\"hGF$!\"\" I'lambdaGF$\"\"##F*F,*(I%Lt21GF$F(F+F(F)F*F**(I%Lt20GF$F(F)F*-I#lnG6$% *protectedGI(_syslibGF$6#F+F(F*I%LA20GF$F(*&I%LA22GF$F(F+F*F**&I%LA23G F$F(F+!\"#F-*&I%LA24GF$F(F+!\"$#F*\"\"$*&I%LA25GF$F(F+!\"%#F*\"\"%*&I% LA26GF$F(F+!\"&#F*\"\"&*&I%LA27GF$F(F+!\"'#F*\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I+Lpsi1InftyG6\"*(-I$expG6$%*protectedGI(_syslibGF$6#*& I\"hGF$!\"\",6*&I$t12GF$\"\"\"I'lambdaGF$\"\"##F.F4*&I$t11GF$F2F3F2F.* &I$t10GF$F2-I#lnGF(6#F3F2F.*&F-F2I$A10GF$F2F2*(F-F2I$A12GF$F2F3F.F.*(F -F2I$A13GF$F2F3!\"#F5*(F-F2I$A14GF$F2F3!\"$#F.\"\"$*(F-F2I$A15GF$F2F3! \"%#F.\"\"%*(F-F2I$A16GF$F2F3!\"&#F.\"\"&*(F-F2I$A17GF$F2F3!\"'#F.\"\" 'F2F2F-F.,6*&I%Lt12GF$F2F3F4F5*&I%Lt11GF$F2F3F2F.*&I%Lt10GF$F2F:F2F.*& F-F2I%LA10GF$F2F2*(F-F2I%LA12GF$F2F3F.F.*(F-F2I%LA13GF$F2F3FCF5*(F-F2I %LA14GF$F2F3FFFG*(F-F2I%LA15GF$F2F3FKFL*(F-F2I%LA16GF$F2F3FPFQ*(F-F2I% LA17GF$F2F3FUFVF2" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I+Lpsi2InftyG6\"*( -I$expG6$%*protectedGI(_syslibGF$6#*&I\"hGF$!\"\",8*&I$t22GF$\"\"\"I'l ambdaGF$\"\"##F.F4*&I$t21GF$F2F3F2F.*&I$t20GF$F2-I#lnGF(6#F3F2F.*&F-F2 F:F2F.*&F-F2I$A20GF$F2F2*(F-F2I$A22GF$F2F3F.F.*(F-F2I$A23GF$F2F3!\"#F5 *(F-F2I$A24GF$F2F3!\"$#F.\"\"$*(F-F2I$A25GF$F2F3!\"%#F.\"\"%*(F-F2I$A2 6GF$F2F3!\"&#F.\"\"&*(F-F2I$A27GF$F2F3!\"'#F.\"\"'F2F2F-F.,6*&I%Lt22GF $F2F3F4F5*&I%Lt21GF$F2F3F2F.*&I%Lt20GF$F2F:F2F.*&F-F2I%LA20GF$F2F2*(F- F2I%LA22GF$F2F3F.F.*(F-F2I%LA23GF$F2F3FDF5*(F-F2I%LA24GF$F2F3FGFH*(F-F 2I%LA25GF$F2F3FLFM*(F-F2I%LA26GF$F2F3FQFR*(F-F2I%LA27GF$F2F3FVFWF2" }} {PARA 11 "" 1 "" {XPPMATH 20 ">I*psi1InftyG6\"-I$expG6$%*protectedGI(_ syslibGF$6#,6*(I$t12GF$\"\"\"I\"hGF$!\"\"I'lambdaGF$\"\"##F0F2*(I$t11G F$F.F/F0F1F.F0*(I$t10GF$F.F/F0-I#lnGF'6#F1F.F0I$A10GF$F.*&I$A12GF$F.F1 F0F0*&I$A13GF$F.F1!\"#F3*&I$A14GF$F.F1!\"$#F0\"\"$*&I$A15GF$F.F1!\"%#F 0\"\"%*&I$A16GF$F.F1!\"&#F0\"\"&*&I$A17GF$F.F1!\"'#F0\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I*psi2InftyG6\"-I$expG6$%*protectedGI(_syslibGF $6#,8*(I$t22GF$\"\"\"I\"hGF$!\"\"I'lambdaGF$\"\"##F0F2*(I$t21GF$F.F1F. F/F0F0*(I$t20GF$F.F/F0-I#lnGF'6#F1F.F0F8F0I$A20GF$F.*&I$A22GF$F.F1F0F0 *&I$A23GF$F.F1!\"#F3*&I$A24GF$F.F1!\"$#F0\"\"$*&I$A25GF$F.F1!\"%#F0\" \"%*&I$A26GF$F.F1!\"&#F0\"\"&*&I$A27GF$F.F1!\"'#F0\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I(Vinfty1G6\",(*&I$t12GF$\"\"\"I'lambdaGF$!\"##!\" \"\"\"#*&I$t11GF$F(F)F,F,*&I$t10GF$F(-I#lnG6$%*protectedGI(_syslibGF$6 #F)F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I(Vinfty2G6\",(*&I$t22GF$\" \"\"I'lambdaGF$!\"##!\"\"\"\"#*&I$t21GF$F(F)F,F,*&I$t20GF$F(-I#lnG6$%* protectedGI(_syslibGF$6#F)F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "L21Infty:=factor(simplify(WronskianTildeLambdaInfty/WronskianL ambdabisInfty)):\n" }{MPLTEXT 1 0 75 "L21InftyOrderlambda3:=factor(-re sidue(L21Infty/lambda^4,lambda=infinity));\n" }{MPLTEXT 1 0 75 "L21Inf tyOrderlambda2:=factor(-residue(L21Infty/lambda^3,lambda=infinity));\n " }{MPLTEXT 1 0 75 "L21InftyOrderlambda1:=factor(-residue(L21Infty/lam bda^2,lambda=infinity));\n" }{MPLTEXT 1 0 75 "L21InftyOrderlambda0:=fa ctor(-residue(L21Infty/lambda^1,lambda=infinity));\n" }{MPLTEXT 1 0 80 "L21InftyOrderlambdaMinus1:=factor(-residue(L21Infty/lambda^0,lambd a=infinity)):\n" }{MPLTEXT 1 0 46 "factor(series(L21InftyOrderlambdaMi nus1,h=0));" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I5L21InftyOrderlambda3G6 \"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I5L21InftyOrderlambda2G6\",$ *&I$t22GF$\"\"\"I$t12GF$F(!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I5L2 1InftyOrderlambda1G6\",&*&I$t11GF$\"\"\"I$t22GF$F(!\"\"*&I$t12GF$F(I$t 21GF$F(F*" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I5L21InftyOrderlambda0G6\" ,**&I\"hGF$\"\"\"I$t12GF$F(!\"\"*&I$t10GF$F(I$t22GF$F(F**&I$t11GF$F(I$ t21GF$F(F**&F)F(I$t20GF$F(F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "+'I\"hG6 \",&*&I$t10GF$\"\"\"I$t21GF$F(!\"\"*&I$t11GF$F(I$t20GF$F(F*\"\"!*&,.*( I$A12GF$F(I$t12GF$F(I$t22GF$F(F(*&F2F(F4\"\"#F**&I$A22GF$F(F3F6F(*(F8F (F3F(F4F(F**&F,F(F3F(F**&F3F(F)F(F(F(,&F4F*F3F(F*F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 251 132 "We get that L_\{2,1\} behaves at infinity like -t22*t11*lambda^2-(t21*t12+t22*t11)*lambda-(t12*t20+t22*t10+t11*t21+h *t12) +O(1/lambda)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "L22In fty:=factor(h*simplify(diff(WronskianLambdabisInfty,lambda)/WronskianL ambdabisInfty)):\n" }{MPLTEXT 1 0 75 "L22InftyOrderlambda3:=factor(-re sidue(L22Infty/lambda^4,lambda=infinity));\n" }{MPLTEXT 1 0 75 "L22Inf tyOrderlambda2:=factor(-residue(L22Infty/lambda^3,lambda=infinity));\n " }{MPLTEXT 1 0 75 "L22InftyOrderlambda1:=factor(-residue(L22Infty/lam bda^2,lambda=infinity));\n" }{MPLTEXT 1 0 75 "L22InftyOrderlambda0:=fa ctor(-residue(L22Infty/lambda^1,lambda=infinity));\n" }{MPLTEXT 1 0 80 "L22InftyOrderlambdaMinus1:=factor(-residue(L22Infty/lambda^0,lambd a=infinity));\n" }{MPLTEXT 1 0 83 "L22InftyOrderlambdaMinus2:=factor(- residue(L22Infty/lambda^(-1),lambda=infinity));\n" }{MPLTEXT 1 0 82 "L 22InftyOrderlambdaMinus3:=factor(-residue(L22Infty/lambda^(-2),lambda= infinity)):" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I5L22InftyOrderlambda3G6 \"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I5L22InftyOrderlambda2G6\"\" \"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I5L22InftyOrderlambda1G6\",&I$t1 2GF$!\"\"I$t22GF$F'" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I5L22InftyOrderl ambda0G6\",&I$t11GF$!\"\"I$t21GF$F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "> I:L22InftyOrderlambdaMinus1G6\",&I$t10GF$!\"\"I$t20GF$F'" }}{PARA 11 " " 1 "" {XPPMATH 20 ">I:L22InftyOrderlambdaMinus2G6\"*(I\"hGF$\"\"\",.* &I$A12GF$F'I$t12GF$F'F'*&F*F'I$t22GF$F'!\"\"*&I$A22GF$F'F+F'F'*&F0F'F- F'F.I$t11GF$F.I$t21GF$F'F',&F-F.F+F'F." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 251 106 "We get that L_\{2,2\} behaves at infinity like -(t12+t2 2)*lambda-(t11+t21)-(t10+t20)/lambda+h*O(1/lambda^2) " }}}{EXCHG {PARA 209 "" 0 "" {TEXT 252 17 "Study at lambda=t" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 43 "logPsi1T:=s10/h*ln(lambda-t)+B10+B12/(2-1)*" }{MPLTEXT 1 0 10 "(lambda-t)" }{MPLTEXT 1 0 17 "^(2-1)+B13/(3-1)*" } {MPLTEXT 1 0 10 "(lambda-t)" }{MPLTEXT 1 0 17 "^(3-1)+B14/(4-1)*" } {MPLTEXT 1 0 10 "(lambda-t)" }{MPLTEXT 1 0 17 "^(4-1)+B15/(5-1)*" } {MPLTEXT 1 0 10 "(lambda-t)" }{MPLTEXT 1 0 17 "^(5-1)+B16/(6-1)*" } {MPLTEXT 1 0 10 "(lambda-t)" }{MPLTEXT 1 0 17 "^(6-1)+B17/(7-1)*" } {MPLTEXT 1 0 10 "(lambda-t)" }{MPLTEXT 1 0 10 "^(7-1) ;\n" }{MPLTEXT 1 0 18 "logPsi2T:=s20/h*ln" }{MPLTEXT 1 0 10 "(lambda-t)" }{MPLTEXT 1 0 15 "+B20+B22/(2-1)*" }{MPLTEXT 1 0 10 "(lambda-t)" }{MPLTEXT 1 0 17 "^(2-1)+B23/(3-1)*" }{MPLTEXT 1 0 10 "(lambda-t)" }{MPLTEXT 1 0 17 "^( 3-1)+B24/(4-1)*" }{MPLTEXT 1 0 10 "(lambda-t)" }{MPLTEXT 1 0 17 "^(4-1 )+B25/(5-1)*" }{MPLTEXT 1 0 10 "(lambda-t)" }{MPLTEXT 1 0 17 "^(5-1)+B 26/(6-1)*" }{MPLTEXT 1 0 10 "(lambda-t)" }{MPLTEXT 1 0 17 "^(6-1)+B27/ (7-1)*" }{MPLTEXT 1 0 10 "(lambda-t)" }{MPLTEXT 1 0 10 "^(7-1) ;\n" } {MPLTEXT 1 0 20 "Llogpsi1T:=Ls10/h*ln" }{MPLTEXT 1 0 10 "(lambda-t)" } {MPLTEXT 1 0 17 "+LB10+LB12/(2-1)*" }{MPLTEXT 1 0 10 "(lambda-t)" } {MPLTEXT 1 0 18 "^(2-1)+LB13/(3-1)*" }{MPLTEXT 1 0 10 "(lambda-t)" } {MPLTEXT 1 0 18 "^(3-1)+LB14/(4-1)*" }{MPLTEXT 1 0 10 "(lambda-t)" } {MPLTEXT 1 0 18 "^(4-1)+LB15/(5-1)*" }{MPLTEXT 1 0 10 "(lambda-t)" } {MPLTEXT 1 0 18 "^(5-1)+LB16*(6-1)/" }{MPLTEXT 1 0 10 "(lambda-t)" } {MPLTEXT 1 0 18 "^(6-1)+LB17/(7-1)*" }{MPLTEXT 1 0 10 "(lambda-t)" } {MPLTEXT 1 0 8 "^(7-1) \n" }{MPLTEXT 1 0 1 "-" }{MPLTEXT 1 0 19 "s10/h /(lambda-t)*Lt" }{MPLTEXT 1 0 5 "-B12*" }{MPLTEXT 1 0 10 "(lambda-t)" }{MPLTEXT 1 0 11 "^(2-1-1)*Lt" }{MPLTEXT 1 0 5 "-B13*" }{MPLTEXT 1 0 18 "(lambda-t)^(3-1-1)" }{MPLTEXT 1 0 3 "*Lt" }{MPLTEXT 1 0 15 "-B14*( lambda-t)" }{MPLTEXT 1 0 8 "^(4-1-1)" }{MPLTEXT 1 0 3 "*Lt" }{MPLTEXT 1 0 3 " ;\n" }{MPLTEXT 1 0 20 "Llogpsi2T:=Ls20/h*ln" }{MPLTEXT 1 0 10 "(lambda-t)" }{MPLTEXT 1 0 17 "+LB20+LB22/(2-1)*" }{MPLTEXT 1 0 10 "(l ambda-t)" }{MPLTEXT 1 0 18 "^(2-1)+LB23/(3-1)*" }{MPLTEXT 1 0 10 "(lam bda-t)" }{MPLTEXT 1 0 18 "^(3-1)+LB24/(4-1)*" }{MPLTEXT 1 0 10 "(lambd a-t)" }{MPLTEXT 1 0 18 "^(4-1)+LB25/(5-1)*" }{MPLTEXT 1 0 10 "(lambda- t)" }{MPLTEXT 1 0 18 "^(5-1)+LB26/(6-1)*" }{MPLTEXT 1 0 10 "(lambda-t) " }{MPLTEXT 1 0 18 "^(6-1)+LB27/(7-1)*" }{MPLTEXT 1 0 10 "(lambda-t)" }{MPLTEXT 1 0 7 "^(7-1)\n" }{MPLTEXT 1 0 1 "-" }{MPLTEXT 1 0 97 "s20/h /(lambda-t)*Lt-B22*(lambda-t)^(2-1-1)*Lt-B23*(lambda-t)^(3-1-1)*Lt-B24 *(lambda-t)^(4-1-1)*Lt" }{MPLTEXT 1 0 2 " ;" }{MPLTEXT 1 0 4 " ;\n" } {MPLTEXT 1 0 23 "Lpsi1T := exp((s10/h*ln" }{MPLTEXT 1 0 10 "(lambda-t) " }{MPLTEXT 1 0 15 "+B10+B12/(2-1)*" }{MPLTEXT 1 0 10 "(lambda-t)" } {MPLTEXT 1 0 17 "^(2-1)+B13/(3-1)*" }{MPLTEXT 1 0 10 "(lambda-t)" } {MPLTEXT 1 0 17 "^(3-1)+B14/(4-1)*" }{MPLTEXT 1 0 10 "(lambda-t)" } {MPLTEXT 1 0 17 "^(4-1)+B15/(5-1)*" }{MPLTEXT 1 0 10 "(lambda-t)" } {MPLTEXT 1 0 17 "^(5-1)+B16/(6-1)*" }{MPLTEXT 1 0 10 "(lambda-t)" } {MPLTEXT 1 0 17 "^(6-1)+B17/(7-1)*" }{MPLTEXT 1 0 10 "(lambda-t)" } {MPLTEXT 1 0 9 "^(7-1)))\n" }{MPLTEXT 1 0 11 "*(Ls10/h*ln" }{MPLTEXT 1 0 10 "(lambda-t)" }{MPLTEXT 1 0 17 "+LB10+LB12/(2-1)*" }{MPLTEXT 1 0 10 "(lambda-t)" }{MPLTEXT 1 0 18 "^(2-1)+LB13/(3-1)*" }{MPLTEXT 1 0 10 "(lambda-t)" }{MPLTEXT 1 0 18 "^(3-1)+LB14/(4-1)*" }{MPLTEXT 1 0 10 "(lambda-t)" }{MPLTEXT 1 0 18 "^(4-1)+LB15/(5-1)*" }{MPLTEXT 1 0 10 "(lambda-t)" }{MPLTEXT 1 0 18 "^(5-1)+LB16/(6-1)*" }{MPLTEXT 1 0 10 "(lambda-t)" }{MPLTEXT 1 0 18 "^(6-1)+LB17/(7-1)*" }{MPLTEXT 1 0 10 "(lambda-t)" }{MPLTEXT 1 0 6 "^(7-1)" }{MPLTEXT 1 0 1 "-" }{MPLTEXT 1 0 97 "s10/h/(lambda-t)*Lt-B12*(lambda-t)^(2-1-1)*Lt-B13*(lambda-t)^ (3-1-1)*Lt-B14*(lambda-t)^(4-1-1)*Lt" }{MPLTEXT 1 0 3 ");\n" }{MPLTEXT 1 0 23 "Lpsi2T := exp((s20/h*ln" }{MPLTEXT 1 0 10 "(lambda-t)" } {MPLTEXT 1 0 15 "+B20+B22/(2-1)*" }{MPLTEXT 1 0 10 "(lambda-t)" } {MPLTEXT 1 0 17 "^(2-1)+B23/(3-1)*" }{MPLTEXT 1 0 10 "(lambda-t)" } {MPLTEXT 1 0 17 "^(3-1)+B24/(4-1)*" }{MPLTEXT 1 0 10 "(lambda-t)" } {MPLTEXT 1 0 17 "^(4-1)+B25/(5-1)*" }{MPLTEXT 1 0 10 "(lambda-t)" } {MPLTEXT 1 0 17 "^(5-1)+B26/(6-1)*" }{MPLTEXT 1 0 10 "(lambda-t)" } {MPLTEXT 1 0 17 "^(6-1)+B27/(7-1)*" }{MPLTEXT 1 0 10 "(lambda-t)" } {MPLTEXT 1 0 9 "^(7-1)))\n" }{MPLTEXT 1 0 11 "*(Ls20/h*ln" }{MPLTEXT 1 0 10 "(lambda-t)" }{MPLTEXT 1 0 17 "+LB20+LB22/(2-1)*" }{MPLTEXT 1 0 10 "(lambda-t)" }{MPLTEXT 1 0 18 "^(2-1)+LB23/(3-1)*" }{MPLTEXT 1 0 10 "(lambda-t)" }{MPLTEXT 1 0 18 "^(3-1)+LB24/(4-1)*" }{MPLTEXT 1 0 10 "(lambda-t)" }{MPLTEXT 1 0 18 "^(4-1)+LB25/(5-1)*" }{MPLTEXT 1 0 10 "(lambda-t)" }{MPLTEXT 1 0 18 "^(5-1)+LB26/(6-1)*" }{MPLTEXT 1 0 10 "(lambda-t)" }{MPLTEXT 1 0 18 "^(6-1)+LB27/(7-1)*" }{MPLTEXT 1 0 10 "(lambda-t)" }{MPLTEXT 1 0 6 "^(7-1)" }{MPLTEXT 1 0 98 "-s20/h/(lam bda-t)*Lt-B22*(lambda-t)^(2-1-1)*Lt-B23*(lambda-t)^(3-1-1)*Lt-B24*(lam bda-t)^(4-1-1)*Lt" }{MPLTEXT 1 0 3 ");\n" }{MPLTEXT 1 0 22 "psi1T:=exp (logPsi1T);\n" }{MPLTEXT 1 0 22 "psi2T:=exp(logPsi2T);\n" }{MPLTEXT 1 0 35 "dpsi1dlambdaT:=diff(psi1T,lambda):\n" }{MPLTEXT 1 0 35 "dpsi2dla mbdaT:=diff(psi2T,lambda):\n" }{MPLTEXT 1 0 39 "d2psi1dlambda2T:=diff( psi1T,lambda$2):\n" }{MPLTEXT 1 0 39 "d2psi2dlambda2T:=diff(psi2T,lamb da$2):\n" }{MPLTEXT 1 0 11 "VT1:=s10*ln" }{MPLTEXT 1 0 10 "(lambda-t)" }{MPLTEXT 1 0 2 ";\n" }{MPLTEXT 1 0 11 "VT2:=s20*ln" }{MPLTEXT 1 0 10 "(lambda-t)" }{MPLTEXT 1 0 2 ";\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 69 "WronskianLambdaT:=h*factor(psi1T*dpsi2dlambdaT-psi2T*dpsi1dlam bdaT):\n" }{MPLTEXT 1 0 112 "WronskianLambdabisT:=h*simplify(factor( ( diff(logPsi2T,lambda)-diff(logPsi1T,lambda))*exp(logPsi1T+logPsi2T))): \n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 95 "WronskianTildeLambdaT:=h^3* factor(dpsi2dlambdaT*d2psi1dlambda2T-dpsi1dlambdaT*d2psi2dlambda2T):" }{MPLTEXT 1 0 1 "\n" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I)logPsi1TG6\",2 *(I$s10GF$\"\"\"I\"hGF$!\"\"-I#lnG6$%*protectedGI(_syslibGF$6#,&I'lamb daGF$F(I\"tGF$F*F(F(I$B10GF$F(*&I$B12GF$F(F1F(F(*&I$B13GF$F(F1\"\"##F( F9*&I$B14GF$F(F1\"\"$#F(F=*&I$B15GF$F(F1\"\"%#F(FA*&I$B16GF$F(F1\"\"&# F(FE*&I$B17GF$F(F1\"\"'#F(FI" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I)logPs i2TG6\",2*(I$s20GF$\"\"\"I\"hGF$!\"\"-I#lnG6$%*protectedGI(_syslibGF$6 #,&I'lambdaGF$F(I\"tGF$F*F(F(I$B20GF$F(*&I$B22GF$F(F1F(F(*&I$B23GF$F(F 1\"\"##F(F9*&I$B24GF$F(F1\"\"$#F(F=*&I$B25GF$F(F1\"\"%#F(FA*&I$B26GF$F (F1\"\"&#F(FE*&I$B27GF$F(F1\"\"'#F(FI" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I*Llogpsi1TG6\",:*(I%Ls10GF$\"\"\"I\"hGF$!\"\"-I#lnG6$%*protectedGI( _syslibGF$6#,&I'lambdaGF$F(I\"tGF$F*F(F(I%LB10GF$F(*&I%LB12GF$F(F1F(F( *&I%LB13GF$F(F1\"\"##F(F9*&I%LB14GF$F(F1\"\"$#F(F=*&I%LB15GF$F(F1\"\"% #F(FA*&I%LB16GF$F(F1!\"&\"\"&*&I%LB17GF$F(F1\"\"'#F(FI**I$s10GF$F(F)F* F1F*I#LtGF$F(F**&I$B12GF$F(FMF(F**(I$B13GF$F(F1F(FMF(F**(I$B14GF$F(F1F 9FMF(F*" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I*Llogpsi2TG6\",:*(I%Ls20GF$ \"\"\"I\"hGF$!\"\"-I#lnG6$%*protectedGI(_syslibGF$6#,&I'lambdaGF$F(I\" tGF$F*F(F(I%LB20GF$F(*&I%LB22GF$F(F1F(F(*&I%LB23GF$F(F1\"\"##F(F9*&I%L B24GF$F(F1\"\"$#F(F=*&I%LB25GF$F(F1\"\"%#F(FA*&I%LB26GF$F(F1\"\"&#F(FE *&I%LB27GF$F(F1\"\"'#F(FI**I$s20GF$F(F)F*F1F*I#LtGF$F(F**&I$B22GF$F(FM F(F**(I$B23GF$F(F1F(FMF(F**(I$B24GF$F(F1F9FMF(F*" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I'Lpsi1TG6\"*&-I$expG6$%*protectedGI(_syslibGF$6#,2*(I$s 10GF$\"\"\"I\"hGF$!\"\"-I#lnGF(6#,&I'lambdaGF$F/I\"tGF$F1F/F/I$B10GF$F /*&I$B12GF$F/F5F/F/*&I$B13GF$F/F5\"\"##F/F=*&I$B14GF$F/F5\"\"$#F/FA*&I $B15GF$F/F5\"\"%#F/FE*&I$B16GF$F/F5\"\"&#F/FI*&I$B17GF$F/F5\"\"'#F/FMF /,:*(I%Ls10GF$F/F0F1F2F/F/I%LB10GF$F/*&I%LB12GF$F/F5F/F/*&I%LB13GF$F/F 5F=F>*&I%LB14GF$F/F5FAFB*&I%LB15GF$F/F5FEFF*&I%LB16GF$F/F5FIFJ*&I%LB17 GF$F/F5FMFN**F.F/F0F1F5F1I#LtGF$F/F1*&F:F/FjnF/F1*(FI'Lpsi2TG6\"*&-I$exp G6$%*protectedGI(_syslibGF$6#,2*(I$s20GF$\"\"\"I\"hGF$!\"\"-I#lnGF(6#, &I'lambdaGF$F/I\"tGF$F1F/F/I$B20GF$F/*&I$B22GF$F/F5F/F/*&I$B23GF$F/F5 \"\"##F/F=*&I$B24GF$F/F5\"\"$#F/FA*&I$B25GF$F/F5\"\"%#F/FE*&I$B26GF$F/ F5\"\"&#F/FI*&I$B27GF$F/F5\"\"'#F/FMF/,:*(I%Ls20GF$F/F0F1F2F/F/I%LB20G F$F/*&I%LB22GF$F/F5F/F/*&I%LB23GF$F/F5F=F>*&I%LB24GF$F/F5FAFB*&I%LB25G F$F/F5FEFF*&I%LB26GF$F/F5FIFJ*&I%LB27GF$F/F5FMFN**F.F/F0F1F5F1I#LtGF$F /F1*&F:F/FjnF/F1*(FI&psi1TG6\"-I$expG6$%*protectedGI(_syslibGF$6#,2*(I$s 10GF$\"\"\"I\"hGF$!\"\"-I#lnGF'6#,&I'lambdaGF$F.I\"tGF$F0F.F.I$B10GF$F .*&I$B12GF$F.F4F.F.*&I$B13GF$F.F4\"\"##F.F<*&I$B14GF$F.F4\"\"$#F.F@*&I $B15GF$F.F4\"\"%#F.FD*&I$B16GF$F.F4\"\"&#F.FH*&I$B17GF$F.F4\"\"'#F.FL" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I&psi2TG6\"-I$expG6$%*protectedGI(_s yslibGF$6#,2*(I$s20GF$\"\"\"I\"hGF$!\"\"-I#lnGF'6#,&I'lambdaGF$F.I\"tG F$F0F.F.I$B20GF$F.*&I$B22GF$F.F4F.F.*&I$B23GF$F.F4\"\"##F.F<*&I$B24GF$ F.F4\"\"$#F.F@*&I$B25GF$F.F4\"\"%#F.FD*&I$B26GF$F.F4\"\"&#F.FH*&I$B27G F$F.F4\"\"'#F.FL" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I$VT1G6\"*&I$s10GF$ \"\"\"-I#lnG6$%*protectedGI(_syslibGF$6#,&I'lambdaGF$F'I\"tGF$!\"\"F'" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I$VT2G6\"*&I$s20GF$\"\"\"-I#lnG6$%*p rotectedGI(_syslibGF$6#,&I'lambdaGF$F'I\"tGF$!\"\"F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "L22T:=factor(h*simplify(diff(WronskianLam bdabisT,lambda)/WronskianLambdabisT)):\n" }{MPLTEXT 1 0 43 "L22TOrderl ambdaMinus3:=factor(residue(L22T*" }{MPLTEXT 1 0 10 "(lambda-t)" } {MPLTEXT 1 0 15 "^2,lambda=t));\n" }{MPLTEXT 1 0 43 "L22TOrderlambdaMi nus2:=factor(residue(L22T*" }{MPLTEXT 1 0 10 "(lambda-t)" }{MPLTEXT 1 0 15 "^1,lambda=t));\n" }{MPLTEXT 1 0 43 "L22TOrderlambdaMinus1:=facto r(residue(L22T*" }{MPLTEXT 1 0 10 "(lambda-t)" }{MPLTEXT 1 0 15 "^0,la mbda=t));\n" }{MPLTEXT 1 0 38 "L22TOrderlambda0:=factor(residue(L22T*" }{MPLTEXT 1 0 10 "(lambda-t)" }{MPLTEXT 1 0 18 "^(-1),lambda=t));\n" }{MPLTEXT 1 0 38 "L22TOrderlambda1:=factor(residue(L22T*" }{MPLTEXT 1 0 10 "(lambda-t)" }{MPLTEXT 1 0 18 "^(-2),lambda=t)):\n" }{MPLTEXT 1 0 38 "L22TOrderlambda2:=factor(residue(L22T*" }{MPLTEXT 1 0 10 "(lambd a-t)" }{MPLTEXT 1 0 17 "^(-3),lambda=t)):" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I6L22TOrderlambdaMinus3G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I6L22TOrderlambdaMinus2G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I6L22TOrderlambdaMinus1G6\",(I$s10GF$\"\"\"I\"hGF$!\"\"I $s20GF$F'" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I1L22TOrderlambda0G6\"*(I \"hGF$\"\"\",.*&I$B12GF$F'F&F'F'*&F*F'I$s10GF$F'F'*&F*F'I$s20GF$F'!\" \"*&I$B22GF$F'F&F'F/*&F1F'F,F'F'*&F1F'F.F'F/F',&F,F'F.F/F/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 251 59 "We get that L_\{2,2\} behaves at lambda= t like (s20+s10-h)/" }{TEXT 251 10 "(lambda-t)" }{TEXT 251 7 " +O(1) \n" }{TEXT 251 85 " Combining with the asympotics at infinity and the \+ apparent singularity we get that " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 251 20 "L_\{2,2\}=(s20+s10-h)/" }{TEXT 251 10 "(lambda-t)" }{TEXT 251 65 " -(t12+t22)*lambda-(t11+t21) +(h-s20-s10-t10-t20)/(lambda-q)\n " }{TEXT 251 4 "i.e." }{TEXT 216 1 "\n" }{TEXT 217 7 "L_\{2,2\}" } {TEXT 218 1 "=" }{TEXT 219 11 "P_1(lambda)" }{TEXT 220 1 "+" }{TEXT 221 12 "h/(lambda-q)" }{TEXT 222 1 "-" }{TEXT 223 2 "h/" }{TEXT 224 10 "(lambda-t)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "L22:" } {MPLTEXT 1 0 1 "=" }{MPLTEXT 1 0 10 "P1(lambda)" }{MPLTEXT 1 0 1 "+" } {MPLTEXT 1 0 12 "h/(lambda-q)" }{MPLTEXT 1 0 1 "-" }{MPLTEXT 1 0 2 "h/ " }{MPLTEXT 1 0 11 "(lambda-t);" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I$L2 2G6\",.*&,&I$s10GF$\"\"\"I$s20GF$F)F),&I'lambdaGF$F)I\"tGF$!\"\"F.F)I$ t11GF$F.I$t21GF$F.*&,&I$t12GF$F.I$t22GF$F.F)F,F)F)*&I\"hGF$F),&F,F)I\" qGF$F.F.F)*&F6F)F+F.F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "L 21T:=factor(simplify(WronskianTildeLambdaT/WronskianLambdabisT)):\n" } {MPLTEXT 1 0 68 "L21TOrderlambdaMinus5:=factor(residue(L21T*(lambda-t) ^4,lambda=t));\n" }{MPLTEXT 1 0 43 "L21TOrderlambdaMinus4:=factor(resi due(L21T*" }{MPLTEXT 1 0 10 "(lambda-t)" }{MPLTEXT 1 0 15 "^3,lambda=t ));\n" }{MPLTEXT 1 0 43 "L21TOrderlambdaMinus3:=factor(residue(L21T*" }{MPLTEXT 1 0 10 "(lambda-t)" }{MPLTEXT 1 0 15 "^2,lambda=t));\n" } {MPLTEXT 1 0 43 "L21TOrderlambdaMinus2:=factor(residue(L21T*" } {MPLTEXT 1 0 10 "(lambda-t)" }{MPLTEXT 1 0 15 "^1,lambda=t));\n" } {MPLTEXT 1 0 43 "L21TOrderlambdaMinus1:=factor(residue(L21T*" } {MPLTEXT 1 0 10 "(lambda-t)" }{MPLTEXT 1 0 15 "^0,lambda=t));\n" } {MPLTEXT 1 0 38 "L21TOrderlambda0:=factor(residue(L21T*" }{MPLTEXT 1 0 10 "(lambda-t)" }{MPLTEXT 1 0 18 "^(-1),lambda=t));\n" }{MPLTEXT 1 0 38 "L21TOrderlambda1:=factor(residue(L21T*" }{MPLTEXT 1 0 10 "(lambd a-t)" }{MPLTEXT 1 0 18 "^(-2),lambda=t)):\n" }{MPLTEXT 1 0 38 "L21TOrd erlambda2:=factor(residue(L21T*" }{MPLTEXT 1 0 10 "(lambda-t)" } {MPLTEXT 1 0 17 "^(-3),lambda=t)):" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I 6L21TOrderlambdaMinus5G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I6L2 1TOrderlambdaMinus4G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I6L21TO rderlambdaMinus3G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I6L21TOrde rlambdaMinus2G6\",$*&I$s10GF$\"\"\"I$s20GF$F(!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I6L21TOrderlambdaMinus1G6\",$*(I\"hGF$\"\"\",.*(I$B12GF $F(F'F(I$s20GF$F(F(*(F+F(I$s10GF$F(F,F(F(*&F+F(F,\"\"#!\"\"*(I$B22GF$F (F'F(F.F(F1*&F3F(F.F0F(*(F3F(F.F(F,F(F1F(,&F.F(F,F1F1F1" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I1L21TOrderlambda0G6\"*(I\"hGF$\"\"\",D*(I$B12GF$ \"\"#F&F+I$s20GF$F'F'**F*F'I$B22GF$F'F&F+I$s10GF$F'!\"\"**F*F'F.F'F&F+ F,F'F0**F*F'F.F'F&F'F/F+F0*,F*F'F.F'F&F'F/F'F,F'F+**F*F'F.F'F&F'F,F+F0 *(F.F+F&F+F/F'F'**I$B13GF$F'F&F'F/F'F,F'!\"#*(F7F'F&F'F,F+F+*(F7F'F/F+ F,F'F0*(F7F'F/F'F,F+F+*&F7F'F,\"\"$F0*(I$B23GF$F'F&F'F/F+F+**F?F'F&F'F /F'F,F'F8*&F?F'F/F=F0*(F?F'F/F+F,F'F+*(F?F'F/F'F,F+F0F',&F/F'F,F0F8" } }}{EXCHG {PARA 0 "" 0 "" {TEXT 251 54 "We get that L_\{2,1\} behaves a t lambda=0 like -s10s20/" }{TEXT 251 10 "(lambda-t)" }{TEXT 251 8 "^2 + O(1/" }{TEXT 251 10 "(lambda-t)" }{TEXT 251 2 ")\n" }{TEXT 251 84 "C ombining with the asympotics at infinity and the apparent singularity \+ we get that " }{TEXT 251 1 "\n" }{TEXT 251 82 "L_\{2,1\}=-t22*t11*lam bda^2-(t21*t12+t22*t11)*lambda-(t12*t20+t22*t10+t11*t21+h*t12)" }{TEXT 251 8 "-s10s20/" }{TEXT 251 10 "(lambda-t)" }{TEXT 251 3 "^2 " }{TEXT 251 3 "+C/" }{TEXT 251 10 "(lambda-t)" }{TEXT 251 2 " " }{TEXT 251 19 "- p*h/(lambda-q) \n" }{TEXT 251 5 "i.e.\n" }{TEXT 225 23 "L_\{2,1 \}=-P_2(\\lambda)+P" }{TEXT 226 4 "012/" }{TEXT 227 10 "(lambda-t)" } {TEXT 228 1 " " }{TEXT 229 3 "+C/" }{TEXT 230 10 "(lambda-t)" }{TEXT 231 12 " -h*t12 + -h" }{TEXT 232 13 "*p/(lambda-q)" }{TEXT 233 1 " " } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "L21:=-P2(lambda)+P" } {MPLTEXT 1 0 4 "012/" }{MPLTEXT 1 0 10 "(lambda-t)" }{MPLTEXT 1 0 4 " \+ +H/" }{MPLTEXT 1 0 10 "(lambda-t)" }{MPLTEXT 1 0 25 " -h*t12 -h*p/(lam bda-q);\n" }{MPLTEXT 1 0 31 "L21:=-tdP2(lambda)+H/(lambda-t)" } {MPLTEXT 1 0 24 " -h*t12 -h*p/(lambda-q);" }{MPLTEXT 1 0 1 " " } {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I$L21G6\",8*(I$s10G F$\"\"\"I$s20GF$F(,&I'lambdaGF$F(I\"tGF$!\"\"!\"#F-*&I$t12GF$F(F'F(F(* &F0F(F)F(F(*&F0F(I$t10GF$F(F(*&F3F(I$t22GF$F(F-*&I$t11GF$F(I$t21GF$F(F -*&,&*&F7F(F5F(F(*&F0F(F8F(F(F(F+F(F-*(F0F(F5F(F+\"\"#F-*&I\"HGF$F(F*F -F(*&I\"hGF$F(F0F(F-*(FBF(I\"pGF$F(,&F+F(I\"qGF$F-F-F-" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I$L21G6\",8*(I$s10GF$\"\"\"I$s20GF$F(,&I'lambdaGF$F (I\"tGF$!\"\"!\"#F-*&I$t12GF$F(F'F(F(*&F0F(F)F(F(*&F0F(I$t10GF$F(F(*&F 3F(I$t22GF$F(F-*&I$t11GF$F(I$t21GF$F(F-*&,&*&F7F(F5F(F(*&F0F(F8F(F(F(F +F(F-*(F0F(F5F(F+\"\"#F-*&I\"HGF$F(F*F-F(*&I\"hGF$F(F0F(F-*(FBF(I\"pGF $F(,&F+F(I\"qGF$F-F-F-" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 249 36 "Auxi liary matrix A in the oper gauge" }{TEXT 249 0 "" }}{EXCHG }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "WronskianLInfty:=factor(psi1Infty*L psi2Infty-psi2Infty*Lpsi1Infty):\n" }{MPLTEXT 1 0 47 "WronskianLT:=fac tor(psi1T*Lpsi2T-psi2T*Lpsi1T):" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 66 "A12Infty:=factor(simplify(WronskianLInfty/WronskianLambdaInfty)): \n" }{MPLTEXT 1 0 54 "A12T:=factor(simplify(WronskianLT/WronskianLambd aT)):\n" }{MPLTEXT 1 0 48 "Y1Infty:=h*factor(dpsi1dlambdaInfty/psi1Inf ty):\n" }{MPLTEXT 1 0 48 "Y2Infty:=h*factor(dpsi2dlambdaInfty/psi2Inft y):\n" }{MPLTEXT 1 0 36 "Y1T:=h*factor(dpsi1dlambdaT/psi1T):\n" } {MPLTEXT 1 0 35 "Y2T:=h*factor(dpsi2dlambdaT/psi2T):" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 39 "Z1Infty:=factor(Lpsi1Infty/psi1Infty):\n" } {MPLTEXT 1 0 39 "Z2Infty:=factor(Lpsi2Infty/psi2Infty):\n" }{MPLTEXT 1 0 27 "Z1T:=factor(Lpsi1T/psi1T):\n" }{MPLTEXT 1 0 26 "Z2T:=factor(Lp si2T/psi2T):" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 68 "A12bisInfty:=fact or(simplify((Z2Infty-Z1Infty)/(Y2Infty-Y1Infty))):\n" }{MPLTEXT 1 0 47 "A12bisT:=factor(simplify((Z2T-Z1T)/(Y2T-Y1T))):" }{MPLTEXT 1 0 1 " \n" }{MPLTEXT 1 0 83 "A11Infty:=factor(simplify( (Y2Infty*Z1Infty-Y1In fty*Z2Infty)/(Y2Infty-Y1Infty) )):\n" }{MPLTEXT 1 0 55 "A11T:=factor(s implify( (Y2T*Z1T-Y1T*Z2T)/(Y2T-Y1T) )):\n" }{MPLTEXT 1 0 1 "\n" } {MPLTEXT 1 0 40 "factor(simplify(A12bisInfty-A12Infty));\n" }{MPLTEXT 1 0 32 "factor(simplify(A12bisT-A12T));\n" }{MPLTEXT 1 0 31 "factor(si mplify(A12bisT-A12T));" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 205 81 "The deformation operator is defined as h*(alpha12 \\partial_ \{t_\{\\infty^\{(1)\},2\}\}+ " }{TEXT 205 7 "alpha22" }{TEXT 205 31 " \+ \\partial_\{t_\{\\infty^\{(2)\},2\}\}+" }{TEXT 205 7 "alpha11" }{TEXT 205 31 "\\partial_\{t_\{\\infty^\{(1)\},1\}\}+ " }{TEXT 205 7 "alpha21 " }{TEXT 205 29 "\\partial_\{t_\{\\infty^\{(2)\},1\}\}" }{TEXT 205 16 " +Lt *partial_t)" }{TEXT 205 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "Lt20:=0:\n" }{MPLTEXT 1 0 8 "Lt10:=0:" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 17 "Lt12:=h*alpha12:\n" }{MPLTEXT 1 0 8 "Lt22:=h* " }{MPLTEXT 1 0 7 "alpha22" }{MPLTEXT 1 0 2 ":\n" }{MPLTEXT 1 0 8 "Lt1 1:=h*" }{MPLTEXT 1 0 7 "alpha11" }{MPLTEXT 1 0 2 ":\n" }{MPLTEXT 1 0 8 "Lt21:=h*" }{MPLTEXT 1 0 8 "alpha21:" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 9 "Ls10:=0:\n" }{MPLTEXT 1 0 9 "Ls20:=0:\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "A12InftyLambda3:=factor(-residue(A12Infty/lamb da^4,lambda=infinity));\n" }{MPLTEXT 1 0 70 "A12InftyLambda2:=factor(- residue(A12Infty/lambda^3,lambda=infinity));\n" }{MPLTEXT 1 0 70 "A12I nftyLambda1:=factor(-residue(A12Infty/lambda^2,lambda=infinity));\n" } {MPLTEXT 1 0 70 "A12InftyLambda0:=factor(-residue(A12Infty/lambda^1,la mbda=infinity));\n" }{MPLTEXT 1 0 74 "A12InftyLambdaMinus1:=factor(-re sidue(A12Infty/lambda^0,lambda=infinity)):" }{MPLTEXT 1 0 0 "" }} {PARA 11 "" 1 "" {XPPMATH 20 ">I0A12InftyLambda3G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I0A12InftyLambda2G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I0A12InftyLambda1G6\",$*&,&I(alpha12GF$\"\"\"I(alpha22GF $!\"\"F),&I$t22GF$F+I$t12GF$F)F+#F)\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I0A12InftyLambda0G6\",$*&,2*&I(alpha11GF$\"\"\"I$t12GF$F*\"\"#*&F )F*I$t22GF$F*!\"#*&I(alpha12GF$F*I$t11GF$F*!\"\"*&F1F*I$t21GF$F*F**&I( alpha21GF$F*F+F*F/*&F7F*F.F*F,*&I(alpha22GF$F*F2F*F**&F:F*F5F*F3F*,&F. F3F+F*F/#F*F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "A12InftyLa mbda0bis:=(1/2)*(t21-t11)/(" }{MPLTEXT 1 0 3 "t12" }{MPLTEXT 1 0 27 "- t22)^2*(alpha12-alpha22)+(" }{MPLTEXT 1 0 7 "alpha11" }{MPLTEXT 1 0 1 "-" }{MPLTEXT 1 0 7 "alpha21" }{MPLTEXT 1 0 3 ")/(" }{MPLTEXT 1 0 3 "t 12" }{MPLTEXT 1 0 7 "-t22):\n" }{MPLTEXT 1 0 16 "simplify(series(" } {MPLTEXT 1 0 16 "A12InftyLambda0-" }{MPLTEXT 1 0 27 "A12InftyLambda0bi s,alpha));" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 63 "A12TLambdaMinus3:=factor(residue(A12T*(lambd a-t)^2,lambda=t));\n" }{MPLTEXT 1 0 38 "A12TLambdaMinus2:=factor(resid ue(A12T*" }{MPLTEXT 1 0 10 "(lambda-t)" }{MPLTEXT 1 0 15 "^1,lambda=t) );\n" }{MPLTEXT 1 0 38 "A12TLambdaMinus1:=factor(residue(A12T*" } {MPLTEXT 1 0 10 "(lambda-t)" }{MPLTEXT 1 0 15 "^0,lambda=t));\n" } {MPLTEXT 1 0 33 "A12TLambda0:=factor(residue(A12T*" }{MPLTEXT 1 0 10 " (lambda-t)" }{MPLTEXT 1 0 18 "^(-1),lambda=t));\n" }{MPLTEXT 1 0 62 "A 12ZeroLambda1:=factor(residue(A12Zero*lambda^(-2),lambda=0));" } {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I1A12TLambdaMinus3G 6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I1A12TLambdaMinus2G6\"\"\"! " }}{PARA 11 "" 1 "" {XPPMATH 20 ">I1A12TLambdaMinus1G6\"\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 ">I,A12TLambda0G6\",$*&I#LtGF$\"\"\"I\"hG F$!\"\"F*" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I/A12ZeroLambda1G6\"\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {TEXT 205 21 "A_\{1,2\} behaves like " } {XPPEDIT 2 0 "Typesetting:-mrow(Typesetting:-mo(\" \", mathvariant = \+ \"normal\", fence = \"false\", separator = \"false\", stretchy = \"fal se\", symmetric = \"false\", largeop = \"false\", movablelimits = \"fa lse\", accent = \"false\", lspace = \"0.0em\", rspace = \"0.0em\"), Ty pesetting:-mrow(Typesetting:-mo(\"&uminus0;\", mathvariant = \"normal \", fence = \"false\", separator = \"false\", stretchy = \"false\", sy mmetric = \"false\", largeop = \"false\", movablelimits = \"false\", a ccent = \"false\", lspace = \"0.2222222em\", rspace = \"0.2222222em\") , Typesetting:-mfrac(Typesetting:-mrow(Typesetting:-mi(\"Lt\", italic \+ = \"true\", mathvariant = \"italic\")), Typesetting:-mrow(Typesetting: -mi(\"h\", italic = \"true\", mathvariant = \"italic\")), linethicknes s = \"1\", denomalign = \"center\", numalign = \"center\", bevelled = \+ \"false\")), Typesetting:-mi(\"\"));" "-I%mrowG6#/I+modulenameG6\"I,Ty pesettingGI(_syslibGF'6%-I#moGF$6-Q\"~F'/%,mathvariantGQ'normalF'/%&fe nceGQ&falseF'/%*separatorGF4/%)stretchyGF4/%*symmetricGF4/%(largeopGF4 /%.movablelimitsGF4/%'accentGF4/%'lspaceGQ&0.0emF'/%'rspaceGFC-F#6$-F, 6-Q*&uminus0;F'F/F2F5F7F9F;F=F?/FBQ,0.2222222emF'/FEFL-I&mfracGF$6(-F# 6#-I#miGF$6%Q#LtF'/%'italicGQ%trueF'/F0Q'italicF'-F#6#-FT6%Q\"hF'FWFZ/ %.linethicknessGQ\"1F'/%+denomalignGQ'centerF'/%)numalignGF`o/%)bevell edGF4-FT6#Q!F'" }{TEXT 205 71 " at \\lambda=t and A_\{1,2\}=(alpha-alp ha22)/2/(tinfty12-tinfty22)*lambda+" }{TEXT 205 17 "(1/2)*(t21-t11)/(" }{TEXT 205 51 "t12-t22)^2*(alpha12-alpha22)+(alpha11-alpha21)/(t12" } {TEXT 205 8 "-t22) + " }{TEXT 205 31 "O(1/lambda) at \\lambda->\\infty \n" }{TEXT 205 18 "Thus, we get that\n" }{TEXT 234 55 "A_\{1,2\}=(alph a12-alpha22)/2/(tinfty12-tinfty22)*lambda+" }{TEXT 235 19 "nu0+q*mu/(l ambda-q)" }{TEXT 236 1 "\n" }{TEXT 237 4 "nu0=" }{TEXT 238 68 "(1/2)*( t21-t11)/(t12-t22)^2*(alpha12-alpha22)+(alpha11-alpha21)/(t12" }{TEXT 239 6 "-t22) " }{TEXT 240 1 " " }{TEXT 241 1 "\n" }{TEXT 242 8 "avec m u=" }{TEXT 243 17 "(1/2)*(t21-t11)/(" }{TEXT 244 51 "t12-t22)^2*(alpha 12-alpha22)+(alpha11-alpha21)/(t12" }{TEXT 245 5 "-t22)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "nuMinus1:=(alpha12-alpha22)/(2*(-t2 2+t12));\n" }{MPLTEXT 1 0 5 "nu0:=" }{MPLTEXT 1 0 17 "(1/2)*(t21-t11)/ (" }{MPLTEXT 1 0 51 "t12-t22)^2*(alpha12-alpha22)+(alpha11-alpha21)/(t 12" }{MPLTEXT 1 0 6 "-t22);" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 17 "A1 2Form:=lambda->" }{MPLTEXT 1 0 9 "nuMinus1*" }{MPLTEXT 1 0 7 "lambda+" }{MPLTEXT 1 0 3 "nu0" }{MPLTEXT 1 0 15 "+mu/(lambda-q);" }{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I)nuMinus1G6\"*&,&I(alpha12GF$ \"\"\"I(alpha22GF$!\"\"F(,&I$t22GF$!\"#I$t12GF$\"\"#F*" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I$nu0G6\",&*(,&I$t21GF$\"\"\"I$t11GF$!\"\"F),&I$t22 GF$F+I$t12GF$F)!\"#,&I(alpha12GF$F)I(alpha22GF$F+F)#F)\"\"#*&,&I(alpha 11GF$F)I(alpha21GF$F+F)F,F+F)" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I(A12F ormG6\"f*6#I'lambdaGF$F$6$I)operatorGF$I&arrowGF$F$,(*&I)nuMinus1GF$\" \"\"9$F.F.I$nu0GF$F.*&I#muGF$F.,&F/F.I\"qGF$!\"\"F5F.F$F$F$" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "mu:=factor(solve(" }{MPLTEXT 1 0 10 "A12Form(t)" }{MPLTEXT 1 0 13 "=-Lt/h,mu)):\n" }{MPLTEXT 1 0 118 "mubis:=(1/2)*(t*(t12-t22)-t11+t21)*(q-t)/(-t22+t12)^2*(alpha12-al pha22)+(alpha11-alpha21)*(q-t)/(t12-t22)+(q-t)*Lt/h;\n" }{MPLTEXT 1 0 33 "factor(series(mu-mubis,alpha=0));" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I&mubisG6\",(**,(*&I\"tGF$\" \"\",&I$t22GF$!\"\"I$t12GF$F*F*F*I$t11GF$F-I$t21GF$F*F*,&F)F-I\"qGF$F* F*F+!\"#,&I(alpha12GF$F*I(alpha22GF$F-F*#F*\"\"#*(,&I(alpha11GF$F*I(al pha21GF$F-F*F1F*F+F-F**(F1F*I#LtGF$F*I\"hGF$F-F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "A12For m:=" }{MPLTEXT 1 0 19 "nuMinus1*lambda+nu0" }{MPLTEXT 1 0 18 "+mubis/( lambda-q):" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 31 "series(simplify(ser ies(A12Form-" }{MPLTEXT 1 0 23 "A12InftyLambda1*lambda-" }{MPLTEXT 1 0 32 "A12InftyLambda0,lambda=infinity)" }{MPLTEXT 1 0 22 "),lambda=inf inity,2);\n" }{MPLTEXT 1 0 56 "series(simplify(series(A12Form,lambda=t ,2)),lambda=t,1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I\"OG6$%*protected GI(_syslibG6\"6#*$I'lambdaGF'!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "+' ,&I'lambdaG6\"\"\"\"I\"tGF%!\"\",$*&I#LtGF%F&I\"hGF%F(F(\"\"!-I\"OG6$% *protectedGI(_syslibGF%6#F&F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 205 49 "Asymptotics of A_\{1,1\} at infinity and \\lambda=t." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "A11InftyLambda3:=factor(-residue(A1 1Infty/lambda^4,lambda=infinity));\n" }{MPLTEXT 1 0 70 "A11InftyLambda 2:=factor(-residue(A11Infty/lambda^3,lambda=infinity));\n" }{MPLTEXT 1 0 70 "A11InftyLambda1:=factor(-residue(A11Infty/lambda^2,lambda=infi nity));\n" }{MPLTEXT 1 0 70 "A11InftyLambda0:=factor(-residue(A11Infty /lambda^1,lambda=infinity));\n" }{MPLTEXT 1 0 75 "A11InftyLambdaMinus1 :=factor(-residue(A11Infty/lambda^0,lambda=infinity)):\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 63 "A11TLambdaMinus3:=factor(residue(A11T*(lamb da-t)^2,lambda=t));\n" }{MPLTEXT 1 0 38 "A11TLambdaMinus2:=factor(resi due(A11T*" }{MPLTEXT 1 0 10 "(lambda-t)" }{MPLTEXT 1 0 15 "^1,lambda=t ));\n" }{MPLTEXT 1 0 38 "A11TLambdaMinus1:=factor(residue(A11T*" } {MPLTEXT 1 0 10 "(lambda-t)" }{MPLTEXT 1 0 15 "^0,lambda=t));\n" } {MPLTEXT 1 0 33 "A11TLambda0:=factor(residue(A11T*" }{MPLTEXT 1 0 10 " (lambda-t)" }{MPLTEXT 1 0 18 "^(-1),lambda=t));\n" }{MPLTEXT 1 0 33 "A 11TLambda1:=factor(residue(A11T*" }{MPLTEXT 1 0 10 "(lambda-t)" } {MPLTEXT 1 0 17 "^(-2),lambda=t)):" }{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I0A11InftyLambda3G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I0A11InftyLambda2G6\",$*&,&*&I(alpha12GF$\"\"\"I$t22GF$F *F**&I(alpha22GF$F*I$t12GF$F*!\"\"F*,&F+F/F.F*F/#F*\"\"#" }}{PARA 11 " " 1 "" {XPPMATH 20 ">I0A11InftyLambda1G6\",$*&,2*(I(alpha11GF$\"\"\"I$ t12GF$F*I$t22GF$F*\"\"#*&F)F*F,F-!\"#*(I(alpha12GF$F*I$t11GF$F*F,F*!\" \"*(F1F*F+F*I$t21GF$F*F**&I(alpha21GF$F*F+F-F/*(F7F*F+F*F,F*F-*(I(alph a22GF$F*F2F*F,F*F**(F:F*F+F*F5F*F3F*,&F,F3F+F*F/#F*F-" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I0A11InftyLambda0G6\",$*&,`o*(I%LA10GF$\"\"\"I$t12GF $\"\"#I$t22GF$F*F,*(F)F*F+F*F-F,!\"%*&F)F*F-\"\"$F,*&I%LA20GF$F*F+F1! \"#*(F3F*F+F,F-F*\"\"%*(F3F*F+F*F-F,F4**I(alpha11GF$F*I$t11GF$F*F+F*F- F*F,*(F9F*F:F*F-F,F4*(F9F*F+F,I$t21GF$F*F4**F9F*F+F*F=F*F-F*F,*(I(alph a12GF$F*I\"hGF$F*F+F,!\"\"**F@F*FAF*F+F*F-F*F***F@F*I$t10GF$F*F+F*F-F* F**(F@F*FEF*F-F,FB*(F@F*F:F,F-F*FB**F@F*F:F*F+F*F=F*F***F@F*F:F*F=F*F- F*F**(F@F*F+F,I$t20GF$F*FB**F@F*F+F*FKF*F-F*F**(F@F*F+F*F=F,FB**I(alph a21GF$F*F:F*F+F*F-F*F4*(FOF*F:F*F-F,F,*(FOF*F+F,F=F*F,**FOF*F+F*F=F*F- F*F4*(I(alpha22GF$F*FAF*F+F,F***FTF*FAF*F+F*F-F*FB**FTF*FEF*F+F*F-F*FB *(FTF*FEF*F-F,F**(FTF*F:F,F-F*F***FTF*F:F*F+F*F=F*FB**FTF*F:F*F=F*F-F* FB*(FTF*F+F,FKF*F***FTF*F+F*FKF*F-F*FB*(FTF*F+F*F=F,F*F*,&F-FBF+F*!\"$ #FBF," }}{PARA 11 "" 1 "" {XPPMATH 20 ">I1A11TLambdaMinus3G6\"\"\"!" } }{PARA 11 "" 1 "" {XPPMATH 20 ">I1A11TLambdaMinus2G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I1A11TLambdaMinus1G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I,A11TLambda0G6\",$*&,&*&I%LB10GF$\"\"\"I$s20GF$F*F** &I%LB20GF$F*I$s10GF$F*!\"\"F*,&F.F*F+F/F/F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "A11InftyLambda1bis:=(1/2)*(" }{MPLTEXT 1 0 8 "t12* t21-" }{MPLTEXT 1 0 10 "t11*t22)/(" }{MPLTEXT 1 0 3 "t12" }{MPLTEXT 1 0 54 "-t22)^2*(alpha12-alpha22)+ (alpha11*t22-alpha21*t12)/(" } {MPLTEXT 1 0 3 "t12" }{MPLTEXT 1 0 7 "-t22):\n" }{MPLTEXT 1 0 69 "fact or(simplify(series(A11InftyLambda1-A11InftyLambda1bis,alpha22)));" }} {PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 205 52 "A_\{1,1\} behaves like O(1) en \\lambda=t and A_\{1,1\}=" } {TEXT 205 152 "(alpha12*t22-alpha22*t12)/(2*(t12-t22))*lambda^2+(1/2)* (t12*t21-t11*t22)/(t12-t22)^2*(alpha12-alpha22)+ (alpha11*t22-alpha21* t12)/(t12-t22)*lambda+ O(1)" }{TEXT 205 23 " quand \\lambda->\\infty\n " }{TEXT 205 6 "Thus,\n" }{TEXT 246 51 "A_\{1,1\}=c2*lambda^2 +c1*lamb da+ c0+ rho/(lambda-q)\n" }{TEXT 247 53 "with c_2=(alpha12*t22-alpha22 *t12)/(2*(t12-t22)) and\n" }{TEXT 248 93 "c1=(1/2)*(t12*t21-t11*t22)/( t12-t22)^2*(alpha12-alpha22)+ (alpha11*t22-alpha21*t12)/(t12-t22)" } {TEXT 205 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "c2:=(alpha 12*t22-alpha22*t12)/(2*(" }{MPLTEXT 1 0 3 "t12" }{MPLTEXT 1 0 8 "-t22) );\n" }{MPLTEXT 1 0 4 "c1:=" }{MPLTEXT 1 0 91 "(1/2)*(t12*t21-t11*t22) /(t12-t22)^2*(alpha12-alpha22)+ (alpha11*t22-alpha21*t12)/(t12-t22);" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 46 "A11Form:=c2*lambda^2+c1*lambda+ rho/(lambda-q);" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I#c2G6\"*&,&*&I(alph a12GF$\"\"\"I$t22GF$F)F)*&I(alpha22GF$F)I$t12GF$F)!\"\"F),&F*!\"#F-\" \"#F." }}{PARA 11 "" 1 "" {XPPMATH 20 ">I#c1G6\",&*(,&*&I$t11GF$\"\"\" I$t22GF$F*!\"\"*&I$t12GF$F*I$t21GF$F*F*F*,&F+F,F.F*!\"#,&I(alpha12GF$F *I(alpha22GF$F,F*#F*\"\"#*&,&*&I(alpha11GF$F*F+F*F**&I(alpha21GF$F*F.F *F,F*F0F,F*" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I(A11FormG6\",(*(,&*&I(a lpha12GF$\"\"\"I$t22GF$F*F**&I(alpha22GF$F*I$t12GF$F*!\"\"F*,&F+!\"#F. \"\"#F/I'lambdaGF$F2F**&,&*(,&*&I$t11GF$F*F+F*F/*&F.F*I$t21GF$F*F*F*,& F+F/F.F*F1,&F)F*F-F/F*#F*F2*&,&*&I(alpha11GF$F*F+F*F**&I(alpha21GF$F*F .F*F/F*F " 0 "" {MPLTEXT 1 0 31 "series(simplify(series(A11Form-" } {MPLTEXT 1 0 24 "A11InftyLambda2*lambda^2" }{MPLTEXT 1 0 62 "-A11Infty Lambda1*lambda,lambda=infinity)),lambda=infinity,2);\n" }{MPLTEXT 1 0 56 "series(simplify(series(A11Form,lambda=t,1)),lambda=t,1);" } {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I\"OG6$%*protectedG I(_syslibG6\"6#*$I'lambdaGF'!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "+', &I'lambdaG6\"\"\"\"I\"tGF%!\"\",$*(,**(,&I$t22GF%F(I$t12GF%F&F&,&*&I(a lpha12GF%F&F.F&F(*&I(alpha22GF%F&F/F&F&F&F'\"\"$F&*&,(*&,&*&F4F&I\"qGF %F&F(I(alpha21GF%\"\"#F&F/F=F&*&,&*&,(*&,&F4F&F2F&F&F;F&F&F " 0 "" {MPLTEXT 1 0 0 "" }}}} {MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }