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}{CSTYLE "" -1 249 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 250 "T imes" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 251 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{PSTYLE "" -1 214 1 {CSTYLE "" -1 -1 "Ti mes" 1 12 0 0 0 1 2 2 2 2 1 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 } {CSTYLE "" -1 252 "Courier" 1 12 255 0 0 1 2 1 2 2 2 2 0 0 0 1 } {CSTYLE "" -1 253 "Courier" 1 12 255 0 0 1 2 1 2 2 2 2 0 0 0 1 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT 213 151 "In this Maple sheet, we compute the Lax matrices using the asymptotics of the wave functions \+ and the local diagonalization for the Painlev\351 5 equation." }{TEXT 205 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 205 118 "We first use the exp ression of the coefficients of the spectral curve in terms of the irre gular times and monodromies." }{TEXT 205 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "restart:\n" }{MPLTEXT 1 0 19 "P011 := t010+t020;\n " }{MPLTEXT 1 0 19 "P111 := t110+t120;\n" }{MPLTEXT 1 0 19 "P121 := t1 11+t121;\n" }{MPLTEXT 1 0 19 "P142 := t111*t121;\n" }{MPLTEXT 1 0 29 " P132 := t110*t121+t111*t120;\n" }{MPLTEXT 1 0 19 "P022 := t010*t020;\n " }{MPLTEXT 1 0 59 "CoherenceEquation1:=tinfty10+tinfty20+t010+t020+t1 10+t120;\n" }{MPLTEXT 1 0 20 "CoherenceEquation2:=" }{MPLTEXT 1 0 11 " P012+P112;\n" }{MPLTEXT 1 0 59 "CoherenceEquation3:=t010*t020-tinfty10 *tinfty20+P112+P122;\n" }{MPLTEXT 1 0 40 "P1:=x-> P011/x+P121/(x-1)^2+ P111/(x-1);\n" }{MPLTEXT 1 0 76 "P2:=x-> P022/x^2+P012/x+P142/(x-1)^4+ P132/(x-1)^3+P122/(x-1)^2+P112/(x-1);\n\n" }{MPLTEXT 1 0 38 "P012:=sol ve(CoherenceEquation2,P012);\n" }{MPLTEXT 1 0 37 "P122:=solve(Coherenc eEquation3,P122);" }}{PARA 0 "" 0 "" {TEXT 254 0 "" }{MPLTEXT 1 0 1 " \n" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I%P011G6\",&I%t010GF$\"\"\"I%t020 GF$F'" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I%P111G6\",&I%t110GF$\"\"\"I%t 120GF$F'" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I%P121G6\",&I%t111GF$\"\"\" I%t121GF$F'" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I%P142G6\"*&I%t111GF$\" \"\"I%t121GF$F'" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I%P132G6\",&*&I%t110 GF$\"\"\"I%t121GF$F(F(*&I%t111GF$F(I%t120GF$F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I%P022G6\"*&I%t010GF$\"\"\"I%t020GF$F'" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I3CoherenceEquation1G6\",.I)tinfty10GF$\"\"\"I)tinfty 20GF$F'I%t010GF$F'I%t020GF$F'I%t110GF$F'I%t120GF$F'" }}{PARA 11 "" 1 " " {XPPMATH 20 ">I3CoherenceEquation2G6\",&I%P012GF$\"\"\"I%P112GF$F'" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I3CoherenceEquation3G6\",**&I%t010GF$ \"\"\"I%t020GF$F(F(*&I)tinfty10GF$F(I)tinfty20GF$F(!\"\"I%P112GF$F(I%P 122GF$F(" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I#P1G6\"f*6#I\"xGF$F$6$I)op eratorGF$I&arrowGF$F$,(*&I%P011GF$\"\"\"9$!\"\"F.*&I%P121GF$F.,&F/F.F0 F.!\"#F.*&I%P111GF$F.F3F0F.F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I# P2G6\"f*6#I\"xGF$F$6$I)operatorGF$I&arrowGF$F$,.*&I%P022GF$\"\"\"9$!\" #F.*&I%P012GF$F.F/!\"\"F.*&I%P142GF$F.,&F/F.F3F.!\"%F.*&I%P132GF$F.F6! \"$F.*&I%P122GF$F.F6F0F.*&I%P112GF$F.F6F3F.F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I%P012G6\",$I%P112GF$!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I%P122G6\",(*&I%t010GF$\"\"\"I%t020GF$F(!\"\"*&I)tinfty10GF$F(I)t infty20GF$F(F(I%P112GF$F*" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 255 30 "Ex pression of the Lax matrix L" }}{EXCHG {PARA 210 "" 0 "" {TEXT 256 36 "Study of the asymptotics at infinity" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 182 "logPsi1Infty:=-tinfty10/h*ln(lambda)+A10-A12/(2-1)/l ambda^(2-1)-A13/(3-1)/lambda^(3-1)-A14/(4-1)/lambda^(4-1)-A15/(5-1)/la mbda^(5-1)-A16/(6-1)/lambda^(6-1)-A17/(7-1)/lambda^(7-1) ;\n" } {MPLTEXT 1 0 195 "logPsi2Infty:=-tinfty20/h*ln(lambda)-1*ln(lambda)+A2 0-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)/lambda^(7-1) \+ ;\n" }{MPLTEXT 1 0 191 "Llogpsi1Infty:=-Ltinfty10/h*ln(lambda)+LA10-L A12/(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 0 191 "Llogpsi2Infty:=-Ltinfty20/h*ln(lambda)+LA2 0-LA22/(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 0 310 "Lpsi1Infty := exp(1/h*(-tinfty10*ln(lam bda)+h*A10-h*A12/lambda-1/2*h*A13/lambda^2-1/3*h*A14/lambda^3-1/4*h*A1 5/lambda^4-1/5*h*A16/lambda^5-1/6*h*A17/lambda^6))*1/h*(-Ltinfty10*ln( lambda)+h*LA10-h*LA12/lambda-1/2*h*LA13/lambda^2-1/3*h*LA14/lambda^3-1 /4*h*LA15/lambda^4-1/5*h*LA16/lambda^5-1/6*h*LA17/lambda^6);\n" } {MPLTEXT 1 0 323 "Lpsi2Infty := exp(1/h*(-tinfty20*ln(lambda)-h*ln(lam bda)+h*A20-h*A22/lambda-1/2*h*A23/lambda^2-1/3*h*A24/lambda^3-1/4*h*A2 5/lambda^4-1/5*h*A26/lambda^5-1/6*h*A27/lambda^6))*1/h*(-Ltinfty20*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/lambda^6);\n" } {MPLTEXT 1 0 30 "psi1Infty:=exp(logPsi1Infty);\n" }{MPLTEXT 1 0 30 "ps i2Infty:=exp(logPsi2Infty);\n" }{MPLTEXT 1 0 43 "dpsi1dlambdaInfty:=di ff(psi1Infty,lambda):\n" }{MPLTEXT 1 0 43 "dpsi2dlambdaInfty:=diff(psi 2Infty,lambda):\n" }{MPLTEXT 1 0 47 "d2psi1dlambda2Infty:=diff(psi1Inf ty,lambda$2):\n" }{MPLTEXT 1 0 47 "d2psi2dlambda2Infty:=diff(psi2Infty ,lambda$2):\n" }{MPLTEXT 1 0 30 "Vinfty1:=tinfty10*ln(lambda);\n" } {MPLTEXT 1 0 30 "Vinfty2:=tinfty20*ln(lambda);\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 89 "WronskianLambdaInfty:=h*factor (psi1Infty*dpsi2dlambdaInfty-psi2Infty*dpsi1dlambdaInfty):\n" } {MPLTEXT 1 0 132 "WronskianLambdabisInfty:=h*simplify(factor( (diff(lo gPsi2Infty,lambda)-diff(logPsi1Infty,lambda))*exp(logPsi1Infty+logPsi2 Infty))):\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 115 "WronskianTildeLam bdaInfty:=h^3*factor(dpsi2dlambdaInfty*d2psi1dlambda2Infty-dpsi1dlambd aInfty*d2psi2dlambda2Infty):" }{MPLTEXT 1 0 1 "\n" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I-logPsi1InftyG6\",2*(I)tinfty10GF$\"\"\"I\"hGF$!\"\"-I# lnG6$%*protectedGI(_syslibGF$6#I'lambdaGF$F(F*I$A10GF$F(*&I$A12GF$F(F1 F*F**&I$A13GF$F(F1!\"##F*\"\"#*&I$A14GF$F(F1!\"$#F*\"\"$*&I$A15GF$F(F1 !\"%#F*\"\"%*&I$A16GF$F(F1!\"&#F*\"\"&*&I$A17GF$F(F1!\"'#F*\"\"'" }} {PARA 11 "" 1 "" {XPPMATH 20 ">I-logPsi2InftyG6\",4*(I)tinfty20GF$\"\" \"I\"hGF$!\"\"-I#lnG6$%*protectedGI(_syslibGF$6#I'lambdaGF$F(F*F+F*I$A 20GF$F(*&I$A22GF$F(F1F*F**&I$A23GF$F(F1!\"##F*\"\"#*&I$A24GF$F(F1!\"$# F*\"\"$*&I$A25GF$F(F1!\"%#F*\"\"%*&I$A26GF$F(F1!\"&#F*\"\"&*&I$A27GF$F (F1!\"'#F*\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I.Llogpsi1InftyG6\", 2*(I*Ltinfty10GF$\"\"\"I\"hGF$!\"\"-I#lnG6$%*protectedGI(_syslibGF$6#I 'lambdaGF$F(F*I%LA10GF$F(*&I%LA12GF$F(F1F*F**&I%LA13GF$F(F1!\"##F*\"\" #*&I%LA14GF$F(F1!\"$#F*\"\"$*&I%LA15GF$F(F1!\"%#F*\"\"%*&I%LA16GF$F(F1 !\"&#F*\"\"&*&I%LA17GF$F(F1!\"'#F*\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I.Llogpsi2InftyG6\",2*(I*Ltinfty20GF$\"\"\"I\"hGF$!\"\"-I#lnG6$%* protectedGI(_syslibGF$6#I'lambdaGF$F(F*I%LA20GF$F(*&I%LA22GF$F(F1F*F** &I%LA23GF$F(F1!\"##F*\"\"#*&I%LA24GF$F(F1!\"$#F*\"\"$*&I%LA25GF$F(F1! \"%#F*\"\"%*&I%LA26GF$F(F1!\"&#F*\"\"&*&I%LA27GF$F(F1!\"'#F*\"\"'" }} {PARA 11 "" 1 "" {XPPMATH 20 ">I+Lpsi1InftyG6\"*(-I$expG6$%*protectedG I(_syslibGF$6#*&I\"hGF$!\"\",2*&I)tinfty10GF$\"\"\"-I#lnGF(6#I'lambdaG F$F2F.*&F-F2I$A10GF$F2F2*(F-F2I$A12GF$F2F6F.F.*(F-F2I$A13GF$F2F6!\"##F .\"\"#*(F-F2I$A14GF$F2F6!\"$#F.\"\"$*(F-F2I$A15GF$F2F6!\"%#F.\"\"%*(F- F2I$A16GF$F2F6!\"&#F.\"\"&*(F-F2I$A17GF$F2F6!\"'#F.\"\"'F2F2F-F.,2*&I* Ltinfty10GF$F2F3F2F.*&F-F2I%LA10GF$F2F2*(F-F2I%LA12GF$F2F6F.F.*(F-F2I% LA13GF$F2F6F=F>*(F-F2I%LA14GF$F2F6FBFC*(F-F2I%LA15GF$F2F6FGFH*(F-F2I%L A16GF$F2F6FLFM*(F-F2I%LA17GF$F2F6FQFRF2" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I+Lpsi2InftyG6\"*(-I$expG6$%*protectedGI(_syslibGF$6#*&I\"hGF$!\" \",4*&I)tinfty20GF$\"\"\"-I#lnGF(6#I'lambdaGF$F2F.*&F-F2F3F2F.*&F-F2I$ A20GF$F2F2*(F-F2I$A22GF$F2F6F.F.*(F-F2I$A23GF$F2F6!\"##F.\"\"#*(F-F2I$ A24GF$F2F6!\"$#F.\"\"$*(F-F2I$A25GF$F2F6!\"%#F.\"\"%*(F-F2I$A26GF$F2F6 !\"&#F.\"\"&*(F-F2I$A27GF$F2F6!\"'#F.\"\"'F2F2F-F.,2*&I*Ltinfty20GF$F2 F3F2F.*&F-F2I%LA20GF$F2F2*(F-F2I%LA22GF$F2F6F.F.*(F-F2I%LA23GF$F2F6F>F ?*(F-F2I%LA24GF$F2F6FCFD*(F-F2I%LA25GF$F2F6FHFI*(F-F2I%LA26GF$F2F6FMFN *(F-F2I%LA27GF$F2F6FRFSF2" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I*psi1Inft yG6\"-I$expG6$%*protectedGI(_syslibGF$6#,2*(I)tinfty10GF$\"\"\"I\"hGF$ !\"\"-I#lnGF'6#I'lambdaGF$F.F0I$A10GF$F.*&I$A12GF$F.F4F0F0*&I$A13GF$F. F4!\"##F0\"\"#*&I$A14GF$F.F4!\"$#F0\"\"$*&I$A15GF$F.F4!\"%#F0\"\"%*&I$ A16GF$F.F4!\"&#F0\"\"&*&I$A17GF$F.F4!\"'#F0\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I*psi2InftyG6\"-I$expG6$%*protectedGI(_syslibGF$6#,4*(I) tinfty20GF$\"\"\"I\"hGF$!\"\"-I#lnGF'6#I'lambdaGF$F.F0F1F0I$A20GF$F.*& I$A22GF$F.F4F0F0*&I$A23GF$F.F4!\"##F0\"\"#*&I$A24GF$F.F4!\"$#F0\"\"$*& I$A25GF$F.F4!\"%#F0\"\"%*&I$A26GF$F.F4!\"&#F0\"\"&*&I$A27GF$F.F4!\"'#F 0\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I(Vinfty1G6\"*&I)tinfty10GF$ \"\"\"-I#lnG6$%*protectedGI(_syslibGF$6#I'lambdaGF$F'" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I(Vinfty2G6\"*&I)tinfty20GF$\"\"\"-I#lnG6$%*protecte dGI(_syslibGF$6#I'lambdaGF$F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "L21Infty:=factor(simplify(WronskianTildeLambdaInfty/WronskianLam bdabisInfty)):" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 75 "L21InftyOrderla mbda1:=factor(-residue(L21Infty/lambda^2,lambda=infinity));\n" } {MPLTEXT 1 0 75 "L21InftyOrderlambda0:=factor(-residue(L21Infty/lambda ^1,lambda=infinity));\n" }{MPLTEXT 1 0 80 "L21InftyOrderlambdaMinus1:= factor(-residue(L21Infty/lambda^0,lambda=infinity));\n" }{MPLTEXT 1 0 83 "L21InftyOrderlambdaMinus2:=factor(-residue(L21Infty/lambda^(-1),la mbda=infinity));\n" }{MPLTEXT 1 0 82 "L21InftyOrderlambdaMinus3:=facto r(-residue(L21Infty/lambda^(-2),lambda=infinity));" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I5L21InftyOrderlambda1G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I5L21InftyOrderlambda0G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I:L21InftyOrderlambdaMinus1G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I:L21InftyOrderlambdaMinus2G6\",$*&,&I\"hGF$\"\"\"I)tin fty20GF$F)F)I)tinfty10GF$F)!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I:L 21InftyOrderlambdaMinus3G6\",$*(I\"hGF$\"\"\",0*(I$A12GF$F(F'F(I)tinft y10GF$F(F(*(F+F(F'F(I)tinfty20GF$F(!\"\"*(F+F(F,F(F.F(F(*&F+F(F.\"\"#F /*(I$A22GF$F(F'F(F,F(!\"#*&F4F(F,F2F(*(F4F(F,F(F.F(F/F(,(F'F(F,F/F.F(F /F/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 254 45 "We get that L_\{2,1\} beh aves at infinity like " }{XPPEDIT 2 0 "Typesetting:-mrow(Typesetting:- mo(\"&uminus0;\", mathvariant = \"normal\", fence = \"false\", separat or = \"false\", stretchy = \"false\", symmetric = \"false\", largeop = \"false\", movablelimits = \"false\", accent = \"false\", lspace = \" 0.2222222em\", rspace = \"0.2222222em\"), Typesetting:-mrow(Typesettin g:-mfenced(Typesetting:-mrow(Typesetting:-mi(\"h\", italic = \"true\", mathvariant = \"italic\"), Typesetting:-mo(\"+\", mathvariant = \+ \"normal\", fence = \"false\", separator = \"false\", stretchy = \"fal se\", symmetric = \"false\", largeop = \"false\", movablelimits = \"fa lse\", accent = \"false\", lspace = \"0.2222222em\", rspace = \"0.2222 222em\"), Typesetting:-mi(\"tinfty20\", italic = \"true\", mathvariant = \"italic\")), mathvariant = \"normal\"), Typesetting:-mo(\"&Invisib leTimes;\", mathvariant = \"normal\", fence = \"false\", separator = \+ \"false\", stretchy = \"false\", symmetric = \"false\", largeop = \"fa lse\", movablelimits = \"false\", accent = \"false\", lspace = \"0.0em \", rspace = \"0.0em\"), Typesetting:-mi(\"tinfty10\", italic = \"true \", mathvariant = \"italic\")), Typesetting:-mi(\"\"));" "-I%mrowG6#/I +modulenameG6\"I,TypesettingGI(_syslibGF'6%-I#moGF$6-Q*&uminus0;F'/%,m athvariantGQ'normalF'/%&fenceGQ&falseF'/%*separatorGF4/%)stretchyGF4/% *symmetricGF4/%(largeopGF4/%.movablelimitsGF4/%'accentGF4/%'lspaceGQ,0 .2222222emF'/%'rspaceGFC-F#6%-I(mfencedGF$6$-F#6%-I#miGF$6%Q\"hF'/%'it alicGQ%trueF'/F0Q'italicF'-F,6-Q'+F'F/F2F5F7F9F;F=F?FAFD-FN6%Q)ti nfty20F'FQFTF/-F,6-Q1⁢F'F/F2F5F7F9F;F=F?/FBQ&0.0emF'/FE Fjn-FN6%Q)tinfty10F'FQFT-FN6#Q!F'" }{TEXT 254 23 "/lambda^2+O(1/lambda ^3)" }{TEXT 254 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "L22I nfty:=factor(h*simplify(diff(WronskianLambdabisInfty,lambda)/Wronskian LambdabisInfty)):\n" }{MPLTEXT 1 0 75 "L22InftyOrderlambda1:=factor(-r esidue(L22Infty/lambda^2,lambda=infinity));\n" }{MPLTEXT 1 0 75 "L22In ftyOrderlambda0:=factor(-residue(L22Infty/lambda^1,lambda=infinity)); \n" }{MPLTEXT 1 0 80 "L22InftyOrderlambdaMinus1:=factor(-residue(L22In fty/lambda^0,lambda=infinity));\n" }{MPLTEXT 1 0 83 "L22InftyOrderlamb daMinus2:=factor(-residue(L22Infty/lambda^(-1),lambda=infinity));\n" } {MPLTEXT 1 0 82 "L22InftyOrderlambdaMinus3:=factor(-residue(L22Infty/l ambda^(-2),lambda=infinity)):" }{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I5L22InftyOrderlambda1G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I5L22InftyOrderlambda0G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I:L22InftyOrderlambdaMinus1G6\",(I\"hGF$!\"#I)tinfty10GF $!\"\"I)tinfty20GF$F)" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I:L22InftyOrde rlambdaMinus2G6\",$*(I\"hGF$\"\"\",,*&I$A12GF$F(I)tinfty10GF$F(F(*&F+F (I)tinfty20GF$F(!\"\"*&I$A22GF$F(F'F(!\"#*&F1F(F,F(F(*&F1F(F.F(F/F(,(F 'F(F,F/F.F(F/F/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 254 44 "We get that L _\{2,2\} behaves at infinity like" }{XPPEDIT 2 0 "Typesetting:-mrow(Ty pesetting:-mo(\"(\", mathvariant = \"normal\", fence = \"true\", separ ator = \"false\", stretchy = \"true\", symmetric = \"false\", largeop \+ = \"false\", movablelimits = \"false\", accent = \"false\", lspace = \+ \"0.1666667em\", rspace = \"0.1666667em\"), Typesetting:-mo(\"&uminus0 ;\", mathvariant = \"normal\", fence = \"false\", separator = \"false \", stretchy = \"false\", symmetric = \"false\", largeop = \"false\", \+ movablelimits = \"false\", accent = \"false\", lspace = \"0.2222222em \", rspace = \"0.2222222em\"), Typesetting:-mn(\"2\", mathvariant = \" normal\"), Typesetting:-mo(\" \", mathvariant = \"normal\", fence = \" false\", separator = \"false\", stretchy = \"false\", symmetric = \"fa lse\", largeop = \"false\", movablelimits = \"false\", accent = \"fals e\", lspace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mi(\"h\", i talic = \"true\", mathvariant = \"italic\"), Typesetting:-mo(\"− \", mathvariant = \"normal\", fence = \"false\", separator = \"false\" , stretchy = \"false\", symmetric = \"false\", largeop = \"false\", mo vablelimits = \"false\", accent = \"false\", lspace = \"0.2222222em\", rspace = \"0.2222222em\"), Typesetting:-mi(\"tinfty10\", italic = \"t rue\", mathvariant = \"italic\"), Typesetting:-mo(\"−\", mathvar iant = \"normal\", fence = \"false\", separator = \"false\", stretchy \+ = \"false\", symmetric = \"false\", largeop = \"false\", movablelimits = \"false\", accent = \"false\", lspace = \"0.2222222em\", rspace = \+ \"0.2222222em\"), Typesetting:-mi(\"tinfty20\", italic = \"true\", mat hvariant = \"italic\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI( _syslibGF'6+-I#moGF$6-Q\"(F'/%,mathvariantGQ'normalF'/%&fenceGQ%trueF' /%*separatorGQ&falseF'/%)stretchyGF4/%*symmetricGF7/%(largeopGF7/%.mov ablelimitsGF7/%'accentGF7/%'lspaceGQ,0.1666667emF'/%'rspaceGFD-F,6-Q*& uminus0;F'F//F3F7F5/F9F7F:FF@/FCQ,0.2222222emF'/FFFM-I#mnGF$6$Q\"2F 'F/-F,6-Q\"~F'F/FJF5FKF:FF@/FCQ&0.0emF'/FFFW-I#miGF$6%Q\"hF'/%'ital icGF4/F0Q'italicF'-F,6-Q(−F'F/FJF5FKF:FF@FLFN-FZ6%Q)tinfty10F 'FgnFinF[o-FZ6%Q)tinfty20F'FgnFin" }{TEXT 254 25 ")/lambda+h*O(1/lambd a^2) " }{TEXT 254 0 "" }}}{EXCHG {PARA 209 "" 0 "" {TEXT 257 37 "Study of the asymptotics at lambda= 0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 176 "logPsi1Zero:=t010/h*ln(lambda)+B10+B12/(2-1)*lambda^(2-1)+B1 3/(3-1)*lambda^(3-1)+B14/(4-1)*lambda^(4-1)+B15/(5-1)*lambda^(5-1)+B16 *(6-1)*lambda^(6-1)+B17*(7-1)*lambda^(7-1) ;\n" }{MPLTEXT 1 0 176 "lo gPsi2Zero:=t020/h*ln(lambda)+B20+B22/(2-1)*lambda^(2-1)+B23/(3-1)*lamb da^(3-1)+B24/(4-1)*lambda^(4-1)+B25/(5-1)*lambda^(5-1)+B26*(6-1)*lambd a^(6-1)+B27*(7-1)*lambda^(7-1) ;\n" }{MPLTEXT 1 0 185 "Llogpsi1Zero:= Lt010/h*ln(lambda)+LB10+LB12/(2-1)*lambda^(2-1)+LB13/(3-1)*lambda^(3-1 )+LB14/(4-1)*lambda^(4-1)+LB15/(5-1)*lambda^(5-1)+LB16*(6-1)*lambda^(6 -1)+LB17*(7-1)*lambda^(7-1) ;\n" }{MPLTEXT 1 0 185 "Llogpsi2Zero:=Lt0 20/h*ln(lambda)+LB20+LB22/(2-1)*lambda^(2-1)+LB23/(3-1)*lambda^(3-1)+L B24/(4-1)*lambda^(4-1)+LB25/(5-1)*lambda^(5-1)+LB26*(6-1)*lambda^(6-1) +LB27*(7-1)*lambda^(7-1) ;\n" }{MPLTEXT 1 0 180 "Lpsi1Zero := exp((t0 10/h*ln(lambda)+B10+B12/(2-1)*lambda^(2-1)+B13/(3-1)*lambda^(3-1)+B14/ (4-1)*lambda^(4-1)+B15/(5-1)*lambda^(5-1)+B16*(6-1)*lambda^(6-1)+B17*( 7-1)*lambda^(7-1)))\n" }{MPLTEXT 1 0 172 "*(Lt010/h*ln(lambda)+LB10+LB 12/(2-1)*lambda^(2-1)+LB13/(3-1)*lambda^(3-1)+LB14/(4-1)*lambda^(4-1)+ LB15/(5-1)*lambda^(5-1)+LB16*(6-1)*lambda^(6-1)+LB17*(7-1)*lambda^(7-1 ));\n" }{MPLTEXT 1 0 180 "Lpsi2Zero := exp((t020/h*ln(lambda)+B20+B22/ (2-1)*lambda^(2-1)+B23/(3-1)*lambda^(3-1)+B24/(4-1)*lambda^(4-1)+B25/( 5-1)*lambda^(5-1)+B26*(6-1)*lambda^(6-1)+B27*(7-1)*lambda^(7-1)))\n" } {MPLTEXT 1 0 172 "*(Lt020/h*ln(lambda)+LB20+LB22/(2-1)*lambda^(2-1)+LB 23/(3-1)*lambda^(3-1)+LB24/(4-1)*lambda^(4-1)+LB25/(5-1)*lambda^(5-1)+ LB26*(6-1)*lambda^(6-1)+LB27*(7-1)*lambda^(7-1));\n" }{MPLTEXT 1 0 28 "psi1Zero:=exp(logPsi1Zero);\n" }{MPLTEXT 1 0 28 "psi2Zero:=exp(logPsi 2Zero);\n" }{MPLTEXT 1 0 41 "dpsi1dlambdaZero:=diff(psi1Zero,lambda): \n" }{MPLTEXT 1 0 41 "dpsi2dlambdaZero:=diff(psi2Zero,lambda):\n" } {MPLTEXT 1 0 45 "d2psi1dlambda2Zero:=diff(psi1Zero,lambda$2):\n" } {MPLTEXT 1 0 45 "d2psi2dlambda2Zero:=diff(psi2Zero,lambda$2):\n" } {MPLTEXT 1 0 25 "VZero1:=t010*ln(lambda);\n" }{MPLTEXT 1 0 25 "VZero2: =t020*ln(lambda);\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" } {MPLTEXT 1 0 84 "WronskianLambdaZero:=h*factor(psi1Zero*dpsi2dlambdaZe ro-psi2Zero*dpsi1dlambdaZero):\n" }{MPLTEXT 1 0 127 "WronskianLambdabi sZero:=h*simplify(factor( (diff(logPsi2Zero,lambda)-diff(logPsi1Zero,l ambda))*exp(logPsi1Zero+logPsi2Zero))):\n" }{MPLTEXT 1 0 1 "\n" } {MPLTEXT 1 0 111 "WronskianTildeLambdaZero:=h^3*factor(dpsi2dlambdaZer o*d2psi1dlambda2Zero-dpsi1dlambdaZero*d2psi2dlambda2Zero):\n" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I,logPsi1ZeroG6\",2*(I%t010GF$\"\"\"I\"hGF$! \"\"-I#lnG6$%*protectedGI(_syslibGF$6#I'lambdaGF$F(F(I$B10GF$F(*&I$B12 GF$F(F1F(F(*&I$B13GF$F(F1\"\"##F(F7*&I$B14GF$F(F1\"\"$#F(F;*&I$B15GF$F (F1\"\"%#F(F?*&I$B16GF$F(F1\"\"&FC*&I$B17GF$F(F1\"\"'FF" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I,logPsi2ZeroG6\",2*(I%t020GF$\"\"\"I\"hGF$!\"\"-I #lnG6$%*protectedGI(_syslibGF$6#I'lambdaGF$F(F(I$B20GF$F(*&I$B22GF$F(F 1F(F(*&I$B23GF$F(F1\"\"##F(F7*&I$B24GF$F(F1\"\"$#F(F;*&I$B25GF$F(F1\" \"%#F(F?*&I$B26GF$F(F1\"\"&FC*&I$B27GF$F(F1\"\"'FF" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I-Llogpsi1ZeroG6\",2*(I&Lt010GF$\"\"\"I\"hGF$!\"\"-I#ln G6$%*protectedGI(_syslibGF$6#I'lambdaGF$F(F(I%LB10GF$F(*&I%LB12GF$F(F1 F(F(*&I%LB13GF$F(F1\"\"##F(F7*&I%LB14GF$F(F1\"\"$#F(F;*&I%LB15GF$F(F1 \"\"%#F(F?*&I%LB16GF$F(F1\"\"&FC*&I%LB17GF$F(F1\"\"'FF" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I-Llogpsi2ZeroG6\",2*(I&Lt020GF$\"\"\"I\"hGF$!\"\"- I#lnG6$%*protectedGI(_syslibGF$6#I'lambdaGF$F(F(I%LB20GF$F(*&I%LB22GF$ F(F1F(F(*&I%LB23GF$F(F1\"\"##F(F7*&I%LB24GF$F(F1\"\"$#F(F;*&I%LB25GF$F (F1\"\"%#F(F?*&I%LB26GF$F(F1\"\"&FC*&I%LB27GF$F(F1\"\"'FF" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I*Lpsi1ZeroG6\"*&-I$expG6$%*protectedGI(_syslibG F$6#,2*(I%t010GF$\"\"\"I\"hGF$!\"\"-I#lnGF(6#I'lambdaGF$F/F/I$B10GF$F/ *&I$B12GF$F/F5F/F/*&I$B13GF$F/F5\"\"##F/F;*&I$B14GF$F/F5\"\"$#F/F?*&I$ B15GF$F/F5\"\"%#F/FC*&I$B16GF$F/F5\"\"&FG*&I$B17GF$F/F5\"\"'FJF/,2*(I& Lt010GF$F/F0F1F2F/F/I%LB10GF$F/*&I%LB12GF$F/F5F/F/*&I%LB13GF$F/F5F;F<* &I%LB14GF$F/F5F?F@*&I%LB15GF$F/F5FCFD*&I%LB16GF$F/F5FGFG*&I%LB17GF$F/F 5FJFJF/" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I*Lpsi2ZeroG6\"*&-I$expG6$%* protectedGI(_syslibGF$6#,2*(I%t020GF$\"\"\"I\"hGF$!\"\"-I#lnGF(6#I'lam bdaGF$F/F/I$B20GF$F/*&I$B22GF$F/F5F/F/*&I$B23GF$F/F5\"\"##F/F;*&I$B24G F$F/F5\"\"$#F/F?*&I$B25GF$F/F5\"\"%#F/FC*&I$B26GF$F/F5\"\"&FG*&I$B27GF $F/F5\"\"'FJF/,2*(I&Lt020GF$F/F0F1F2F/F/I%LB20GF$F/*&I%LB22GF$F/F5F/F/ *&I%LB23GF$F/F5F;F<*&I%LB24GF$F/F5F?F@*&I%LB25GF$F/F5FCFD*&I%LB26GF$F/ F5FGFG*&I%LB27GF$F/F5FJFJF/" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I)psi1Ze roG6\"-I$expG6$%*protectedGI(_syslibGF$6#,2*(I%t010GF$\"\"\"I\"hGF$!\" \"-I#lnGF'6#I'lambdaGF$F.F.I$B10GF$F.*&I$B12GF$F.F4F.F.*&I$B13GF$F.F4 \"\"##F.F:*&I$B14GF$F.F4\"\"$#F.F>*&I$B15GF$F.F4\"\"%#F.FB*&I$B16GF$F. F4\"\"&FF*&I$B17GF$F.F4\"\"'FI" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I)psi 2ZeroG6\"-I$expG6$%*protectedGI(_syslibGF$6#,2*(I%t020GF$\"\"\"I\"hGF$ !\"\"-I#lnGF'6#I'lambdaGF$F.F.I$B20GF$F.*&I$B22GF$F.F4F.F.*&I$B23GF$F. F4\"\"##F.F:*&I$B24GF$F.F4\"\"$#F.F>*&I$B25GF$F.F4\"\"%#F.FB*&I$B26GF$ F.F4\"\"&FF*&I$B27GF$F.F4\"\"'FI" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I'V Zero1G6\"*&I%t010GF$\"\"\"-I#lnG6$%*protectedGI(_syslibGF$6#I'lambdaGF $F'" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I'VZero2G6\"*&I%t020GF$\"\"\"-I# lnG6$%*protectedGI(_syslibGF$6#I'lambdaGF$F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "L22Zero:=factor(h*simplify(diff(WronskianLambdabi sZero,lambda)/WronskianLambdabisZero)):\n" }{MPLTEXT 1 0 70 "L22ZeroOr derlambdaMinus3:=factor(residue(L22Zero*lambda^2,lambda=0));\n" } {MPLTEXT 1 0 70 "L22ZeroOrderlambdaMinus2:=factor(residue(L22Zero*lamb da^1,lambda=0));\n" }{MPLTEXT 1 0 70 "L22ZeroOrderlambdaMinus1:=factor (residue(L22Zero*lambda^0,lambda=0));\n" }{MPLTEXT 1 0 68 "L22ZeroOrde rlambda0:=factor(residue(L22Zero*lambda^(-1),lambda=0));\n" }{MPLTEXT 1 0 68 "L22ZeroOrderlambda1:=factor(residue(L22Zero*lambda^(-2),lambda =0)):\n" }{MPLTEXT 1 0 67 "L22ZeroOrderlambda2:=factor(residue(L22Zero *lambda^(-3),lambda=0)):" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I9L22ZeroOr derlambdaMinus3G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I9L22ZeroOr derlambdaMinus2G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I9L22ZeroOr derlambdaMinus1G6\",(I%t010GF$\"\"\"I\"hGF$!\"\"I%t020GF$F'" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I4L22ZeroOrderlambda0G6\"*(I\"hGF$\"\"\",.*&I $B12GF$F'F&F'F'*&F*F'I%t010GF$F'F'*&F*F'I%t020GF$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 254 71 "We get that L_\{2,2\} behaves at lambda=0 like (t010+t02 0-h)/lambda +O(1)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "L21Zer o:=factor(simplify(WronskianTildeLambdaZero/WronskianLambdabisZero)): \n" }{MPLTEXT 1 0 70 "L21ZeroOrderlambdaMinus5:=factor(residue(L21Zero *lambda^4,lambda=0));\n" }{MPLTEXT 1 0 70 "L21ZeroOrderlambdaMinus4:=f actor(residue(L21Zero*lambda^3,lambda=0));\n" }{MPLTEXT 1 0 70 "L21Zer oOrderlambdaMinus3:=factor(residue(L21Zero*lambda^2,lambda=0));\n" } {MPLTEXT 1 0 70 "L21ZeroOrderlambdaMinus2:=factor(residue(L21Zero*lamb da^1,lambda=0));\n" }{MPLTEXT 1 0 70 "L21ZeroOrderlambdaMinus1:=factor (residue(L21Zero*lambda^0,lambda=0));\n" }{MPLTEXT 1 0 68 "L21ZeroOrde rlambda0:=factor(residue(L21Zero*lambda^(-1),lambda=0));\n" }{MPLTEXT 1 0 68 "L21ZeroOrderlambda1:=factor(residue(L21Zero*lambda^(-2),lambda =0)):\n" }{MPLTEXT 1 0 67 "L21ZeroOrderlambda2:=factor(residue(L21Zero *lambda^(-3),lambda=0)):" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I9L21ZeroOr derlambdaMinus5G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I9L21ZeroOr derlambdaMinus4G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I9L21ZeroOr derlambdaMinus3G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I9L21ZeroOr derlambdaMinus2G6\",$*&I%t010GF$\"\"\"I%t020GF$F(!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I9L21ZeroOrderlambdaMinus1G6\",$*(I\"hGF$\"\"\",.*( I$B12GF$F(F'F(I%t020GF$F(F(*(F+F(I%t010GF$F(F,F(F(*&F+F(F,\"\"#!\"\"*( I$B22GF$F(F'F(F.F(F1*&F3F(F.F0F(*(F3F(F.F(F,F(F1F(,&F.F(F,F1F1F1" }} {PARA 11 "" 1 "" {XPPMATH 20 ">I4L21ZeroOrderlambda0G6\"*(I\"hGF$\"\" \",D*(I$B12GF$\"\"#F&F+I%t020GF$F'F'**F*F'I$B22GF$F'F&F+I%t010GF$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 254 77 "We get that L_ \{2,1\} behaves at lambda=0 like -t010*t020/lambda^2+ O(1/lambda)" }}} {EXCHG {PARA 210 "" 0 "" {TEXT 256 36 "Study of the asymptotics at lam bda=1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "logPsi1Un:=-" } {MPLTEXT 1 0 17 "t111/h/(lambda-1)" }{MPLTEXT 1 0 190 "+t110/h*ln(lamb da-1)+C10+C12/(2-1)*(lambda-1)^(2-1)+C13/(3-1)*(lambda-1)^(3-1)+C14/(4 -1)*(lambda-1)^(4-1)+C15/(5-1)*(lambda-1)^(5-1)+C16*(6-1)*(lambda-1)^( 6-1)+C17*(7-1)*(lambda-1)^(7-1) ;\n" }{MPLTEXT 1 0 12 "logPsi2Un:=-" }{MPLTEXT 1 0 18 "t121/h/(lambda-1)+" }{MPLTEXT 1 0 189 "t120/h*ln(lam bda-1)+C20+C22/(2-1)*(lambda-1)^(2-1)+C23/(3-1)*(lambda-1)^(3-1)+C24/( 4-1)*(lambda-1)^(4-1)+C25/(5-1)*(lambda-1)^(5-1)+C26*(6-1)*(lambda-1)^ (6-1)+C27*(7-1)*(lambda-1)^(7-1) ;\n" }{MPLTEXT 1 0 14 "Llogpsi1Un:=- L" }{MPLTEXT 1 0 17 "t111/h/(lambda-1)" }{MPLTEXT 1 0 198 "+Lt110/h*ln (lambda-1)+LC10+LC12/(2-1)*(lambda-1)^(2-1)+LC13/(3-1)*(lambda-1)^(3-1 )+LC14/(4-1)*(lambda-1)^(4-1)+LC15/(5-1)*(lambda-1)^(5-1)+LC16*(6-1)*( lambda-1)^(6-1)+LC17*(7-1)*(lambda-1)^(7-1) ;\n" }{MPLTEXT 1 0 13 "Ll ogpsi2Un:=-" }{MPLTEXT 1 0 1 "L" }{MPLTEXT 1 0 17 "t121/h/(lambda-1)" }{MPLTEXT 1 0 1 "+" }{MPLTEXT 1 0 197 "Lt120/h*ln(lambda-1)+LC20+LC22/ (2-1)*(lambda-1)^(2-1)+LC23/(3-1)*(lambda-1)^(3-1)+LC24/(4-1)*(lambda- 1)^(4-1)+LC25/(5-1)*(lambda-1)^(5-1)+LC26*(6-1)*(lambda-1)^(6-1)+LC27* (7-1)*(lambda-1)^(7-1) ;\n" }{MPLTEXT 1 0 17 "Lpsi1Un := exp((-" } {MPLTEXT 1 0 18 "t111/h/(lambda-1)+" }{MPLTEXT 1 0 188 "t110/h*ln(lamb da-1)+C10+C12/(2-1)*(lambda-1)^(2-1)+C13/(3-1)*(lambda-1)^(3-1)+C14/(4 -1)*(lambda-1)^(4-1)+C15/(5-1)*(lambda-1)^(5-1)+C16*(6-1)*(lambda-1)^( 6-1)+C17*(7-1)*(lambda-1)^(7-1)))\n" }{MPLTEXT 1 0 3 "*(-" }{MPLTEXT 1 0 1 "L" }{MPLTEXT 1 0 18 "t111/h/(lambda-1)+" }{MPLTEXT 1 0 196 "Lt1 10/h*ln(lambda-1)+LC10+LC12/(2-1)*(lambda-1)^(2-1)+LC13/(3-1)*(lambda- 1)^(3-1)+LC14/(4-1)*(lambda-1)^(4-1)+LC15/(5-1)*(lambda-1)^(5-1)+LC16* (6-1)*(lambda-1)^(6-1)+LC17*(7-1)*(lambda-1)^(7-1));\n" }{MPLTEXT 1 0 17 "Lpsi2Un := exp((-" }{MPLTEXT 1 0 18 "t121/h/(lambda-1)+" }{MPLTEXT 1 0 188 "t120/h*ln(lambda-1)+C20+C22/(2-1)*(lambda-1)^(2-1)+C23/(3-1) *(lambda-1)^(3-1)+C24/(4-1)*(lambda-1)^(4-1)+C25/(5-1)*(lambda-1)^(5-1 )+C26*(6-1)*(lambda-1)^(6-1)+C27*(7-1)*(lambda-1)^(7-1)))\n" }{MPLTEXT 1 0 3 "*(-" }{MPLTEXT 1 0 19 "Lt121/h/(lambda-1)+" }{MPLTEXT 1 0 196 "Lt120/h*ln(lambda-1)+LC20+LC22/(2-1)*(lambda-1)^(2-1)+LC23/(3-1)*(lam bda-1)^(3-1)+LC24/(4-1)*(lambda-1)^(4-1)+LC25/(5-1)*(lambda-1)^(5-1)+L C26*(6-1)*(lambda-1)^(6-1)+LC27*(7-1)*(lambda-1)^(7-1));\n" }{MPLTEXT 1 0 24 "psi1Un:=exp(logPsi1Un);\n" }{MPLTEXT 1 0 24 "psi2Un:=exp(logPs i2Un);\n" }{MPLTEXT 1 0 37 "dpsi1dlambdaUn:=diff(psi1Un,lambda):\n" } {MPLTEXT 1 0 37 "dpsi2dlambdaUn:=diff(psi2Un,lambda):\n" }{MPLTEXT 1 0 41 "d2psi1dlambda2Un:=diff(psi1Un,lambda$2):\n" }{MPLTEXT 1 0 41 "d2 psi2dlambda2Un:=diff(psi2Un,lambda$2):\n" }{MPLTEXT 1 0 1 "\n" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 74 "WronskianLambdaUn:=h*factor(psi1 Un*dpsi2dlambdaUn-psi2Un*dpsi1dlambdaUn):\n" }{MPLTEXT 1 0 117 "Wronsk ianLambdabisUn:=h*simplify(factor( (diff(logPsi2Un,lambda)-diff(logPsi 1Un,lambda))*exp(logPsi1Un+logPsi2Un))):\n" }{MPLTEXT 1 0 1 "\n" } {MPLTEXT 1 0 101 "WronskianTildeLambdaUn:=h^3*factor(dpsi2dlambdaUn*d2 psi1dlambda2Un-dpsi1dlambdaUn*d2psi2dlambda2Un):\n" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I*logPsi1UnG6\",4*(I%t111GF$\"\"\"I\"hGF$!\"\",&I'lambd aGF$F(F*F(F*F**(I%t110GF$F(F)F*-I#lnG6$%*protectedGI(_syslibGF$6#F+F(F (I$C10GF$F(*&I$C12GF$F(F+F(F(*&I$C13GF$F(F+\"\"##F(F:*&I$C14GF$F(F+\" \"$#F(F>*&I$C15GF$F(F+\"\"%#F(FB*&I$C16GF$F(F+\"\"&FF*&I$C17GF$F(F+\" \"'FI" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I*logPsi2UnG6\",4*(I%t121GF$\" \"\"I\"hGF$!\"\",&I'lambdaGF$F(F*F(F*F**(I%t120GF$F(F)F*-I#lnG6$%*prot ectedGI(_syslibGF$6#F+F(F(I$C20GF$F(*&I$C22GF$F(F+F(F(*&I$C23GF$F(F+\" \"##F(F:*&I$C24GF$F(F+\"\"$#F(F>*&I$C25GF$F(F+\"\"%#F(FB*&I$C26GF$F(F+ \"\"&FF*&I$C27GF$F(F+\"\"'FI" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I+Llogp si1UnG6\",4*(I&Lt111GF$\"\"\"I\"hGF$!\"\",&I'lambdaGF$F(F*F(F*F**(I&Lt 110GF$F(F)F*-I#lnG6$%*protectedGI(_syslibGF$6#F+F(F(I%LC10GF$F(*&I%LC1 2GF$F(F+F(F(*&I%LC13GF$F(F+\"\"##F(F:*&I%LC14GF$F(F+\"\"$#F(F>*&I%LC15 GF$F(F+\"\"%#F(FB*&I%LC16GF$F(F+\"\"&FF*&I%LC17GF$F(F+\"\"'FI" }} {PARA 11 "" 1 "" {XPPMATH 20 ">I+Llogpsi2UnG6\",4*(I&Lt121GF$\"\"\"I\" hGF$!\"\",&I'lambdaGF$F(F*F(F*F**(I&Lt120GF$F(F)F*-I#lnG6$%*protectedG I(_syslibGF$6#F+F(F(I%LC20GF$F(*&I%LC22GF$F(F+F(F(*&I%LC23GF$F(F+\"\"# #F(F:*&I%LC24GF$F(F+\"\"$#F(F>*&I%LC25GF$F(F+\"\"%#F(FB*&I%LC26GF$F(F+ \"\"&FF*&I%LC27GF$F(F+\"\"'FI" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I(Lpsi 1UnG6\"*&-I$expG6$%*protectedGI(_syslibGF$6#,4*(I%t111GF$\"\"\"I\"hGF$ !\"\",&I'lambdaGF$F/F1F/F1F1*(I%t110GF$F/F0F1-I#lnGF(6#F2F/F/I$C10GF$F /*&I$C12GF$F/F2F/F/*&I$C13GF$F/F2\"\"##F/F>*&I$C14GF$F/F2\"\"$#F/FB*&I $C15GF$F/F2\"\"%#F/FF*&I$C16GF$F/F2\"\"&FJ*&I$C17GF$F/F2\"\"'FMF/,4*(I &Lt111GF$F/F0F1F2F1F1*(I&Lt110GF$F/F0F1F6F/F/I%LC10GF$F/*&I%LC12GF$F/F 2F/F/*&I%LC13GF$F/F2F>F?*&I%LC14GF$F/F2FBFC*&I%LC15GF$F/F2FFFG*&I%LC16 GF$F/F2FJFJ*&I%LC17GF$F/F2FMFMF/" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I(L psi2UnG6\"*&-I$expG6$%*protectedGI(_syslibGF$6#,4*(I%t121GF$\"\"\"I\"h GF$!\"\",&I'lambdaGF$F/F1F/F1F1*(I%t120GF$F/F0F1-I#lnGF(6#F2F/F/I$C20G F$F/*&I$C22GF$F/F2F/F/*&I$C23GF$F/F2\"\"##F/F>*&I$C24GF$F/F2\"\"$#F/FB *&I$C25GF$F/F2\"\"%#F/FF*&I$C26GF$F/F2\"\"&FJ*&I$C27GF$F/F2\"\"'FMF/,4 *(I&Lt121GF$F/F0F1F2F1F1*(I&Lt120GF$F/F0F1F6F/F/I%LC20GF$F/*&I%LC22GF$ F/F2F/F/*&I%LC23GF$F/F2F>F?*&I%LC24GF$F/F2FBFC*&I%LC25GF$F/F2FFFG*&I%L C26GF$F/F2FJFJ*&I%LC27GF$F/F2FMFMF/" }}{PARA 11 "" 1 "" {XPPMATH 20 "> I'psi1UnG6\"-I$expG6$%*protectedGI(_syslibGF$6#,4*(I%t111GF$\"\"\"I\"h GF$!\"\",&I'lambdaGF$F.F0F.F0F0*(I%t110GF$F.F/F0-I#lnGF'6#F1F.F.I$C10G F$F.*&I$C12GF$F.F1F.F.*&I$C13GF$F.F1\"\"##F.F=*&I$C14GF$F.F1\"\"$#F.FA *&I$C15GF$F.F1\"\"%#F.FE*&I$C16GF$F.F1\"\"&FI*&I$C17GF$F.F1\"\"'FL" }} {PARA 11 "" 1 "" {XPPMATH 20 ">I'psi2UnG6\"-I$expG6$%*protectedGI(_sys libGF$6#,4*(I%t121GF$\"\"\"I\"hGF$!\"\",&I'lambdaGF$F.F0F.F0F0*(I%t120 GF$F.F/F0-I#lnGF'6#F1F.F.I$C20GF$F.*&I$C22GF$F.F1F.F.*&I$C23GF$F.F1\" \"##F.F=*&I$C24GF$F.F1\"\"$#F.FA*&I$C25GF$F.F1\"\"%#F.FE*&I$C26GF$F.F1 \"\"&FI*&I$C27GF$F.F1\"\"'FL" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "L22Un:=factor(h*simplify(diff(WronskianLambdabisUn,lambda)/Wrons kianLambdabisUn)):\n" }{MPLTEXT 1 0 70 "L22UnOrderlambdaMinus3:=factor (residue(L22Un*(lambda-1)^2,lambda=1));\n" }{MPLTEXT 1 0 70 "L22UnOrde rlambdaMinus2:=factor(residue(L22Un*(lambda-1)^1,lambda=1));\n" } {MPLTEXT 1 0 70 "L22UnOrderlambdaMinus1:=factor(residue(L22Un*(lambda- 1)^0,lambda=1));\n" }{MPLTEXT 1 0 68 "L22UnOrderlambda0:=factor(residu e(L22Un*(lambda-1)^(-1),lambda=1));\n" }{MPLTEXT 1 0 68 "L22UnOrderlam bda1:=factor(residue(L22Un*(lambda-1)^(-2),lambda=1)):\n" }{MPLTEXT 1 0 67 "L22UnOrderlambda2:=factor(residue(L22Un*(lambda-1)^(-3),lambda=1 )):" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I7L22UnOrderlambdaMinus3G6\"\"\" !" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I7L22UnOrderlambdaMinus2G6\",&I%t1 11GF$\"\"\"I%t121GF$F'" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I7L22UnOrderl ambdaMinus1G6\",(I\"hGF$!\"#I%t110GF$\"\"\"I%t120GF$F)" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I2L22UnOrderlambda0G6\"*(I\"hGF$\"\"\",.*&I$C12GF$F 'I%t111GF$F'F'*&F*F'I%t121GF$F'!\"\"*&I$C22GF$F'F+F'F'*&F0F'F-F'F.I%t1 10GF$F'I%t120GF$F.F',&F+F'F-F.F." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 254 43 "We get that L_\{2,2\} behave at lambda=1 like" }{XPPEDIT 2 0 "Type setting:-mrow(Typesetting:-mo(\"(\", mathvariant = \"normal\", fence = \"true\", separator = \"false\", stretchy = \"true\", symmetric = \"f alse\", largeop = \"false\", movablelimits = \"false\", accent = \"fal se\", lspace = \"0.1666667em\", rspace = \"0.1666667em\"), Typesetting :-mi(\"t111\", italic = \"true\", mathvariant = \"italic\"), Typesetti ng:-mo(\"+\", mathvariant = \"normal\", fence = \"false\", separator = \"false\", stretchy = \"false\", symmetric = \"false\", largeop = \"f alse\", movablelimits = \"false\", accent = \"false\", lspace = \"0.22 22222em\", rspace = \"0.2222222em\"), Typesetting:-mi(\"t121\", italic = \"true\", mathvariant = \"italic\"));" "-I%mrowG6#/I+modulenameG6\" I,TypesettingGI(_syslibGF'6&-I#moGF$6-Q\"(F'/%,mathvariantGQ'normalF'/ %&fenceGQ%trueF'/%*separatorGQ&falseF'/%)stretchyGF4/%*symmetricGF7/%( largeopGF7/%.movablelimitsGF7/%'accentGF7/%'lspaceGQ,0.1666667emF'/%'r spaceGFD-I#miGF$6%Q%t111F'/%'italicGF4/F0Q'italicF'-F,6-Q\"+F'F//F3F7F 5/F9F7F:FF@/FCQ,0.2222222emF'/FFFU-FH6%Q%t121F'FKFM" }{TEXT 254 47 ")/(lambda-1)^2+ (t110+t120-2h)/(lambda-1) +O(1)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "L21Un:=factor(simplify(WronskianTildeLambdaUn /WronskianLambdabisUn)):\n" }{MPLTEXT 1 0 70 "L21UnOrderlambdaMinus5:= factor(residue(L21Un*(lambda-1)^4,lambda=1));\n" }{MPLTEXT 1 0 70 "L21 UnOrderlambdaMinus4:=factor(residue(L21Un*(lambda-1)^3,lambda=1));\n" }{MPLTEXT 1 0 70 "L21UnOrderlambdaMinus3:=factor(residue(L21Un*(lambda -1)^2,lambda=1));\n" }{MPLTEXT 1 0 69 "L21UnOrderlambdaMinus2:=factor( residue(L21Un*(lambda-1)^1,lambda=1));" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I7L21UnOrderlambdaMinus5G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 " >I7L21UnOrderlambdaMinus4G6\",$*&I%t111GF$\"\"\"I%t121GF$F(!\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 ">I7L21UnOrderlambdaMinus3G6\",&*&I%t110G F$\"\"\"I%t121GF$F(!\"\"*&I%t111GF$F(I%t120GF$F(F*" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I7L21UnOrderlambdaMinus2G6\",$*&,2**I$C12GF$\"\"\"I\"hG F$F*I%t111GF$F*I%t121GF$F*F**(F)F*F+F*F-\"\"#!\"\"*(I$C22GF$F*F+F*F,F/ F***F2F*F+F*F,F*F-F*F0*(F+F*I%t110GF$F*F-F*F**(F+F*F,F*I%t120GF$F*F0*( F5F*F,F*F7F*F**(F5F*F7F*F-F*F0F*,&F,F*F-F0F0F0" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 254 45 "We get that L_\{2,1\} behave at lambda=1 like " } {XPPEDIT 2 0 "Typesetting:-mrow(Typesetting:-mo(\"&uminus0;\", mathvar iant = \"normal\", fence = \"false\", separator = \"false\", stretchy \+ = \"false\", symmetric = \"false\", largeop = \"false\", movablelimits = \"false\", accent = \"false\", lspace = \"0.2222222em\", rspace = \+ \"0.2222222em\"), Typesetting:-mrow(Typesetting:-mi(\"t111\", italic = \"true\", mathvariant = \"italic\"), Typesetting:-mo(\"&InvisibleTime s;\", mathvariant = \"normal\", fence = \"false\", separator = \"false \", stretchy = \"false\", symmetric = \"false\", largeop = \"false\", \+ movablelimits = \"false\", accent = \"false\", lspace = \"0.0em\", rsp ace = \"0.0em\"), Typesetting:-mi(\"t121\", italic = \"true\", mathvar iant = \"italic\")), Typesetting:-mi(\"\"));" "-I%mrowG6#/I+modulename G6\"I,TypesettingGI(_syslibGF'6%-I#moGF$6-Q*&uminus0;F'/%,mathvariantG Q'normalF'/%&fenceGQ&falseF'/%*separatorGF4/%)stretchyGF4/%*symmetricG F4/%(largeopGF4/%.movablelimitsGF4/%'accentGF4/%'lspaceGQ,0.2222222emF '/%'rspaceGFC-F#6%-I#miGF$6%Q%t111F'/%'italicGQ%trueF'/F0Q'italicF'-F, 6-Q1⁢F'F/F2F5F7F9F;F=F?/FBQ&0.0emF'/FEFU-FI6%Q%t121F'FL FO-FI6#Q!F'" }{TEXT 254 16 "/(lambda-1)^4- (" }{XPPEDIT 2 0 "Typesetti ng:-mrow(Typesetting:-mi(\"\"), Typesetting:-mrow(Typesetting:-mi(\"t1 21\", italic = \"true\", mathvariant = \"italic\"), Typesetting:-mo(\" ⁢\", mathvariant = \"normal\", fence = \"false\", separ ator = \"false\", stretchy = \"false\", symmetric = \"false\", largeop = \"false\", movablelimits = \"false\", accent = \"false\", lspace = \+ \"0.0em\", rspace = \"0.0em\"), Typesetting:-mi(\"t110\", italic = \"t rue\", mathvariant = \"italic\")), Typesetting:-mo(\"+\", mathvar iant = \"normal\", fence = \"false\", separator = \"false\", stretchy \+ = \"false\", symmetric = \"false\", largeop = \"false\", movablelimits = \"false\", accent = \"false\", lspace = \"0.2222222em\", rspace = \+ \"0.2222222em\"), Typesetting:-mrow(Typesetting:-mi(\"t120\", italic = \"true\", mathvariant = \"italic\"), Typesetting:-mo(\"&InvisibleTime s;\", mathvariant = \"normal\", fence = \"false\", separator = \"false \", stretchy = \"false\", symmetric = \"false\", largeop = \"false\", \+ movablelimits = \"false\", accent = \"false\", lspace = \"0.0em\", rsp ace = \"0.0em\"), Typesetting:-mi(\"t111\", italic = \"true\", mathvar iant = \"italic\")), Typesetting:-mi(\"\"));" "-I%mrowG6#/I+modulename G6\"I,TypesettingGI(_syslibGF'6'-I#miGF$6#Q!F'-F#6%-F,6%Q%t121F'/%'ita licGQ%trueF'/%,mathvariantGQ'italicF'-I#moGF$6-Q1⁢F'/F8 Q'normalF'/%&fenceGQ&falseF'/%*separatorGFB/%)stretchyGFB/%*symmetricG FB/%(largeopGFB/%.movablelimitsGFB/%'accentGFB/%'lspaceGQ&0.0emF'/%'rs paceGFQ-F,6%Q%t110F'F4F7-F;6-Q'+F'F>F@FCFEFGFIFKFM/FPQ,0.2222222e mF'/FSFen-F#6%-F,6%Q%t120F'F4F7F:-F,6%Q%t111F'F4F7F+" }{TEXT 254 34 ") /(lambda-1)^3+ O((lambda-1)^\{-2\})" }}}{EXCHG {PARA 211 "> " 0 "" {MPLTEXT 1 0 9 "-residue(" }{MPLTEXT 1 0 3 " (t" }{MPLTEXT 1 0 51 "111 +t121)/(lambda-1)^2+ (t110+t120-2*h)/(lambda-1)" }{MPLTEXT 1 205 74 " + (t010+t020-h)/lambda +h/(lambda-q),lambda=infinity)-CoherenceEquati on1;" }}{PARA 11 "" 1 "" {XPPMATH 20 ",(I\"hG6\"!\"#I)tinfty10GF$!\"\" I)tinfty20GF$F'" }}}{EXCHG {PARA 209 "" 0 "" {TEXT 257 32 "Formulas fo r L_\{2,2\} and L_\{2,1\}" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 254 42 "We \+ have L_\{2,2\} behaves at lambda=1 like (" }{TEXT 254 25 "t111+t121)/( lambda-1)^2+ " }{TEXT 254 33 " (t110+t120-2h)/(lambda-1) +O(1)\n" } {TEXT 254 23 " L_\{2,2\} " }{TEXT 254 11 "behaves at " } {TEXT 254 10 " lambda=0 " }{TEXT 254 4 "like" }{TEXT 254 28 " (t010+t0 20-h)/lambda +O(1)\n" }{TEXT 254 23 " L_\{2,2\} " }{TEXT 254 28 "behaves at lambda=infinity " }{TEXT 254 4 "like" }{TEXT 254 53 "-(tinfty10+tinfty20+2h)/lambda+ O(1/lambda^2) \nThus, " }}{PARA 0 "" 0 "" {TEXT 214 8 "L_\{2,2\}=" }{TEXT 254 1 " " }{TEXT 215 1 "(" } {TEXT 216 1 "t" }{TEXT 217 24 "111+t121)/(lambda-1)^2+ " }{TEXT 218 26 " (t110+t120-2h)/(lambda-1)" }{TEXT 219 2 "+ " }{TEXT 220 35 "(t010 +t020-h)/lambda +h/(lambda-q)" }}{PARA 0 "" 0 "" {TEXT 221 16 " \+ =" }{TEXT 222 47 "P1(lambda)+h/(lambda-q)-h/lambda-2*h/(lambd a-1)" }{TEXT 223 1 "\n" }{TEXT 254 128 "with the additional condition \+ that t110+t120+t010+t020+tinfty10+tinfty20=0 to get the correct asympt otic behavior at infinity. \n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 254 16 "We have L_\{2,1\} " }{TEXT 254 10 "behaves at" }{TEXT 254 10 " lambda =1 " }{TEXT 254 4 "like" }{TEXT 254 81 " -t111*t121/(lambda-1)^4-(t121 *t110+t120*t111)/(lambda-1)^3+ O((lambda-1)^\{-2\}) \n" }{TEXT 254 23 " L_\{2,1\} " }{TEXT 254 10 "behaves at" }{TEXT 254 10 " lambda=0 " }{TEXT 254 4 "like" }{TEXT 254 34 " -t010*t020/lambda^2+ O (1/lambda)\n" }{TEXT 254 23 " L_\{2,1\} " }{TEXT 254 10 "behaves at" }{TEXT 254 17 " lambda=infinity " }{TEXT 254 4 "like" } {TEXT 254 48 " -tinfty10*(tinfty20+h)/lambda^2+ O(1/lambda^3) " }{TEXT 254 6 "\nThus," }}{PARA 0 "" 0 "" {TEXT 224 7 "L_\{2,1\}" }{TEXT 225 1 "=" }{TEXT 226 59 "-t111*t121/(lambda-1)^4-(t121*t110+t120*t111)/(la mbda-1)^3 " }{TEXT 227 15 "-a_2/(lambda-1)" }{TEXT 228 4 " ^2 " }{TEXT 229 66 "-a_1/(lambda-1) -t010*t020/lambda^2 -a_0/lambda - p*h/(lamb da-q)" }{TEXT 230 13 "\n " }{TEXT 231 1 "=" }{TEXT 232 23 " -P2(lambda)+P012/lambda" }{TEXT 233 18 "+P122/(lambda-1)^2" }{TEXT 234 27 "+P112/(lambda-1)-a0/lambda " }{TEXT 235 16 "-a2/(lambda-1)^2" }{TEXT 236 31 " -a1/(lambda-1) -h*p/(lambda-q)" }{TEXT 237 1 "\n" } {TEXT 254 30 "with the additional relation " }{TEXT 238 1 "a" }{TEXT 239 56 "_0+a_1+h*p=0 to ensure the correct behavior at infinity\n" }}} {EXCHG {PARA 212 "> " 0 "" {MPLTEXT 1 0 48 "P1(lambda)+h/(lambda-q)-h/ lambda-2*h/(lambda-1);" }{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 ",.*&,&I%t010G6\"\"\"\"I%t020GF&F'F'I'lambdaGF&!\"\"F'*&,& I%t111GF&F'I%t121GF&F'F',&F)F'F*F'!\"#F'*&,&I%t110GF&F'I%t120GF&F'F'F/ F*F'*&I\"hGF&F',&F)F'I\"qGF&F*F*F'*&F6F'F)F*F**&F6F'F/F*F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "-P2(lambda)+P012/lambda" }{MPLTEXT 1 0 18 "+P122/(lambda-1)^2" }{MPLTEXT 1 0 27 "+P112/(lambda-1)-a0/lamb da " }{MPLTEXT 1 0 16 "-a2/(lambda-1)^2" }{MPLTEXT 1 0 32 " -a1/(lambd a-1) -h*p/(lambda-q);" }{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 ",0*(I%t010G6\"\"\"\"I%t020GF%F&I'lambdaGF%!\"#!\"\"*(I%t111GF%F&I% t121GF%F&,&F(F&F*F&!\"%F**&,&*&I%t110GF%F&F-F&F&*&F,F&I%t120GF%F&F&F&F .!\"$F**&I#a0GF%F&F(F*F**&I#a2GF%F&F.F)F**&I#a1GF%F&F.F*F**(I\"hGF%F&I \"pGF%F&,&F(F&I\"qGF%F*F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "EqCoeffLambdaMinus1Infinity:= residue(-P2(lambda)+P012/lambda" } {MPLTEXT 1 0 18 "+P122/(lambda-1)^2" }{MPLTEXT 1 0 27 "+P112/(lambda-1 )-a0/lambda " }{MPLTEXT 1 0 16 "-a2/(lambda-1)^2" }{MPLTEXT 1 0 49 " - a1/(lambda-1) -h*p/(lambda-q),lambda=infinity);" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 29 "EqCoeffLambdaMinus1Infinity:=" }{MPLTEXT 1 0 1 "-" } {MPLTEXT 1 0 32 "residue((-P2(lambda)+P012/lambda" }{MPLTEXT 1 0 18 "+ P122/(lambda-1)^2" }{MPLTEXT 1 0 27 "+P112/(lambda-1)-a0/lambda " } {MPLTEXT 1 0 16 "-a2/(lambda-1)^2" }{MPLTEXT 1 0 59 " -a1/(lambda-1) - h*p/(lambda-q))*lambda,lambda=infinity) -(" }{MPLTEXT 1 0 23 "-tinfty1 0*(tinfty20+h))" }{MPLTEXT 1 0 1 ";" }}{PARA 11 "" 1 "" {XPPMATH 20 "> II " 0 "" {MPLTEXT 1 0 5 "L22:=" }{MPLTEXT 1 0 49 "P1(l ambda)+h/(lambda-q)-h/lambda-2*h/(lambda-1);\n" }{MPLTEXT 1 0 5 "L21:= " }{MPLTEXT 1 0 23 "-P2(lambda)+P012/lambda" }{MPLTEXT 1 0 18 "+P122/( lambda-1)^2" }{MPLTEXT 1 0 27 "+P112/(lambda-1)-a0/lambda " }{MPLTEXT 1 0 16 "-a2/(lambda-1)^2" }{MPLTEXT 1 0 32 " -a1/(lambda-1) -h*p/(lamb da-q);" }{MPLTEXT 1 0 1 " " }}{PARA 11 "" 1 "" {XPPMATH 20 ">I$L22G6\" ,.*&,&I%t010GF$\"\"\"I%t020GF$F)F)I'lambdaGF$!\"\"F)*&,&I%t111GF$F)I%t 121GF$F)F),&F+F)F,F)!\"#F)*&,&I%t110GF$F)I%t120GF$F)F)F1F,F)*&I\"hGF$F ),&F+F)I\"qGF$F,F,F)*&F8F)F+F,F,*&F8F)F1F,F2" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I$L21G6\",0*(I%t010GF$\"\"\"I%t020GF$F(I'lambdaGF$!\"#! \"\"*(I%t111GF$F(I%t121GF$F(,&F*F(F,F(!\"%F,*&,&*&I%t110GF$F(F/F(F(*&F .F(I%t120GF$F(F(F(F0!\"$F,*&I#a0GF$F(F*F,F,*&I#a2GF$F(F0F+F,*&I#a1GF$F (F0F,F,*(I\"hGF$F(I\"pGF$F(,&F*F(I\"qGF$F,F,F," }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 255 19 "Auxiliary Matrix A." }{TEXT 255 0 "" }}{EXCHG {PARA 213 "" 0 "" {TEXT 257 94 "The deformation operator is \n \\mathc al\{L\}=\\hbar*(alpha111*\\partial_\{t_\{1^\{(1)\},1\}\}+ alpha121* " }{TEXT 257 25 "\\partial_\{t_\{1^\{(2)\},1\}\})" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "WronskianLInfty:=factor(psi1Infty*Lpsi2Infty-p si2Infty*Lpsi1Infty):\n" }{MPLTEXT 1 0 63 "WronskianLZero:=factor(psi1 Zero*Lpsi2Zero-psi2Zero*Lpsi1Zero):\n" }{MPLTEXT 1 0 53 "WronskianLUn: =factor(psi1Un*Lpsi2Un-psi2Un*Lpsi1Un):\n" }{MPLTEXT 1 0 66 "A12Infty: =factor(simplify(WronskianLInfty/WronskianLambdaInfty)):\n" }{MPLTEXT 1 0 63 "A12Zero:=factor(simplify(WronskianLZero/WronskianLambdaZero)): \n" }{MPLTEXT 1 0 57 "A12Un:=factor(simplify(WronskianLUn/WronskianLam bdaUn)):\n" }{MPLTEXT 1 0 48 "Y1Infty:=h*factor(dpsi1dlambdaInfty/psi1 Infty):\n" }{MPLTEXT 1 0 48 "Y2Infty:=h*factor(dpsi2dlambdaInfty/psi2I nfty):\n" }{MPLTEXT 1 0 45 "Y1Zero:=h*factor(dpsi1dlambdaZero/psi1Zero ):\n" }{MPLTEXT 1 0 45 "Y2Zero:=h*factor(dpsi2dlambdaZero/psi2Zero):\n " }{MPLTEXT 1 0 39 "Y1Un:=h*factor(dpsi1dlambdaUn/psi1Un):\n" } {MPLTEXT 1 0 39 "Y2Un:=h*factor(dpsi2dlambdaUn/psi2Un):\n" }{MPLTEXT 1 0 39 "Z1Infty:=factor(Lpsi1Infty/psi1Infty):\n" }{MPLTEXT 1 0 39 "Z2 Infty:=factor(Lpsi2Infty/psi2Infty):\n" }{MPLTEXT 1 0 36 "Z1Zero:=fact or(Lpsi1Zero/psi1Zero):\n" }{MPLTEXT 1 0 36 "Z2Zero:=factor(Lpsi2Zero/ psi2Zero):\n" }{MPLTEXT 1 0 30 "Z1Un:=factor(Lpsi1Un/psi1Un):\n" } {MPLTEXT 1 0 30 "Z2Un:=factor(Lpsi2Un/psi2Un):\n" }{MPLTEXT 1 0 68 "A1 2bisInfty:=factor(simplify((Z2Infty-Z1Infty)/(Y2Infty-Y1Infty))):\n" } {MPLTEXT 1 0 63 "A12bisZero:=factor(simplify((Z2Zero-Z1Zero)/(Y2Zero-Y 1Zero))):\n" }{MPLTEXT 1 0 53 "A12bisUn:=factor(simplify((Z2Un-Z1Un)/( Y2Un-Y1Un))):\n" }{MPLTEXT 1 0 83 "A11Infty:=factor(simplify( (Y2Infty *Z1Infty-Y1Infty*Z2Infty)/(Y2Infty-Y1Infty) )):\n" }{MPLTEXT 1 0 76 "A 11Zero:=factor(simplify( (Y2Zero*Z1Zero-Y1Zero*Z2Zero)/(Y2Zero-Y1Zero) )):\n" }{MPLTEXT 1 0 62 "A11Un:=factor(simplify( (Y2Un*Z1Un-Y1Un*Z2Un )/(Y2Un-Y1Un) )):\n" }{MPLTEXT 1 0 40 "factor(simplify(A12bisInfty-A12 Infty));\n" }{MPLTEXT 1 0 38 "factor(simplify(A12bisZero-A12Zero));\n" }{MPLTEXT 1 0 37 "factor(simplify(A12bisZero-A12Zero));" }{MPLTEXT 1 0 1 "\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "Lt020:=0:\n" }{MPLTEXT 1 0 10 "Lt12 0:=0:\n" }{MPLTEXT 1 0 10 "Lt010:=0:\n" }{MPLTEXT 1 0 10 "Lt110:=0:\n" }{MPLTEXT 1 0 14 "Ltinfty20:=0:\n" }{MPLTEXT 1 0 13 "Ltinfty10:=0:" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "L" }{MPLTEXT 1 0 18 "t111:=h*alph a111:\n" }{MPLTEXT 1 0 18 "Lt121:=h*alpha121:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "A12InftyLambda3:=factor(-residue(A12Infty/lambda ^4,lambda=infinity));\n" }{MPLTEXT 1 0 70 "A12InftyLambda2:=factor(-re sidue(A12Infty/lambda^3,lambda=infinity));\n" }{MPLTEXT 1 0 70 "A12Inf tyLambda1:=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%LA10GF$\"\"\"I%LA20GF$!\"\"F( ,(I\"hGF$F(I)tinfty10GF$F*I)tinfty20GF$F(F*" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I0A12InftyLambda0G6\",$*&,6*(I$A12GF$\"\"\"I%LA10GF$F*I \"hGF$F*F**(F)F*I%LA20GF$F*F,F*!\"\"*(I$A22GF$F*F+F*F,F*F/*(F1F*F.F*F, F*F**&I%LA12GF$F*F,F*F**&F4F*I)tinfty10GF$F*F/*&F4F*I)tinfty20GF$F*F** &I%LA22GF$F*F,F*F/*&F:F*F6F*F**&F:F*F8F*F/F*,(F,F*F6F/F8F*!\"#F/" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "A12ZeroLambdaMinus3:=factor( residue(A12Zero*lambda^2,lambda=0));\n" }{MPLTEXT 1 0 65 "A12ZeroLambd aMinus2:=factor(residue(A12Zero*lambda^1,lambda=0));\n" }{MPLTEXT 1 0 65 "A12ZeroLambdaMinus1:=factor(residue(A12Zero*lambda^0,lambda=0));\n " }{MPLTEXT 1 0 63 "A12ZeroLambda0:=factor(residue(A12Zero*lambda^(-1) ,lambda=0));\n" }{MPLTEXT 1 0 62 "A12ZeroLambda1:=factor(residue(A12Ze ro*lambda^(-2),lambda=0));" }{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I4A12ZeroLambdaMinus3G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I4A12ZeroLambdaMinus2G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I4A12ZeroLambdaMinus1G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I/A12ZeroLambda0G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I/A12ZeroLambda1G6\"*&,&I%LB10GF$\"\"\"I%LB20GF$!\"\"F(,&I%t010GF $F(I%t020GF$F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "A12UnLa mbdaMinus3:=factor(residue(A12Un*(lambda-1)^2,lambda=1));\n" }{MPLTEXT 1 0 65 "A12UnLambdaMinus2:=factor(residue(A12Un*(lambda-1)^1,lambda=1 ));\n" }{MPLTEXT 1 0 65 "A12UnLambdaMinus1:=factor(residue(A12Un*(lamb da-1)^0,lambda=1));\n" }{MPLTEXT 1 0 63 "A12UnLambda0:=factor(residue( A12Un*(lambda-1)^(-1),lambda=1));\n" }{MPLTEXT 1 0 62 "A12UnLambda1:=f actor(residue(A12Un*(lambda-1)^(-2),lambda=1));" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 62 "A12UnLambda2:=factor(residue(A12Un*(lambda-1)^(-3),l ambda=1));" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I2A12UnLambdaMinus3G6\"\" \"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I2A12UnLambdaMinus2G6\"\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 ">I2A12UnLambdaMinus1G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I-A12UnLambda0G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I-A12UnLambda1G6\",$*&,&I)alpha111GF$\"\"\"I)alpha121GF$ !\"\"F),&I%t111GF$F)I%t121GF$F+F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "> I-A12UnLambda2G6\"*&,2*&I%LC10GF$\"\"\"I%t111GF$F)F)*&F(F)I%t121GF$F)! \"\"*&I%LC20GF$F)F*F)F-*&F/F)F,F)F)*&I%t110GF$F)I)alpha111GF$F)F)*&F2F )I)alpha121GF$F)F-*&I%t120GF$F)F3F)F-*&F7F)F5F)F)F),&F*F)F,F-!\"#" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "A11InftyLambda3:=factor(-res idue(A11Infty/lambda^4,lambda=infinity));\n" }{MPLTEXT 1 0 70 "A11Inft yLambda2:=factor(-residue(A11Infty/lambda^3,lambda=infinity));\n" } {MPLTEXT 1 0 70 "A11InftyLambda1:=factor(-residue(A11Infty/lambda^2,la mbda=infinity));\n" }{MPLTEXT 1 0 70 "A11InftyLambda0:=factor(-residue (A11Infty/lambda^1,lambda=infinity));\n" }{MPLTEXT 1 0 75 "A11InftyLam bdaMinus1:=factor(-residue(A11Infty/lambda^0,lambda=infinity)):\n" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 65 "A11ZeroLambdaMinus3:=factor(resi due(A11Zero*lambda^2,lambda=0));\n" }{MPLTEXT 1 0 65 "A11ZeroLambdaMin us2:=factor(residue(A11Zero*lambda^1,lambda=0));\n" }{MPLTEXT 1 0 65 " A11ZeroLambdaMinus1:=factor(residue(A11Zero*lambda^0,lambda=0));\n" } {MPLTEXT 1 0 63 "A11ZeroLambda0:=factor(residue(A11Zero*lambda^(-1),la mbda=0));\n" }{MPLTEXT 1 0 63 "A11ZeroLambda1:=factor(residue(A11Zero* lambda^(-2),lambda=0)):\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 65 "A11U nLambdaMinus3:=factor(residue(A11Un*(lambda-1)^2,lambda=1));\n" } {MPLTEXT 1 0 65 "A11UnLambdaMinus2:=factor(residue(A11Un*(lambda-1)^1, lambda=1));\n" }{MPLTEXT 1 0 65 "A11UnLambdaMinus1:=factor(residue(A11 Un*(lambda-1)^0,lambda=1));\n" }{MPLTEXT 1 0 63 "A11UnLambda0:=factor( residue(A11Un*(lambda-1)^(-1),lambda=1));\n" }{MPLTEXT 1 0 62 "A11UnLa mbda1:=factor(residue(A11Un*(lambda-1)^(-2),lambda=1)):" }{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I0A11InftyLambda3G6\"\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 ">I0A11InftyLambda2G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I0A11InftyLambda1G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I0A11InftyLambda0G6\"*&,(*&I%LA10GF$\"\"\"I\"hGF$F)F)*&F (F)I)tinfty20GF$F)F)*&I%LA20GF$F)I)tinfty10GF$F)!\"\"F),(F*F)F/F0F,F)F 0" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I4A11ZeroLambdaMinus3G6\"\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 ">I4A11ZeroLambdaMinus2G6\"\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 ">I4A11ZeroLambdaMinus1G6\"\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 ">I/A11ZeroLambda0G6\",$*&,&*&I%LB10GF$\" \"\"I%t020GF$F*F**&I%LB20GF$F*I%t010GF$F*!\"\"F*,&F.F*F+F/F/F/" }} {PARA 11 "" 1 "" {XPPMATH 20 ">I2A11UnLambdaMinus3G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I2A11UnLambdaMinus2G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I2A11UnLambdaMinus1G6\",$*&,&*&I%t111GF$\"\"\"I)alph a121GF$F*F**&I%t121GF$F*I)alpha111GF$F*!\"\"F*,&F)F*F-F/F/F/" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I-A11UnLambda0G6\",$*&,2*(I%LC10GF$\"\"\"I%t1 11GF$F*I%t121GF$F*F**&F)F*F,\"\"#!\"\"*&I%LC20GF$F*F+F.F/*(F1F*F+F*F,F *F**(I%t110GF$F*F,F*I)alpha111GF$F*F**(F4F*F,F*I)alpha121GF$F*F/*(F+F* I%t120GF$F*F5F*F/*(F+F*F9F*F7F*F*F*,&F+F*F,F/!\"#F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "A12Form:=nu0+" }{MPLTEXT 1 0 8 "nuMinus1" }{MPLTEXT 1 0 24 "*lambda+ mu/(lambda-q);\n" }{MPLTEXT 1 0 43 "EQ1:=re sidue(A12Form/(lambda-1),lambda=1);\n" }{MPLTEXT 1 0 33 "EQ2:=residue( A12Form/(lambda-1)^2" }{MPLTEXT 1 0 12 " ,lambda=1) " }{MPLTEXT 1 0 2 "-(" }{MPLTEXT 1 0 32 "-(alpha111-alpha121)/(t111-t121)" }{MPLTEXT 1 0 3 ");\n" }{MPLTEXT 1 0 39 "EQ3:=residue(A12Form/lambda,lambda=0);\n" }{MPLTEXT 1 0 34 "solve(\{EQ1,EQ2,EQ3\},\{nu0,nuMinus1," }{MPLTEXT 1 0 2 "mu" }{MPLTEXT 1 0 3 "\});" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I(A12 FormG6\",(I$nu0GF$\"\"\"*&I)nuMinus1GF$F'I'lambdaGF$F'F'*&I#muGF$F',&F *F'I\"qGF$!\"\"F/F'" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I$EQ1G6\"*&,,*&I $nu0GF$\"\"\"I\"qGF$F)!\"\"*&F*F)I)nuMinus1GF$F)F+I#muGF$F)F(F)F-F)F), &F*F+F)F)F+" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I$EQ2G6\",&*&,**&I\"qGF$ \"\"\"I)nuMinus1GF$F*!\"\"I$nu0GF$F*F+\"\"#*&,,*&F-F*F)F*F,F(F,I#muGF$ F*F-F*F+F*F*,&F)F*F,F*F,F*F*,&F)F,F*F*F,F**&,&I)alpha111GF$F*I)alpha12 1GF$F,F*,&I%t111GF$F*I%t121GF$F,F,F*" }}{PARA 11 "" 1 "" {XPPMATH 20 " >I$EQ3G6\",$*&,&*&I$nu0GF$\"\"\"I\"qGF$F*!\"\"I#muGF$F*F*F+F,F," }} {PARA 11 "" 1 "" {XPPMATH 20 "<%/I#muG6\"**,&I\"qGF%\"\"\"!\"\"F)F),** &F(F)I)alpha111GF%F)F)*&F(F)I)alpha121GF%F)F*F-F*F/F)F),&I%t111GF%F)I% t121GF%F*F*F(F)/I$nu0GF%*(F'F)F+F)F0F*/I)nuMinus1GF%*&F+F)F0F*" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "factor((alpha111*q-alpha121* q-alpha111+alpha121)/(t111-t121));" }}{PARA 11 "" 1 "" {XPPMATH 20 "*( ,&I)alpha111G6\"\"\"\"I)alpha121GF%!\"\"F&,&I\"qGF%F&F(F&F&,&I%t111GF% F&I%t121GF%F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "mu:=(alp ha111-alpha121)*q*(q-1)^2/(t111-t121):\n" }{MPLTEXT 1 0 46 "nu0:=(alph a111-alpha121)*(q-1)^2/(t111-t121);\n" }{MPLTEXT 1 0 49 "nuMinus1:=(al pha111-alpha121)*(q-1)/(t111-t121);\n" }{MPLTEXT 1 0 9 "A12Form:=" } {MPLTEXT 1 0 4 "nu0+" }{MPLTEXT 1 0 8 "nuMinus1" }{MPLTEXT 1 0 22 "*la mbda+ mu/(lambda-q)" }{MPLTEXT 1 0 2 ";\n" }{MPLTEXT 1 0 36 "simplify( series(A12Form,lambda=1));\n" }{MPLTEXT 1 0 36 "simplify(series(A12For m,lambda=0));\n" }{MPLTEXT 1 0 33 "series(A12Form,lambda=infinity);\n" }{MPLTEXT 1 0 16 "factor(A12Form);" }}{PARA 11 "" 1 "" {XPPMATH 20 "> I$nu0G6\"*(,&I)alpha111GF$\"\"\"I)alpha121GF$!\"\"F(,&I\"qGF$F(F*F(\" \"#,&I%t111GF$F(I%t121GF$F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I)nuMi nus1G6\"*(,&I)alpha111GF$\"\"\"I)alpha121GF$!\"\"F(,&I\"qGF$F(F*F(F(,& I%t111GF$F(I%t121GF$F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I(A12FormG6 \",(*(,&I)alpha111GF$\"\"\"I)alpha121GF$!\"\"F),&I\"qGF$F)F+F)\"\"#,&I %t111GF$F)I%t121GF$F+F+F)**F'F)F,F)F/F+I'lambdaGF$F)F)*,F'F)F-F)F,F.F/ F+,&F3F)F-F+F+F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "+/,&I'lambdaG6\"\"\" \"!\"\"F&*&,&I)alpha111GF%F'I)alpha121GF%F&F&,&I%t111GF%F&I%t121GF%F'F 'F&,$**,&F*F&F+F'F&I\"qGF%F&F,F',&F2F&F'F&F'F'\"\"#,$**F1F&F2F&F3!\"#F ,F'F'\"\"$,$**F1F&F2F&F3!\"$F,F'F'\"\"%,$**F1F&F2F&F3!\"%F,F'F'\"\"&-I \"OG6$%*protectedGI(_syslibGF%6#F&\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "+/I'lambdaG6\"**,&I)alpha111GF$\"\"\"I)alpha121GF$!\"\"F(,&I\"qGF$ F(F*F(F(,&I%t111GF$F(I%t121GF$F*F*F,F*F(,$**F&F(F+\"\"#F-F*F,!\"#F*F2, $**F&F(F+F2F-F*F,!\"$F*\"\"$,$**F&F(F+F2F-F*F,!\"%F*\"\"%,$**F&F(F+F2F -F*F,!\"&F*\"\"&-I\"OG6$%*protectedGI(_syslibGF$6#F(\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 ",2**,&I)alpha111G6\"\"\"\"I)alpha121GF&!\"\"F',&I \"qGF&F'F)F'F',&I%t111GF&F'I%t121GF&F)F)I'lambdaGF&F'F'*(F$F'F*\"\"#F, F)F'*,F$F'F+F'F*F1F,F)F/F)F'*,F$F'F+F1F*F1F,F)F/!\"#F'*,F$F'F+\"\"$F*F 1F,F)F/!\"$F'*,F$F'F+\"\"%F*F1F,F)F/!\"%F'*,F$F'F+\"\"&F*F1F,F)F/!\"&F '-I\"OG6$%*protectedGI(_syslibGF&6#*$F/!\"'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "*.,&I)alpha111G6\"\"\"\"I)alpha121GF%!\"\"F&,&I\"qGF%F&F( F&F&I'lambdaGF%F&,&F+F&F(F&F&,&I%t111GF%F&I%t121GF%F(F(,&F+F&F*F(F(" } }}{EXCHG {PARA 0 "" 0 "" {TEXT 205 132 "A_\{1,2\} is O(\\lambda) at la mbda=0 and at lambda=infinity. It is of the form -(alpha111-alpha121)/ (t111-t121)(lambda-1) at lambda=1.\n" }{TEXT 205 17 "Thus we get that \n" }{TEXT 240 78 "A_\{1,2\}= (alpha111-alpha121)*(q-1)*lambda*(lambda -1)/((t111-t121)*(lambda-q))\n" }{TEXT 241 1 "=" }{TEXT 242 34 "nu0+nu Minus1*lambda+ mu/(lambda-q)" }{TEXT 205 1 "\n" }{TEXT 205 4 "with" } {TEXT 243 1 "\n" }{TEXT 244 47 "mu:=(alpha111-alpha121)*q*(q-1)^2/(t11 1-t121):\n" }{TEXT 245 46 "nu0:=(alpha111-alpha121)*(q-1)^2/(t111-t121 );\n" }{TEXT 246 48 "nuMinus1:=(alpha111-alpha121)*(q-1)/(t111-t121);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "A12:=(alpha111-alpha121) *(q-1)*lambda*(lambda-1)/((t111-t121)*(lambda-q));\n" }{MPLTEXT 1 0 16 "simplify(A12- ( " }{MPLTEXT 1 0 34 "nu0+nuMinus1*lambda+ mu/(lambd a-q)" }{MPLTEXT 1 0 3 "));" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I$A12G6\" *.,&I)alpha111GF$\"\"\"I)alpha121GF$!\"\"F(,&I\"qGF$F(F*F(F(I'lambdaGF $F(,&F-F(F*F(F(,&I%t111GF$F(I%t121GF$F*F*,&F-F(F,F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {TEXT 205 69 "A_\{ 1,1\} is O(1) at lambda=0 an at \\lambda=infinity. It behaves like " }{XPPEDIT 2 0 "Typesetting:-mrow(Typesetting:-mi(\"\"), Typesetting:-m frac(Typesetting:-mrow(Typesetting:-mi(\"\1661111\", italic = \"true\" , mathvariant = \"italic\"), Typesetting:-mo(\"⁢\", mat hvariant = \"normal\", fence = \"false\", separator = \"false\", stret chy = \"false\", symmetric = \"false\", largeop = \"false\", movableli mits = \"false\", accent = \"false\", lspace = \"0.0em\", rspace = \"0 .0em\"), Typesetting:-mi(\"t121\", italic = \"true\", mathvariant = \" italic\"), Typesetting:-mo(\"−\", mathvariant = \"normal\", fenc e = \"false\", separator = \"false\", stretchy = \"false\", symmetric \+ = \"false\", largeop = \"false\", movablelimits = \"false\", accent = \+ \"false\", lspace = \"0.2222222em\", rspace = \"0.2222222em\"), Typese tting:-mi(\"\1661121\", italic = \"true\", mathvariant = \"italic\"), \+ Typesetting:-mo(\" \", mathvariant = \"normal\", fence = \"false\", se parator = \"false\", stretchy = \"false\", symmetric = \"false\", larg eop = \"false\", movablelimits = \"false\", accent = \"false\", lspace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mi(\"t111\", italic = \+ \"true\", mathvariant = \"italic\")), Typesetting:-mrow(Typesetting:-m i(\"t111\", italic = \"true\", mathvariant = \"italic\"), Typesetting: -mo(\"−\", mathvariant = \"normal\", fence = \"false\", separato r = \"false\", stretchy = \"false\", symmetric = \"false\", largeop = \+ \"false\", movablelimits = \"false\", accent = \"false\", lspace = \"0 .2222222em\", rspace = \"0.2222222em\"), Typesetting:-mi(\"t121\", ita lic = \"true\", mathvariant = \"italic\")), linethickness = \"1\", den omalign = \"center\", numalign = \"center\", bevelled = \"false\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6$-I#miGF$6#Q!F' -I&mfracGF$6(-F#6)-F,6%Q+α111F'/%'italicGQ%trueF'/%,mathvariantG Q'italicF'-I#moGF$6-Q1⁢F'/F;Q'normalF'/%&fenceGQ&falseF '/%*separatorGFE/%)stretchyGFE/%*symmetricGFE/%(largeopGFE/%.movableli mitsGFE/%'accentGFE/%'lspaceGQ&0.0emF'/%'rspaceGFT-F,6%Q%t121F'F7F:-F> 6-Q(−F'FAFCFFFHFJFLFNFP/FSQ,0.2222222emF'/FVFhn-F,6%Q+α121 F'F7F:-F>6-Q\"~F'FAFCFFFHFJFLFNFPFRFU-F,6%Q%t111F'F7F:-F#6%F`oFZFW/%.l inethicknessGQ\"1F'/%+denomalignGQ'centerF'/%)numalignGFjo/%)bevelledG FE" }{TEXT 205 43 " /(lambda-1) at \\lambda=1 \nThus we get that" } {TEXT 205 1 "\n" }{TEXT 247 8 "A_\{1,1\}=" }{TEXT 248 73 "(alpha111*t1 21-alpha121*t111)/(t111-t121)/(lambda-1) +c_0+ rho/(lambda-q)" }{TEXT 205 1 "\n" }{TEXT 249 8 "A_\{1,1\}=" }{TEXT 250 4 "c1/(" }{TEXT 251 79 "lambda-1) +c0+rho/(lambda-q)\nwith c_1=(alpha111*t121-alpha121*t11 1)/(t111-t121)" }}}{EXCHG {PARA 214 "> " 0 "" {MPLTEXT 1 0 39 "A11:=c1 /(lambda-1) +c0+rho/(lambda-q);\n" }{MPLTEXT 1 0 45 "c1:=(alpha111*t12 1-alpha121*t111)/(t111-t121)" }{MPLTEXT 1 205 2 ";\n" }{MPLTEXT 1 0 23 "series(A11,lambda=1,2);" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 24 "se ries(A11,lambda=0,2);\n" }{MPLTEXT 1 0 30 "series(A11,lambda=infinity, 2);" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I$A11G6\",(*&I#c1GF$\"\"\",&I'la mbdaGF$F(!\"\"F(F+F(I#c0GF$F(*&I$rhoGF$F(,&F*F(I\"qGF$F+F+F(" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I#c1G6\"*&,&*&I%t111GF$\"\"\"I)alpha121GF$F)! \"\"*&I%t121GF$F)I)alpha111GF$F)F)F),&F(F)F-F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "++,&I'lambdaG6\"\"\"\"!\"\"F&*&,&*&I%t111GF%F&I)alpha121G F%F&F'*&I%t121GF%F&I)alpha111GF%F&F&F&,&F+F&F.F'F'F',&I#c0GF%F&*&I$rho GF%F&,&I\"qGF%F'F&F&F'F&\"\"!*(F4F&,&F6F&F'F&F'F5F'F&-I\"OG6$%*protect edGI(_syslibGF%6#F&\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "+)I'lambdaG6 \",(*&,&*&I%t111GF$\"\"\"I)alpha121GF$F*!\"\"*&I%t121GF$F*I)alpha111GF $F*F*F*,&F)F*F.F,F,F,I#c0GF$F**&I$rhoGF$F*I\"qGF$F,F,\"\"!,&*&,&F(F*F- F,F*F0F,F**&F3F*F4!\"#F,F*-I\"OG6$%*protectedGI(_syslibGF$6#F*\"\"#" } }{PARA 11 "" 1 "" {XPPMATH 20 ",(I#c0G6\"\"\"\"*&,&*&,&*&I%t111GF$F%I) alpha121GF$F%!\"\"*&I%t121GF$F%I)alpha111GF$F%F%F%,&F+F%F/F-F-F%I$rhoG F$F%F%I'lambdaGF$F-F%-I\"OG6$%*protectedGI(_syslibGF$6#*$F3!\"#F%" }}} {EXCHG {PARA 210 "" 0 "" {TEXT 256 52 "Summary of the results (c_0 do es not play any role)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "mu :=(alpha111-alpha121)*q*(q-1)^2/(t111-t121):\n" }{MPLTEXT 1 0 46 "nu0: =(alpha111-alpha121)*(q-1)^2/(t111-t121);\n" }{MPLTEXT 1 0 48 "nuMinus 1:=(alpha111-alpha121)*(q-1)/(t111-t121);" }{MPLTEXT 1 0 1 "\n" } {MPLTEXT 1 0 75 "A12:=(alpha111-alpha121)*(q-1)*lambda*(lambda-1)/((t1 11-t121)*(lambda-q));\n" }{MPLTEXT 1 0 16 "simplify(A12- ( " }{MPLTEXT 1 0 34 "nu0+nuMinus1*lambda+ mu/(lambda-q)" }{MPLTEXT 1 0 3 "));" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 39 "A11:=c1/(lambda-1) +c0+rho/(lamb da-q);\n" }{MPLTEXT 1 0 45 "c1:=(alpha111*t121-alpha121*t111)/(t111-t1 21)" }{MPLTEXT 1 205 2 ";\n" }{TEXT 252 13 "a0+a1+h*p=0;\n" }{TEXT 253 46 "h*p*q+t010*t020+a1+a2-(h+tinfty20)*tinfty10=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I$nu0G6\"*(,&I)alpha111GF$\"\"\"I)alpha121GF$!\"\"F (,&I\"qGF$F(F*F(\"\"#,&I%t111GF$F(I%t121GF$F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I)nuMinus1G6\"*(,&I)alpha111GF$\"\"\"I)alpha121GF$!\"\"F (,&I\"qGF$F(F*F(F(,&I%t111GF$F(I%t121GF$F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I$A12G6\"*.,&I)alpha111GF$\"\"\"I)alpha121GF$!\"\"F(,&I \"qGF$F(F*F(F(I'lambdaGF$F(,&F-F(F*F(F(,&I%t111GF$F(I%t121GF$F*F*,&F-F (F,F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I$A11G6\",(*(,&*&I%t111GF$\"\"\"I)alpha121GF$F*!\"\"*&I% t121GF$F*I)alpha111GF$F*F*F*,&F)F*F.F,F,,&I'lambdaGF$F*F,F*F,F*I#c0GF$ F**&I$rhoGF$F*,&F2F*I\"qGF$F,F,F*" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I# c1G6\"*&,&*&I%t111GF$\"\"\"I)alpha121GF$F)!\"\"*&I%t121GF$F)I)alpha111 GF$F)F)F),&F(F)F-F+F+" }}}{EXCHG {PARA 0 "> " 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 }