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{SECT 0 {EXCHG {PARA 211 "" 0 "" {TEXT 209 151 "In this Maple sheet, \+
we compute the Lax matrices using the asymptotics of the wave function
s and the local diagonalization for the Painlev\351 2 equation." }}}
{EXCHG {PARA 0 "" 0 "" {TEXT 209 118 "We first use the expression of t
he coefficients of the spectral curve in terms of the irregular times \+
and monodromies." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "restart:
\n" }{MPLTEXT 1 0 39 "CoherenceEquation1:=tinfty10+tinfty20;\n" }
{MPLTEXT 1 0 21 "tinfty20:=-tinfty10:\n" }{MPLTEXT 1 0 31 "Pinfty42 :=
 tinfty13*tinfty23;\n" }{MPLTEXT 1 0 49 "Pinfty32 := tinfty12*tinfty23
+tinfty13*tinfty22;\n" }{MPLTEXT 1 0 67 "Pinfty22 := tinfty12*tinfty22
+tinfty13*tinfty21+tinfty11*tinfty23;\n" }{MPLTEXT 1 0 85 "Pinfty12 :=
 tinfty20*tinfty13+tinfty12*tinfty21+tinfty10*tinfty23+tinfty11*tinfty
22;\n" }{MPLTEXT 1 0 32 "Pinfty01 := -tinfty11-tinfty21;\n" }{MPLTEXT 
1 0 32 "Pinfty11 := -tinfty12-tinfty22;\n" }{MPLTEXT 1 0 32 "Pinfty21 \+
:= -tinfty13-tinfty23;\n" }{MPLTEXT 1 0 42 "P1:=x-> Pinfty01+Pinfty11*
x+Pinfty21*x^2:\n" }{MPLTEXT 1 0 67 "P2:=x-> Pinfty02+Pinfty12*x+Pinft
y22*x^2+Pinfty32*x^3+Pinfty42*x^4:" }}{PARA 0 "" 0 "" {TEXT 219 0 "" }
{MPLTEXT 1 0 1 "\n" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I3CoherenceEquati
on1G6\",&I)tinfty10GF$\"\"\"I)tinfty20GF$F'" }}{PARA 11 "" 1 "" 
{XPPMATH 20 ">I)Pinfty42G6\"*&I)tinfty13GF$\"\"\"I)tinfty23GF$F'" }}
{PARA 11 "" 1 "" {XPPMATH 20 ">I)Pinfty32G6\",&*&I)tinfty12GF$\"\"\"I)
tinfty23GF$F(F(*&I)tinfty13GF$F(I)tinfty22GF$F(F(" }}{PARA 11 "" 1 "" 
{XPPMATH 20 ">I)Pinfty22G6\",(*&I)tinfty11GF$\"\"\"I)tinfty23GF$F(F(*&
I)tinfty12GF$F(I)tinfty22GF$F(F(*&I)tinfty13GF$F(I)tinfty21GF$F(F(" }}
{PARA 11 "" 1 "" {XPPMATH 20 ">I)Pinfty12G6\",**&I)tinfty10GF$\"\"\"I)
tinfty13GF$F(!\"\"*&F'F(I)tinfty23GF$F(F(*&I)tinfty11GF$F(I)tinfty22GF
$F(F(*&I)tinfty12GF$F(I)tinfty21GF$F(F(" }}{PARA 11 "" 1 "" {XPPMATH 
20 ">I)Pinfty01G6\",&I)tinfty11GF$!\"\"I)tinfty21GF$F'" }}{PARA 11 "" 
1 "" {XPPMATH 20 ">I)Pinfty11G6\",&I)tinfty12GF$!\"\"I)tinfty22GF$F'" 
}}{PARA 11 "" 1 "" {XPPMATH 20 ">I)Pinfty21G6\",&I)tinfty13GF$!\"\"I)t
infty23GF$F'" }}}{EXCHG {PARA 210 "" 0 "" {TEXT 220 36 "Study of the a
symptotics at infinity" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 244 "
logPsi1Infty:=-tinfty13/3/h*lambda^3-tinfty12/2/h*lambda^2-tinfty11/h*
lambda-tinfty10/h*ln(lambda)+A10-A12/(2-1)/lambda^(2-1)-A13/(3-1)/lamb
da^(3-1)-A14/(4-1)/lambda^(4-1)-A15/(5-1)/lambda^(5-1)-A16/(6-1)/lambd
a^(6-1)-A17/(7-1)/lambda^(7-1)  ;\n" }{MPLTEXT 1 0 257 "logPsi2Infty:=
-tinfty23/3/h*lambda^3-tinfty22/2/h*lambda^2-tinfty21/h*lambda-tinfty2
0/h*ln(lambda)-1*ln(lambda)+A20-A22/(2-1)/lambda^(2-1)-A23/(3-1)/lambd
a^(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 0 256 "Llogpsi1Infty:=
-Ltinfty13/3/h*lambda^3-Ltinfty12/2/h*lambda^2-Ltinfty11/h*lambda-Ltin
fty10/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 0 256 "Llogpsi2Infty:=-L
tinfty23/3/h*lambda^3-Ltinfty22/2/h*lambda^2-Ltinfty21/h*lambda-Ltinft
y20/h*ln(lambda)+LA20-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 0 425 "Lpsi1Infty := exp(1
/h*(-tinfty13/3*lambda^3-tinfty12/2*lambda^2-tinfty11*lambda-tinfty10*
ln(lambda)+h*A10-h*A12/lambda-1/2*h*A13/lambda^2-1/3*h*A14/lambda^3-1/
4*h*A15/lambda^4-1/5*h*A16/lambda^5-1/6*h*A17/lambda^6))*1/h*(-Ltinfty
13/3*lambda^3-Ltinfty12/2*lambda^2-Ltinfty11*lambda-Ltinfty10*ln(lambd
a)+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 0 438 "Lpsi2Infty := exp(1/h*(-tinfty23/3*lambda^3-tinfty22/2*lambda
^2-tinfty21*lambda-tinfty20*ln(lambda)-h*ln(lambda)+h*A20-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/la
mbda^5-1/6*h*A27/lambda^6))*1/h*(-Ltinfty23/3*lambda^3-Ltinfty22/2*lam
bda^2-Ltinfty21*lambda-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/lamb
da^5-1/6*h*LA27/lambda^6);\n" }{MPLTEXT 1 0 30 "psi1Infty:=exp(logPsi1
Infty);\n" }{MPLTEXT 1 0 30 "psi2Infty:=exp(logPsi2Infty);\n" }
{MPLTEXT 1 0 43 "dpsi1dlambdaInfty:=diff(psi1Infty,lambda):\n" }
{MPLTEXT 1 0 43 "dpsi2dlambdaInfty:=diff(psi2Infty,lambda):\n" }
{MPLTEXT 1 0 47 "d2psi1dlambda2Infty:=diff(psi1Infty,lambda$2):\n" }
{MPLTEXT 1 0 47 "d2psi2dlambda2Infty:=diff(psi2Infty,lambda$2):\n" }
{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 89 "WronskianLa
mbdaInfty:=h*factor(psi1Infty*dpsi2dlambdaInfty-psi2Infty*dpsi1dlambda
Infty):\n" }{MPLTEXT 1 0 132 "WronskianLambdabisInfty:=h*simplify(fact
or( (diff(logPsi2Infty,lambda)-diff(logPsi1Infty,lambda))*exp(logPsi1I
nfty+logPsi2Infty))):\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 116 "Wrons
kianTildeLambdaInfty:=h^3*factor(dpsi2dlambdaInfty*d2psi1dlambda2Infty
-dpsi1dlambdaInfty*d2psi2dlambda2Infty):\n" }}{PARA 11 "" 1 "" 
{XPPMATH 20 ">I-logPsi1InftyG6\",8**#\"\"\"\"\"$F(I)tinfty13GF$F(I\"hG
F$!\"\")I'lambdaGF$F)F(F,**#F(\"\"#F(I)tinfty12GF$F(F+F,)F.F1F(F,*(I)t
infty11GF$F(F+F,F.F(F,*(I)tinfty10GF$F(F+F,-I#lnG6$%*protectedGI(_sysl
ibGF$6#F.F(F,I$A10GF$F(*&I$A12GF$F(F.F,F,*(F0F(I$A13GF$F(F3F,F,*(F'F(I
$A14GF$F(F-F,F,*(#F(\"\"%F(I$A15GF$F()F.FGF,F,*(#F(\"\"&F(I$A16GF$F()F
.FLF,F,*(#F(\"\"'F(I$A17GF$F()F.FQF,F," }}{PARA 11 "" 1 "" {XPPMATH 20
 ">I-logPsi2InftyG6\",:**#\"\"\"\"\"$F(I)tinfty23GF$F(I\"hGF$!\"\")I'l
ambdaGF$F)F(F,**#F(\"\"#F(I)tinfty22GF$F(F+F,)F.F1F(F,*(I)tinfty21GF$F
(F+F,F.F(F,*(I)tinfty10GF$F(F+F,-I#lnG6$%*protectedGI(_syslibGF$6#F.F(
F(F8F,I$A20GF$F(*&I$A22GF$F(F.F,F,*(F0F(I$A23GF$F(F3F,F,*(F'F(I$A24GF$
F(F-F,F,*(#F(\"\"%F(I$A25GF$F()F.FGF,F,*(#F(\"\"&F(I$A26GF$F()F.FLF,F,
*(#F(\"\"'F(I$A27GF$F()F.FQF,F," }}{PARA 11 "" 1 "" {XPPMATH 20 ">I.Ll
ogpsi1InftyG6\",8**#\"\"\"\"\"$F(I*Ltinfty13GF$F(I\"hGF$!\"\")I'lambda
GF$F)F(F,**#F(\"\"#F(I*Ltinfty12GF$F(F+F,)F.F1F(F,*(I*Ltinfty11GF$F(F+
F,F.F(F,*(I*Ltinfty10GF$F(F+F,-I#lnG6$%*protectedGI(_syslibGF$6#F.F(F,
I%LA10GF$F(*&I%LA12GF$F(F.F,F,*(F0F(I%LA13GF$F(F3F,F,*(F'F(I%LA14GF$F(
F-F,F,*(#F(\"\"%F(I%LA15GF$F()F.FGF,F,*(#F(\"\"&F(I%LA16GF$F()F.FLF,F,
*(#F(\"\"'F(I%LA17GF$F()F.FQF,F," }}{PARA 11 "" 1 "" {XPPMATH 20 ">I.L
logpsi2InftyG6\",8**#\"\"\"\"\"$F(I*Ltinfty23GF$F(I\"hGF$!\"\")I'lambd
aGF$F)F(F,**#F(\"\"#F(I*Ltinfty22GF$F(F+F,)F.F1F(F,*(I*Ltinfty21GF$F(F
+F,F.F(F,*(I*Ltinfty20GF$F(F+F,-I#lnG6$%*protectedGI(_syslibGF$6#F.F(F
,I%LA20GF$F(*&I%LA22GF$F(F.F,F,*(F0F(I%LA23GF$F(F3F,F,*(F'F(I%LA24GF$F
(F-F,F,*(#F(\"\"%F(I%LA25GF$F()F.FGF,F,*(#F(\"\"&F(I%LA26GF$F()F.FLF,F
,*(#F(\"\"'F(I%LA27GF$F()F.FQF,F," }}{PARA 11 "" 1 "" {XPPMATH 20 ">I+
Lpsi1InftyG6\"*(-I$expG6$%*protectedGI(_syslibGF$6#*&I\"hGF$!\"\",8*(#
\"\"\"\"\"$F2I)tinfty13GF$F2)I'lambdaGF$F3F2F.*(#F2\"\"#F2I)tinfty12GF
$F2)F6F9F2F.*&I)tinfty11GF$F2F6F2F.*&I)tinfty10GF$F2-I#lnGF(6#F6F2F.*&
F-F2I$A10GF$F2F2*(F-F2I$A12GF$F2F6F.F.**F8F2F-F2I$A13GF$F2F;F.F.**F1F2
F-F2I$A14GF$F2F5F.F.**#F2\"\"%F2F-F2I$A15GF$F2)F6FMF.F.**#F2\"\"&F2F-F
2I$A16GF$F2)F6FRF.F.**#F2\"\"'F2F-F2I$A17GF$F2)F6FWF.F.F2F2F-F.,8*(F1F
2I*Ltinfty13GF$F2F5F2F.*(F8F2I*Ltinfty12GF$F2F;F2F.*&I*Ltinfty11GF$F2F
6F2F.*&I*Ltinfty10GF$F2F@F2F.*&F-F2I%LA10GF$F2F2*(F-F2I%LA12GF$F2F6F.F
.**F8F2F-F2I%LA13GF$F2F;F.F.**F1F2F-F2I%LA14GF$F2F5F.F.**FLF2F-F2I%LA1
5GF$F2FOF.F.**FQF2F-F2I%LA16GF$F2FTF.F.**FVF2F-F2I%LA17GF$F2FYF.F.F2" 
}}{PARA 11 "" 1 "" {XPPMATH 20 ">I+Lpsi2InftyG6\"*(-I$expG6$%*protecte
dGI(_syslibGF$6#*&I\"hGF$!\"\",:*(#\"\"\"\"\"$F2I)tinfty23GF$F2)I'lamb
daGF$F3F2F.*(#F2\"\"#F2I)tinfty22GF$F2)F6F9F2F.*&I)tinfty21GF$F2F6F2F.
*&I)tinfty10GF$F2-I#lnGF(6#F6F2F2*&F-F2F@F2F.*&F-F2I$A20GF$F2F2*(F-F2I
$A22GF$F2F6F.F.**F8F2F-F2I$A23GF$F2F;F.F.**F1F2F-F2I$A24GF$F2F5F.F.**#
F2\"\"%F2F-F2I$A25GF$F2)F6FNF.F.**#F2\"\"&F2F-F2I$A26GF$F2)F6FSF.F.**#
F2\"\"'F2F-F2I$A27GF$F2)F6FXF.F.F2F2F-F.,8*(F1F2I*Ltinfty23GF$F2F5F2F.
*(F8F2I*Ltinfty22GF$F2F;F2F.*&I*Ltinfty21GF$F2F6F2F.*&I*Ltinfty20GF$F2
F@F2F.*&F-F2I%LA20GF$F2F2*(F-F2I%LA22GF$F2F6F.F.**F8F2F-F2I%LA23GF$F2F
;F.F.**F1F2F-F2I%LA24GF$F2F5F.F.**FMF2F-F2I%LA25GF$F2FPF.F.**FRF2F-F2I
%LA26GF$F2FUF.F.**FWF2F-F2I%LA27GF$F2FZF.F.F2" }}{PARA 11 "" 1 "" 
{XPPMATH 20 ">I*psi1InftyG6\"-I$expG6$%*protectedGI(_syslibGF$6#,8**#
\"\"\"\"\"$F.I)tinfty13GF$F.I\"hGF$!\"\")I'lambdaGF$F/F.F2**#F.\"\"#F.
I)tinfty12GF$F.F1F2)F4F7F.F2*(I)tinfty11GF$F.F1F2F4F.F2*(I)tinfty10GF$
F.F1F2-I#lnGF'6#F4F.F2I$A10GF$F.*&I$A12GF$F.F4F2F2*(F6F.I$A13GF$F.F9F2
F2*(F-F.I$A14GF$F.F3F2F2*(#F.\"\"%F.I$A15GF$F.)F4FJF2F2*(#F.\"\"&F.I$A
16GF$F.)F4FOF2F2*(#F.\"\"'F.I$A17GF$F.)F4FTF2F2" }}{PARA 11 "" 1 "" 
{XPPMATH 20 ">I*psi2InftyG6\"-I$expG6$%*protectedGI(_syslibGF$6#,:**#
\"\"\"\"\"$F.I)tinfty23GF$F.I\"hGF$!\"\")I'lambdaGF$F/F.F2**#F.\"\"#F.
I)tinfty22GF$F.F1F2)F4F7F.F2*(I)tinfty21GF$F.F1F2F4F.F2*(I)tinfty10GF$
F.F1F2-I#lnGF'6#F4F.F.F>F2I$A20GF$F.*&I$A22GF$F.F4F2F2*(F6F.I$A23GF$F.
F9F2F2*(F-F.I$A24GF$F.F3F2F2*(#F.\"\"%F.I$A25GF$F.)F4FJF2F2*(#F.\"\"&F
.I$A26GF$F.)F4FOF2F2*(#F.\"\"'F.I$A27GF$F.)F4FTF2F2" }}}{SECT 0 {PARA 
3 "" 0 "" {TEXT 221 30 "Expression of the Lax matrix L" }}{EXCHG 
{PARA 0 "" 0 "" {TEXT 209 29 "Shape of L_\{2,1\} at infinity." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "L21Infty:=factor(simplify(Wr
onskianTildeLambdaInfty/WronskianLambdabisInfty)):\n" }{MPLTEXT 1 0 
75 "L21InftyOrderlambda5:=factor(-residue(L21Infty/lambda^6,lambda=inf
inity));\n" }{MPLTEXT 1 0 75 "L21InftyOrderlambda4:=factor(-residue(L2
1Infty/lambda^5,lambda=infinity));\n" }{MPLTEXT 1 0 75 "L21InftyOrderl
ambda3:=factor(-residue(L21Infty/lambda^4,lambda=infinity));\n" }
{MPLTEXT 1 0 75 "L21InftyOrderlambda2:=factor(-residue(L21Infty/lambda
^3,lambda=infinity));\n" }{MPLTEXT 1 0 75 "L21InftyOrderlambda1:=facto
r(-residue(L21Infty/lambda^2,lambda=infinity));\n" }{MPLTEXT 1 0 74 "L
21InftyOrderlambda0:=factor(-residue(L21Infty/lambda^1,lambda=infinity
));" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I5L21InftyOrderlambda5G6\"\"\"!"
 }}{PARA 11 "" 1 "" {XPPMATH 20 ">I5L21InftyOrderlambda4G6\",$*&I)tinf
ty13GF$\"\"\"I)tinfty23GF$F(!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I5
L21InftyOrderlambda3G6\",&*&I)tinfty12GF$\"\"\"I)tinfty23GF$F(!\"\"*&I
)tinfty13GF$F(I)tinfty22GF$F(F*" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I5L2
1InftyOrderlambda2G6\",(*&I)tinfty11GF$\"\"\"I)tinfty23GF$F(!\"\"*&I)t
infty12GF$F(I)tinfty22GF$F(F**&I)tinfty13GF$F(I)tinfty21GF$F(F*" }}
{PARA 11 "" 1 "" {XPPMATH 20 ">I5L21InftyOrderlambda1G6\",,*&I\"hGF$\"
\"\"I)tinfty13GF$F(!\"\"*&I)tinfty10GF$F(F)F(F(*&F,F(I)tinfty23GF$F(F*
*&I)tinfty11GF$F(I)tinfty22GF$F(F**&I)tinfty12GF$F(I)tinfty21GF$F(F*" 
}}{PARA 11 "" 1 "" {XPPMATH 20 ">I5L21InftyOrderlambda0G6\"*&,:**I$A12
GF$\"\"\"I\"hGF$F)I)tinfty13GF$F)I)tinfty23GF$F)F)*(F(F)F*F))F,\"\"#F)
!\"\"*(I$A22GF$F)F*F))F+F/F)F)**F2F)F*F)F+F)F,F)F0*(F*F)I)tinfty12GF$F
)F+F)F0*(F*F)F+F)I)tinfty22GF$F)F)*(I)tinfty10GF$F)F6F)F+F)F)*(F:F)F6F
)F,F)F0*(F:F)F+F)F8F)F0*(F:F)F8F)F,F)F)*(I)tinfty11GF$F)F+F)I)tinfty21
GF$F)F0*(F?F)F@F)F,F)F)F),&F,F0F+F)F0" }}}{EXCHG {PARA 0 "" 0 "" {TEXT
 219 49 "We conclude that L_\{2,1\} behaves at infinity like" }}{PARA 
0 "" 0 "" {TEXT 219 240 " -tinfty13*tinfty23*lambda^4-(tinfty22*tinfty
13+tinfty23*tinfty12)*lambda^3-(tinfty21*tinfty13+tinfty23*tinfty11+ti
nfty22*tinfty12)*lambda^2-(tinfty13*tinfty20+tinfty21*tinfty12+tinfty2
3*tinfty10+tinfty22*tinfty11+tinfty13*h)*lambda +O(1) \n" }{TEXT 219 
16 "In other words:\n" }{TEXT 219 57 "L_\{2,1\}= -P_2(lambda)+Pinfty02
+C -h*lambda*tinfty13+ O(1)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
178 "factor(simplify(L21InftyOrderlambda4*lambda^4+L21InftyOrderlambda
3*lambda^3+L21InftyOrderlambda2*lambda^2+L21InftyOrderlambda1*lambda- \+
(-P2(lambda)+Pinfty02-h*lambda*tinfty13)));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 209 29 "Shape of L
_\{2,2\} at infinity." }{TEXT 209 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 92 "L22Infty:=factor(h*simplify(diff(WronskianLambdabisIn
fty,lambda)/WronskianLambdabisInfty)):\n" }{MPLTEXT 1 0 75 "L22InftyOr
derlambda5:=factor(-residue(L22Infty/lambda^6,lambda=infinity));\n" }
{MPLTEXT 1 0 75 "L22InftyOrderlambda4:=factor(-residue(L22Infty/lambda
^5,lambda=infinity));\n" }{MPLTEXT 1 0 75 "L22InftyOrderlambda3:=facto
r(-residue(L22Infty/lambda^4,lambda=infinity));\n" }{MPLTEXT 1 0 75 "L
22InftyOrderlambda2:=factor(-residue(L22Infty/lambda^3,lambda=infinity
));\n" }{MPLTEXT 1 0 75 "L22InftyOrderlambda1:=factor(-residue(L22Inft
y/lambda^2,lambda=infinity));\n" }{MPLTEXT 1 0 75 "L22InftyOrderlambda
0:=factor(-residue(L22Infty/lambda^1,lambda=infinity));\n" }{MPLTEXT 
1 0 80 "L22InftyOrderlambdaMinus1:=factor(-residue(L22Infty/lambda^0,l
ambda=infinity));\n" }{MPLTEXT 1 0 20 "L22InftyOrderlambdaM" }{MPLTEXT
 1 0 4 "inus" }{MPLTEXT 1 0 58 "2:=factor(-residue(L22Infty/lambda^(-1
),lambda=infinity));" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I5L22InftyOrder
lambda5G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I5L22InftyOrderlamb
da4G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I5L22InftyOrderlambda3G
6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I5L22InftyOrderlambda2G6\",
&I)tinfty13GF$!\"\"I)tinfty23GF$F'" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I
5L22InftyOrderlambda1G6\",&I)tinfty12GF$!\"\"I)tinfty22GF$F'" }}{PARA 
11 "" 1 "" {XPPMATH 20 ">I5L22InftyOrderlambda0G6\",&I)tinfty11GF$!\"
\"I)tinfty21GF$F'" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I:L22InftyOrderlam
bdaMinus1G6\"I\"hGF$" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I:L22InftyOrder
lambdaMinus2G6\"*(I\"hGF$\"\"\",.*&I$A12GF$F'I)tinfty13GF$F'F'*&F*F'I)
tinfty23GF$F'!\"\"*&I$A22GF$F'F+F'F'*&F0F'F-F'F.I)tinfty12GF$F.I)tinft
y22GF$F'F',&F-F.F+F'F." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 219 166 "We de
duce that L_\{2,2\} behaves at infinity like -(tinfty13+tinfty23)*lamb
da^2 -(tinfty12+tinfty22)*lambda +h/lambda +O(1/lambda^2) =P1(lambda) \+
+h/lambda+O(1/lambda^2)" }}}{EXCHG {PARA 209 "" 0 "" {TEXT 222 120 "Co
nclusion: Using the additional apparent singularities and the definiti
on of the Darboux coordinates, we conclude that " }{TEXT 214 37 "L_\{2
,2\}=  P_1(lambda)  +h/(lambda-q)," }{TEXT 222 5 "     " }{TEXT 215 7 
"L_\{2,1\}" }{TEXT 216 2 "= " }{TEXT 209 42 "-P_2(lambda)+Pinfty02+C -
h*lambda*tinfty13" }{TEXT 217 18 "   -p*h/(lambda-q)" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 66 "L21Form:=-P2(lambda)+Pinfty02 -h*lambda*t
infty13- p*h/(lambda-q);\n" }{MPLTEXT 1 0 36 "L22Form:=P1(lambda)  +h/
(lambda-q);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I(L21FormG6\",.*&,**&I
)tinfty10GF$\"\"\"I)tinfty13GF$F*!\"\"*&F)F*I)tinfty23GF$F*F**&I)tinft
y11GF$F*I)tinfty22GF$F*F**&I)tinfty12GF$F*I)tinfty21GF$F*F*F*I'lambdaG
F$F*F,*&,(*&F0F*F.F*F**&F3F*F1F*F**&F+F*F4F*F*F*)F5\"\"#F*F,*&,&*&F3F*
F.F*F**&F+F*F1F*F*F*)F5\"\"$F*F,*(F+F*F.F*)F5\"\"%F*F,*(I\"hGF$F*F5F*F
+F*F,*(I\"pGF$F*FGF*,&F5F*I\"qGF$F,F,F," }}{PARA 11 "" 1 "" {XPPMATH 
20 ">I(L22FormG6\",,I)tinfty11GF$!\"\"I)tinfty21GF$F'*&,&I)tinfty12GF$
F'I)tinfty22GF$F'\"\"\"I'lambdaGF$F-F-*&,&I)tinfty13GF$F'I)tinfty23GF$
F'F-)F.\"\"#F-F-*&I\"hGF$F-,&F.F-I\"qGF$F'F'F-" }}}}{SECT 0 {PARA 3 ""
 0 "" {TEXT 221 0 "" }}{EXCHG {PARA 209 "" 0 "" {TEXT 222 18 "Auxiliar
y matrix A" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 219 35 "We define the oper
ator \\mathcal\{L\}=" }{TEXT 219 228 "\\hbar (alpha13\\partial_\{t_\{
\\infty^\{(1)\},3\} +alpha23\\partial_\{t_\{\\infty^\{(2)\},3\}+alpha1
2\\partial_\{t_\{\\infty^\{(1)\},2\} +alpha22\\partial_\{t_\{\\infty^
\{(2)\},2\}+alpha11\\partial_\{t_\{\\infty^\{(1)\},1\} +alpha21\\parti
al_\{t_\{\\infty^\{(2)\},1\})) " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 68 "WronskianLInfty:=factor(psi1Infty*Lpsi2Infty-psi2Infty*Lpsi1In
fty):\n" }{MPLTEXT 1 0 66 "A12Infty:=factor(simplify(WronskianLInfty/W
ronskianLambdaInfty)):\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }
{MPLTEXT 1 0 1 "\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 209 56 "We test th
e alternative formulas for A_\{1,2\} and A_\{1,1\}" }}}{EXCHG {PARA 0 
"> " 0 "" {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 39 "Z1Infty:=factor(Lpsi1Infty/psi1Infty):\n" }
{MPLTEXT 1 0 39 "Z2Infty:=factor(Lpsi2Infty/psi2Infty):\n" }{MPLTEXT 
1 0 68 "A12bisInfty:=factor(simplify((Z2Infty-Z1Infty)/(Y2Infty-Y1Inft
y))):\n" }{MPLTEXT 1 0 83 "A11Infty:=factor(simplify( (Y2Infty*Z1Infty
-Y1Infty*Z2Infty)/(Y2Infty-Y1Infty) )):\n" }{MPLTEXT 1 0 39 "factor(si
mplify(A12bisInfty-A12Infty));" }{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 209 55 "We define \+
the coefficients of the operator \\mathcal\{L\}." }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 22 "Ltinfty13:=h*alpha13:\n" }{MPLTEXT 1 0 22 "Lt
infty23:=h*alpha23:\n" }{MPLTEXT 1 0 22 "Ltinfty12:=h*alpha12:\n" }
{MPLTEXT 1 0 22 "Ltinfty22:=h*alpha22:\n" }{MPLTEXT 1 0 22 "Ltinfty11:
=h*alpha11:\n" }{MPLTEXT 1 0 22 "Ltinfty21:=h*alpha21:\n" }{MPLTEXT 1 
0 14 "Ltinfty10:=0:\n" }{MPLTEXT 1 0 14 "Ltinfty20:=0:\n" }{MPLTEXT 1 
0 1 "\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 209 34 "Asymptotic of A_\{1,2
\} at infinity." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "A12Infty
Lambda3:=factor(-residue(A12Infty/lambda^4,lambda=infinity));\n" }
{MPLTEXT 1 0 70 "A12InftyLambda2:=factor(-residue(A12Infty/lambda^3,la
mbda=infinity));\n" }{MPLTEXT 1 0 70 "A12InftyLambda1:=factor(-residue
(A12Infty/lambda^2,lambda=infinity));\n" }{MPLTEXT 1 0 70 "A12InftyLam
bda0:=factor(-residue(A12Infty/lambda^1,lambda=infinity));\n" }
{MPLTEXT 1 0 75 "A12InftyLambdaMoins1:=factor(-residue(A12Infty/lambda
^0,lambda=infinity));\n" }{MPLTEXT 1 0 78 "A12InftyLambdaMoins2:=facto
r(-residue(A12Infty/lambda^(-1),lambda=infinity)):\n" }}{PARA 11 "" 1 
"" {XPPMATH 20 ">I0A12InftyLambda3G6\"\"\"!" }}{PARA 11 "" 1 "" 
{XPPMATH 20 ">I0A12InftyLambda2G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 
20 ">I0A12InftyLambda1G6\",$*(#\"\"\"\"\"$F(,&I(alpha13GF$F(I(alpha23G
F$!\"\"F(,&I)tinfty23GF$F-I)tinfty13GF$F(F-F(" }}{PARA 11 "" 1 "" 
{XPPMATH 20 ">I0A12InftyLambda0G6\",$*(#\"\"\"\"\"'F(,2*(\"\"$F(I(alph
a12GF$F(I)tinfty13GF$F(F(*(F,F(F-F(I)tinfty23GF$F(!\"\"*(\"\"#F(I(alph
a13GF$F(I)tinfty12GF$F(F1*(F3F(F4F(I)tinfty22GF$F(F(*(F,F(I(alpha22GF$
F(F.F(F1*(F,F(F9F(F0F(F(*(F3F(I(alpha23GF$F(F5F(F(*(F3F(F<F(F7F(F1F(),
&F0F1F.F(F3F1F(" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I5A12InftyLambdaMoin
s1G6\",$*(#\"\"\"\"\"'F(,Z*(F)F(I(alpha11GF$F()I)tinfty13GF$\"\"#F(F(*
*\"#7F(F,F(F.F(I)tinfty23GF$F(!\"\"*(F)F(F,F()F2F/F(F(**\"\"$F(I(alpha
12GF$F(I)tinfty12GF$F(F.F(F3**F7F(F8F(F9F(F2F(F(**F7F(F8F(F.F(I)tinfty
22GF$F(F(**F7F(F8F(F<F(F2F(F3**F/F(I(alpha13GF$F(I)tinfty11GF$F(F.F(F3
**F/F(F?F(F@F(F2F(F(*(F/F(F?F()F9F/F(F(**\"\"%F(F?F(F9F(F<F(F3**F/F(F?
F(F.F(I)tinfty21GF$F(F(**F/F(F?F(FGF(F2F(F3*(F/F(F?F()F<F/F(F(*(F)F(I(
alpha21GF$F(F-F(F3**F1F(FLF(F.F(F2F(F(*(F)F(FLF(F5F(F3**F7F(I(alpha22G
F$F(F9F(F.F(F(**F7F(FPF(F9F(F2F(F3**F7F(FPF(F.F(F<F(F3**F7F(FPF(F<F(F2
F(F(**F/F(I(alpha23GF$F(F@F(F.F(F(**F/F(FUF(F@F(F2F(F3*(F/F(FUF(FCF(F3
**FEF(FUF(F9F(F<F(F(**F/F(FUF(F.F(FGF(F3**F/F(FUF(FGF(F2F(F(*(F/F(FUF(
FJF(F3F(),&F2F3F.F(F7F3F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 219 20 "We \+
thus deduce that " }{TEXT 218 115 "A_\{1,2\}=(alpha13-alpha23)/3/(tinf
ty13-tinfty23)*lambda+nu+mu/(lambda-q). Expressions of (mu,nu) are obt
ained below." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "A12Form:=(a
lpha13-alpha23)/3/(-tinfty23+tinfty13)*lambda+nu+ mu/(lambda-q);\n" }
{MPLTEXT 1 0 71 "simplify(-residue(A12Form/lambda^2,lambda=infinity)-A
12InftyLambda1); \n" }{MPLTEXT 1 0 154 "solve(\{factor(-residue(A12For
m/lambda,lambda=infinity))=A12InftyLambda0,factor(-residue(A12Form,lam
bda=infinity))=factor(A12InftyLambdaMoins1)\},\{mu,nu\});\n" }{MPLTEXT
 1 0 1 "\n" }{MPLTEXT 1 0 814 "mu := -1/6*(-2*alpha23*tinfty21*tinfty2
3+3*alpha22*tinfty13*tinfty22+3*alpha22*tinfty23*tinfty12-3*alpha22*ti
nfty23*tinfty22-12*alpha21*tinfty13*tinfty23+12*alpha11*tinfty13*tinft
y23+2*alpha13*tinfty11*tinfty13-2*alpha13*tinfty11*tinfty23-2*alpha13*
tinfty21*tinfty13+2*alpha13*tinfty21*tinfty23-2*alpha23*tinfty11*tinft
y13+2*alpha23*tinfty11*tinfty23+2*alpha23*tinfty21*tinfty13+4*alpha13*
tinfty22*tinfty12-4*alpha23*tinfty22*tinfty12+3*alpha12*tinfty13*tinft
y12-3*alpha12*tinfty13*tinfty22-3*alpha12*tinfty23*tinfty12+3*alpha12*
tinfty23*tinfty22-3*alpha22*tinfty13*tinfty12-6*alpha11*tinfty23^2-2*a
lpha13*tinfty12^2-2*alpha13*tinfty22^2+2*alpha23*tinfty12^2+2*alpha23*
tinfty22^2+6*alpha21*tinfty13^2+6*alpha21*tinfty23^2-6*alpha11*tinfty1
3^2)/(-tinfty23^3+3*tinfty13*tinfty23^2-3*tinfty23*tinfty13^2+tinfty13
^3);\n" }{MPLTEXT 1 0 208 "nu := 1/6*(3*alpha22*tinfty23-3*alpha22*tin
fty13-3*alpha12*tinfty23+3*alpha12*tinfty13-2*alpha23*tinfty22+2*alpha
23*tinfty12+2*alpha13*tinfty22-2*alpha13*tinfty12)/(tinfty23^2-2*tinft
y13*tinfty23+tinfty13^2);" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }
}{PARA 11 "" 1 "" {XPPMATH 20 ">I(A12FormG6\",(**#\"\"\"\"\"$F(,&I(alp
ha13GF$F(I(alpha23GF$!\"\"F(,&I)tinfty23GF$F-I)tinfty13GF$F(F-I'lambda
GF$F(F(I#nuGF$F(*&I#muGF$F(,&F1F(I\"qGF$F-F-F(" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "<$/I#muG6\",$*(#\"
\"\"\"\"'F),Z*(F*F)I(alpha11GF%F))I)tinfty13GF%\"\"#F)F)**\"#7F)F-F)F/
F)I)tinfty23GF%F)!\"\"*(F*F)F-F))F3F0F)F)**\"\"$F)I(alpha12GF%F)I)tinf
ty12GF%F)F/F)F4**F8F)F9F)F:F)F3F)F)**F8F)F9F)F/F)I)tinfty22GF%F)F)**F8
F)F9F)F=F)F3F)F4**F0F)I(alpha13GF%F)I)tinfty11GF%F)F/F)F4**F0F)F@F)FAF
)F3F)F)*(F0F)F@F))F:F0F)F)**\"\"%F)F@F)F:F)F=F)F4**F0F)F@F)F/F)I)tinft
y21GF%F)F)**F0F)F@F)FHF)F3F)F4*(F0F)F@F))F=F0F)F)*(F*F)I(alpha21GF%F)F
.F)F4**F2F)FMF)F/F)F3F)F)*(F*F)FMF)F6F)F4**F8F)I(alpha22GF%F)F:F)F/F)F
)**F8F)FQF)F:F)F3F)F4**F8F)FQF)F/F)F=F)F4**F8F)FQF)F=F)F3F)F)**F0F)I(a
lpha23GF%F)FAF)F/F)F)**F0F)FVF)FAF)F3F)F4*(F0F)FVF)FDF)F4**FFF)FVF)F:F
)F=F)F)**F0F)FVF)F/F)FHF)F4**F0F)FVF)FHF)F3F)F)*(F0F)FVF)FKF)F4F),**$)
F/F8F)F)*(F8F)F.F)F3F)F4*(F8F)F/F)F6F)F)*$)F3F8F)F4F4F)/I#nuGF%,$*(F(F
),2*(F8F)F9F)F/F)F)*(F8F)F9F)F3F)F4*(F0F)F@F)F:F)F4*(F0F)F@F)F=F)F)*(F
8F)FQF)F/F)F4*(F8F)FQF)F3F)F)*(F0F)FVF)F:F)F)*(F0F)FVF)F=F)F4F),(*$F.F
)F)*(F0F)F/F)F3F)F4*$F6F)F)F4F)" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I#mu
G6\",$*(#\"\"\"\"\"'F(,Z*(F)F(I(alpha11GF$F()I)tinfty13GF$\"\"#F(!\"\"
**\"#7F(F,F(F.F(I)tinfty23GF$F(F(*(F)F(F,F()F3F/F(F0**\"\"$F(I(alpha12
GF$F(I)tinfty12GF$F(F.F(F(**F7F(F8F(F9F(F3F(F0**F7F(F8F(F.F(I)tinfty22
GF$F(F0**F7F(F8F(F<F(F3F(F(**F/F(I(alpha13GF$F(I)tinfty11GF$F(F.F(F(**
F/F(F?F(F@F(F3F(F0*(F/F(F?F()F9F/F(F0**\"\"%F(F?F(F9F(F<F(F(**F/F(F?F(
F.F(I)tinfty21GF$F(F0**F/F(F?F(FGF(F3F(F(*(F/F(F?F()F<F/F(F0*(F)F(I(al
pha21GF$F(F-F(F(**F2F(FLF(F.F(F3F(F0*(F)F(FLF(F5F(F(**F7F(I(alpha22GF$
F(F9F(F.F(F0**F7F(FPF(F9F(F3F(F(**F7F(FPF(F.F(F<F(F(**F7F(FPF(F<F(F3F(
F0**F/F(I(alpha23GF$F(F@F(F.F(F0**F/F(FUF(F@F(F3F(F(*(F/F(FUF(FCF(F(**
FEF(FUF(F9F(F<F(F0**F/F(FUF(F.F(FGF(F(**F/F(FUF(FGF(F3F(F0*(F/F(FUF(FJ
F(F(F(,**$)F.F7F(F(*(F7F(F-F(F3F(F0*(F7F(F.F(F5F(F(*$)F3F7F(F0F0F0" }}
{PARA 11 "" 1 "" {XPPMATH 20 ">I#nuG6\",$*(#\"\"\"\"\"'F(,2*(\"\"$F(I(
alpha12GF$F(I)tinfty13GF$F(F(*(F,F(F-F(I)tinfty23GF$F(!\"\"*(\"\"#F(I(
alpha13GF$F(I)tinfty12GF$F(F1*(F3F(F4F(I)tinfty22GF$F(F(*(F,F(I(alpha2
2GF$F(F.F(F1*(F,F(F9F(F0F(F(*(F3F(I(alpha23GF$F(F5F(F(*(F3F(F<F(F7F(F1
F(,(*$)F.F3F(F(*(F3F(F.F(F0F(F1*$)F0F3F(F(F1F(" }}}{EXCHG {PARA 0 "" 0
 "" {TEXT 209 28 "Study of A_\{1,1\} at infinity" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 70 "A11InftyLambda4:=factor(-residue(A11Infty/lam
bda^5,lambda=infinity));\n" }{MPLTEXT 1 0 70 "A11InftyLambda3:=factor(
-residue(A11Infty/lambda^4,lambda=infinity));\n" }{MPLTEXT 1 0 70 "A11
InftyLambda2:=factor(-residue(A11Infty/lambda^3,lambda=infinity));\n" 
}{MPLTEXT 1 0 70 "A11InftyLambda1:=factor(-residue(A11Infty/lambda^2,l
ambda=infinity));\n" }{MPLTEXT 1 0 70 "A11InftyLambda0:=factor(-residu
e(A11Infty/lambda^1,lambda=infinity)):\n" }{MPLTEXT 1 0 75 "A11InftyLa
mbdaMoins1:=factor(-residue(A11Infty/lambda^0,lambda=infinity)):\n" }}
{PARA 11 "" 1 "" {XPPMATH 20 ">I0A11InftyLambda4G6\"\"\"!" }}{PARA 11 
"" 1 "" {XPPMATH 20 ">I0A11InftyLambda3G6\",$*(#\"\"\"\"\"$F(,&*&I(alp
ha13GF$F(I)tinfty23GF$F(F(*&I(alpha23GF$F(I)tinfty13GF$F(!\"\"F(,&F-F1
F0F(F1F(" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I0A11InftyLambda2G6\",$*(#
\"\"\"\"\"'F(,2**\"\"$F(I(alpha12GF$F(I)tinfty13GF$F(I)tinfty23GF$F(F(
*(F,F(F-F()F/\"\"#F(!\"\"**F2F(I(alpha13GF$F(I)tinfty12GF$F(F/F(F3**F2
F(F5F(F.F(I)tinfty22GF$F(F(*(F,F(I(alpha22GF$F()F.F2F(F3**F,F(F:F(F.F(
F/F(F(**F2F(I(alpha23GF$F(F6F(F/F(F(**F2F(F>F(F.F(F8F(F3F(),&F/F3F.F(F
2F3F(" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I0A11InftyLambda1G6\",$*(#\"\"
\"\"\"'F(,hn**F)F(I(alpha11GF$F()I)tinfty13GF$\"\"#F(I)tinfty23GF$F(F(
**\"#7F(F,F(F.F()F0F/F(!\"\"*(F)F(F,F()F0\"\"$F(F(*,F7F(I(alpha12GF$F(
I)tinfty12GF$F(F.F(F0F(F4**F7F(F9F(F:F(F3F(F(**F7F(F9F(F-F(I)tinfty22G
F$F(F(*,F7F(F9F(F.F(F=F(F0F(F4*,F/F(I(alpha13GF$F(I)tinfty11GF$F(F.F(F
0F(F4**F/F(F@F(FAF(F3F(F(**F/F(F@F()F:F/F(F0F(F(*,F/F(F@F(F:F(F.F(F=F(
F4*,F/F(F@F(F:F(F=F(F0F(F4**F/F(F@F(F-F(I)tinfty21GF$F(F(*,F/F(F@F(F.F
(FHF(F0F(F4**F/F(F@F(F.F()F=F/F(F(*(F)F(I(alpha21GF$F()F.F7F(F4**F2F(F
MF(F-F(F0F(F(**F)F(FMF(F.F(F3F(F4*,F7F(I(alpha22GF$F(F:F(F.F(F0F(F(**F
7F(FRF(F:F(F3F(F4**F7F(FRF(F-F(F=F(F4*,F7F(FRF(F.F(F=F(F0F(F(*,F/F(I(a
lpha23GF$F(FAF(F.F(F0F(F(**F/F(FWF(FAF(F3F(F4**F/F(FWF(FDF(F0F(F4*,F/F
(FWF(F:F(F.F(F=F(F(*,F/F(FWF(F:F(F=F(F0F(F(**F/F(FWF(F-F(FHF(F4*,F/F(F
WF(F.F(FHF(F0F(F(**F/F(FWF(F.F(FKF(F4F(),&F0F4F.F(F7F4F(" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT 209 179 "We deduce that A_\{1,1\}=1/3*(alpha13*t
infty23-alpha23*tinfty13)/(tinfty13-tinfty23)*lambda^3+c2*lambda^2+c1*
lambda +c0+ rho/(lambda-q). Expressions of (c_1,c_2) are obtained belo
w." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 121 "A11Form:=1/3*(alpha1
3*tinfty23-alpha23*tinfty13)/(tinfty13-tinfty23)*lambda^3+c2*lambda^2+
c1*lambda +c0+ rho/(lambda-q);\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 
71 "simplify(-residue(A11Form/lambda^4,lambda=infinity)-A11InftyLambda
3); \n" }{MPLTEXT 1 0 151 "solve(\{factor(-residue(A11Form/lambda^3,la
mbda=infinity))=A11InftyLambda2,factor(-residue(A11Form/lambda^2,lambd
a=infinity))=A11InftyLambda1\},\{c2,c1\});" }}{PARA 11 "" 1 "" 
{XPPMATH 20 ">I(A11FormG6\",,**#\"\"\"\"\"$F(,&*&I(alpha13GF$F(I)tinft
y23GF$F(F(*&I(alpha23GF$F(I)tinfty13GF$F(!\"\"F(,&F-F1F0F(F1)I'lambdaG
F$F)F(F(*&I#c2GF$F()F4\"\"#F(F(*&I#c1GF$F(F4F(F(I#c0GF$F(*&I$rhoGF$F(,
&F4F(I\"qGF$F1F1F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 
"" 1 "" {XPPMATH 20 "<$/I#c1G6\",$*(#\"\"\"\"\"'F),hn**F*F)I(alpha11GF
%F))I)tinfty13GF%\"\"#F)I)tinfty23GF%F)F)**\"#7F)F-F)F/F))F1F0F)!\"\"*
(F*F)F-F))F1\"\"$F)F)*,F8F)I(alpha12GF%F)I)tinfty12GF%F)F/F)F1F)F5**F8
F)F:F)F;F)F4F)F)**F8F)F:F)F.F)I)tinfty22GF%F)F)*,F8F)F:F)F/F)F>F)F1F)F
5*,F0F)I(alpha13GF%F)I)tinfty11GF%F)F/F)F1F)F5**F0F)FAF)FBF)F4F)F)**F0
F)FAF))F;F0F)F1F)F)*,F0F)FAF)F;F)F/F)F>F)F5*,F0F)FAF)F;F)F>F)F1F)F5**F
0F)FAF)F.F)I)tinfty21GF%F)F)*,F0F)FAF)F/F)FIF)F1F)F5**F0F)FAF)F/F))F>F
0F)F)*(F*F)I(alpha21GF%F))F/F8F)F5**F3F)FNF)F.F)F1F)F)**F*F)FNF)F/F)F4
F)F5*,F8F)I(alpha22GF%F)F;F)F/F)F1F)F)**F8F)FSF)F;F)F4F)F5**F8F)FSF)F.
F)F>F)F5*,F8F)FSF)F/F)F>F)F1F)F)*,F0F)I(alpha23GF%F)FBF)F/F)F1F)F)**F0
F)FXF)FBF)F4F)F5**F0F)FXF)FEF)F1F)F5*,F0F)FXF)F;F)F/F)F>F)F)*,F0F)FXF)
F;F)F>F)F1F)F)**F0F)FXF)F.F)FIF)F5*,F0F)FXF)F/F)FIF)F1F)F)**F0F)FXF)F/
F)FLF)F5F),**$FOF)F)*(F8F)F.F)F1F)F5*(F8F)F/F)F4F)F)*$F7F)F5F5F)/I#c2G
F%,$*(F(F),2**F8F)F:F)F/F)F1F)F)*(F8F)F:F)F4F)F5**F0F)FAF)F;F)F1F)F5**
F0F)FAF)F/F)F>F)F)*(F8F)FSF)F.F)F5**F8F)FSF)F/F)F1F)F)**F0F)FXF)F;F)F1
F)F)**F0F)FXF)F/F)F>F)F5F),(*$F.F)F)*(F0F)F/F)F1F)F5*$F4F)F)F5F)" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1058 "c1 := factor(-(2*alpha13*t
infty11*tinfty13*tinfty23-2*alpha13*tinfty11*tinfty23^2-2*alpha13*tinf
ty12^2*tinfty23+2*alpha13*tinfty12*tinfty13*tinfty22+2*alpha13*tinfty1
2*tinfty22*tinfty23-2*alpha13*tinfty13^2*tinfty21+2*alpha13*tinfty13*t
infty21*tinfty23-2*alpha13*tinfty13*tinfty22^2-2*alpha23*tinfty11*tinf
ty13*tinfty23+2*alpha23*tinfty11*tinfty23^2+2*alpha23*tinfty12^2*tinft
y23-2*alpha23*tinfty12*tinfty13*tinfty22-2*alpha23*tinfty12*tinfty22*t
infty23+2*alpha23*tinfty13^2*tinfty21-2*alpha23*tinfty13*tinfty21*tinf
ty23+2*alpha23*tinfty13*tinfty22^2+3*alpha12*tinfty12*tinfty13*tinfty2
3-3*alpha12*tinfty12*tinfty23^2-3*alpha12*tinfty13^2*tinfty22+3*alpha1
2*tinfty13*tinfty22*tinfty23-3*alpha22*tinfty12*tinfty13*tinfty23+3*al
pha22*tinfty12*tinfty23^2+3*alpha22*tinfty13^2*tinfty22-3*alpha22*tinf
ty13*tinfty22*tinfty23-6*alpha11*tinfty13^2*tinfty23+12*alpha11*tinfty
13*tinfty23^2-6*alpha11*tinfty23^3+6*alpha21*tinfty13^3-12*alpha21*tin
fty13^2*tinfty23+6*alpha21*tinfty13*tinfty23^2)/(6*(tinfty13^3-3*tinft
y13^2*tinfty23+3*tinfty13*tinfty23^2-tinfty23^3)));\n" }{MPLTEXT 1 0 
275 "c2 := factor(-(2*alpha13*tinfty12*tinfty23-2*alpha13*tinfty13*tin
fty22-2*alpha23*tinfty12*tinfty23+2*alpha23*tinfty13*tinfty22-3*alpha1
2*tinfty13*tinfty23+3*alpha12*tinfty23^2+3*alpha22*tinfty13^2-3*alpha2
2*tinfty13*tinfty23)/(6*(tinfty13^2-2*tinfty13*tinfty23+tinfty23^2)));
" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I#c1G6\",$*(#\"\"\"\"\"'F(,hn**F)F(
I(alpha11GF$F()I)tinfty13GF$\"\"#F(I)tinfty23GF$F(F(**\"#7F(F,F(F.F()F
0F/F(!\"\"*(F)F(F,F()F0\"\"$F(F(*,F7F(I(alpha12GF$F(I)tinfty12GF$F(F.F
(F0F(F4**F7F(F9F(F:F(F3F(F(**F7F(F9F(F-F(I)tinfty22GF$F(F(*,F7F(F9F(F.
F(F=F(F0F(F4*,F/F(I(alpha13GF$F(I)tinfty11GF$F(F.F(F0F(F4**F/F(F@F(FAF
(F3F(F(**F/F(F@F()F:F/F(F0F(F(*,F/F(F@F(F:F(F.F(F=F(F4*,F/F(F@F(F:F(F=
F(F0F(F4**F/F(F@F(F-F(I)tinfty21GF$F(F(*,F/F(F@F(F.F(FHF(F0F(F4**F/F(F
@F(F.F()F=F/F(F(*(F)F(I(alpha21GF$F()F.F7F(F4**F2F(FMF(F-F(F0F(F(**F)F
(FMF(F.F(F3F(F4*,F7F(I(alpha22GF$F(F:F(F.F(F0F(F(**F7F(FRF(F:F(F3F(F4*
*F7F(FRF(F-F(F=F(F4*,F7F(FRF(F.F(F=F(F0F(F(*,F/F(I(alpha23GF$F(FAF(F.F
(F0F(F(**F/F(FWF(FAF(F3F(F4**F/F(FWF(FDF(F0F(F4*,F/F(FWF(F:F(F.F(F=F(F
(*,F/F(FWF(F:F(F=F(F0F(F(**F/F(FWF(F-F(FHF(F4*,F/F(FWF(F.F(FHF(F0F(F(*
*F/F(FWF(F.F(FKF(F4F(),&F0F4F.F(F7F4F(" }}{PARA 11 "" 1 "" {XPPMATH 20
 ">I#c2G6\",$*(#\"\"\"\"\"'F(,2**\"\"$F(I(alpha12GF$F(I)tinfty13GF$F(I
)tinfty23GF$F(F(*(F,F(F-F()F/\"\"#F(!\"\"**F2F(I(alpha13GF$F(I)tinfty1
2GF$F(F/F(F3**F2F(F5F(F.F(I)tinfty22GF$F(F(*(F,F(I(alpha22GF$F()F.F2F(
F3**F,F(F:F(F.F(F/F(F(**F2F(I(alpha23GF$F(F6F(F/F(F(**F2F(F>F(F.F(F8F(
F3F(),&F/F3F.F(F2F3F(" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "" "%
#%?G" }}}}}
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