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{SECT 0 {EXCHG {PARA 216 "" 0 "" {TEXT 210 149 "In this Maple file, w
e compute the Casimir coefficients of the Lax matrix L associated to t
he Painlev\351 2 equation in relation with the spectral curve" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "restart:\n" }{MPLTEXT 1 0 42 
"P1:=x-> Pinfty01+Pinfty11*x+Pinfty21*x^2;\n" }{MPLTEXT 1 0 68 "P2:=x-
> Pinfty02+Pinfty12*x+Pinfty22*x^2+Pinfty32*x^3+Pinfty42*x^4;\n" }
{MPLTEXT 1 0 45 "SpectralCurve:=unapply( y^2-P1(x)*y+P2(x),y);" }}
{PARA 11 "" 1 "" {XPPMATH 20 ">I#P1G6\"f*6#I\"xGF$F$6$I)operatorGF$I&a
rrowGF$F$,(I)Pinfty01GF$\"\"\"*&I)Pinfty11GF$F-9$F-F-*&I)Pinfty21GF$F-
F0\"\"#F-F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I#P2G6\"f*6#I\"xGF$F
$6$I)operatorGF$I&arrowGF$F$,,I)Pinfty02GF$\"\"\"*&I)Pinfty12GF$F-9$F-
F-*&I)Pinfty22GF$F-F0\"\"#F-*&I)Pinfty32GF$F-F0\"\"$F-*&I)Pinfty42GF$F
-F0\"\"%F-F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I.SpectralCurveG6\"
f*6#I\"yGF$F$6$I)operatorGF$I&arrowGF$F$,0*$9$\"\"#\"\"\"*&,(*&I\"xGF$
F.I)Pinfty21GF$F/F/*&F3F/I)Pinfty11GF$F/F/I)Pinfty01GF$F/F/F-F/!\"\"*&
I)Pinfty42GF$F/F3\"\"%F/*&I)Pinfty32GF$F/F3\"\"$F/*&I)Pinfty22GF$F/F3F
.F/*&I)Pinfty12GF$F/F3F/F/I)Pinfty02GF$F/F$F$F$" }}}{EXCHG {PARA 0 "> 
" 0 "" {MPLTEXT 1 0 75 "DiaginftySheet1:=-tinfty13*x^2-tinfty12*x-tinf
ty11-tinfty10/x+Unknown/x^2;\n" }{MPLTEXT 1 0 15 "DiaginftySheet2" }
{MPLTEXT 1 0 60 ":=-tinfty23*x^2-tinfty22*x-tinfty21-tinfty20/x+Unknow
n2/x^2;" }{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I0Diaginf
tySheet1G6\",,*&I)tinfty13GF$\"\"\"I\"xGF$\"\"#!\"\"*&I)tinfty12GF$F(F
)F(F+I)tinfty11GF$F+*&I)tinfty10GF$F(F)F+F+*&I(UnknownGF$F(F)!\"#F(" }
}{PARA 11 "" 1 "" {XPPMATH 20 ">I0DiaginftySheet2G6\",,*&I)tinfty23GF$
\"\"\"I\"xGF$\"\"#!\"\"*&I)tinfty22GF$F(F)F(F+I)tinfty21GF$F+*&I)tinft
y20GF$F(F)F+F+*&I)Unknown2GF$F(F)!\"#F(" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT 215 67 "Expression of P_1 in terms of the diagonal expansion in \+
both sheets" }}}{EXCHG {PARA 0 "> " 0 "" {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 79 "CoherenceEquation1:=residue(DiaginftySheet1+DiaginftySheet2-P1
(x),x=infinity);\n" }{MPLTEXT 1 0 20 "tinfty20:=-tinfty10;" }{MPLTEXT 
1 0 0 "" }}{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$!\"\"I)tinfty23GF$F'" }}{PARA 11 "" 1 "" 
{XPPMATH 20 ">I3CoherenceEquation1G6\",&I)tinfty10GF$\"\"\"I)tinfty20G
F$F'" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I)tinfty20G6\",$I)tinfty10GF$!
\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 215 17 "Study at infinity" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "series(" }{MPLTEXT 1 0 14 "Sp
ectralCurve(" }{MPLTEXT 1 0 15 "DiaginftySheet1" }{MPLTEXT 1 0 17 "),x
=infinity,6):\n" }{MPLTEXT 1 0 7 "series(" }{MPLTEXT 1 0 14 "SpectralC
urve(" }{MPLTEXT 1 0 15 "DiaginftySheet2" }{MPLTEXT 1 0 17 "),x=infini
ty,6):\n" }{MPLTEXT 1 0 25 "EQinfty1:=residue(x^(-5)*" }{MPLTEXT 1 0 
14 "SpectralCurve(" }{MPLTEXT 1 0 15 "DiaginftySheet1" }{MPLTEXT 1 0 
15 "),x=infinity);\n" }{MPLTEXT 1 0 25 "EQinfty2:=residue(x^(-5)*" }
{MPLTEXT 1 0 14 "SpectralCurve(" }{MPLTEXT 1 0 15 "DiaginftySheet2" }
{MPLTEXT 1 0 15 "),x=infinity);\n" }{MPLTEXT 1 0 25 "EQinfty3:=residue
(x^(-4)*" }{MPLTEXT 1 0 14 "SpectralCurve(" }{MPLTEXT 1 0 15 "Diaginft
ySheet1" }{MPLTEXT 1 0 15 "),x=infinity);\n" }{MPLTEXT 1 0 25 "EQinfty
4:=residue(x^(-4)*" }{MPLTEXT 1 0 14 "SpectralCurve(" }{MPLTEXT 1 0 
15 "DiaginftySheet2" }{MPLTEXT 1 0 15 "),x=infinity);\n" }{MPLTEXT 1 
0 25 "EQinfty5:=residue(x^(-3)*" }{MPLTEXT 1 0 14 "SpectralCurve(" }
{MPLTEXT 1 0 15 "DiaginftySheet1" }{MPLTEXT 1 0 15 "),x=infinity);\n" 
}{MPLTEXT 1 0 25 "EQinfty6:=residue(x^(-3)*" }{MPLTEXT 1 0 14 "Spectra
lCurve(" }{MPLTEXT 1 0 15 "DiaginftySheet2" }{MPLTEXT 1 0 15 "),x=infi
nity);\n" }{MPLTEXT 1 0 25 "EQinfty7:=residue(x^(-2)*" }{MPLTEXT 1 0 
14 "SpectralCurve(" }{MPLTEXT 1 0 15 "DiaginftySheet1" }{MPLTEXT 1 0 
15 "),x=infinity);\n" }{MPLTEXT 1 0 25 "EQinfty8:=residue(x^(-2)*" }
{MPLTEXT 1 0 14 "SpectralCurve(" }{MPLTEXT 1 0 15 "DiaginftySheet2" }
{MPLTEXT 1 0 15 "),x=infinity);\n" }{MPLTEXT 1 0 25 "EQinfty9:=residue
(x^(-1)*" }{MPLTEXT 1 0 14 "SpectralCurve(" }{MPLTEXT 1 0 15 "Diaginft
ySheet1" }{MPLTEXT 1 0 15 "),x=infinity);\n" }{MPLTEXT 1 0 26 "EQinfty
10:=residue(x^(-1)*" }{MPLTEXT 1 0 14 "SpectralCurve(" }{MPLTEXT 1 0 
15 "DiaginftySheet2" }{MPLTEXT 1 0 15 "),x=infinity);\n" }{MPLTEXT 1 
0 1 "\n" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I)EQinfty1G6\",(*$I)tinfty13
GF$\"\"#!\"\"*&,&F'F)I)tinfty23GF$F)\"\"\"F'F-F)I)Pinfty42GF$F)" }}
{PARA 11 "" 1 "" {XPPMATH 20 ">I)EQinfty2G6\",(*$I)tinfty23GF$\"\"#!\"
\"*&,&I)tinfty13GF$F)F'F)\"\"\"F'F-F)I)Pinfty42GF$F)" }}{PARA 11 "" 1 
"" {XPPMATH 20 ">I)EQinfty3G6\",**&I)tinfty13GF$\"\"\"I)tinfty12GF$F(!
\"#*&,&F'!\"\"I)tinfty23GF$F-F(F)F(F-*&,&F)F-I)tinfty22GF$F-F(F'F(F-I)
Pinfty32GF$F-" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I)EQinfty4G6\",**&I)ti
nfty23GF$\"\"\"I)tinfty22GF$F(!\"#*&,&I)tinfty13GF$!\"\"F'F.F(F)F(F.*&
,&I)tinfty12GF$F.F)F.F(F'F(F.I)Pinfty32GF$F." }}{PARA 11 "" 1 "" 
{XPPMATH 20 ">I)EQinfty5G6\",.*&I)tinfty13GF$\"\"\"I)tinfty11GF$F(!\"#
*$I)tinfty12GF$\"\"#!\"\"*&,&F'F.I)tinfty23GF$F.F(F)F(F.*&,&F,F.I)tinf
ty22GF$F.F(F,F(F.*&,&F)F.I)tinfty21GF$F.F(F'F(F.I)Pinfty22GF$F." }}
{PARA 11 "" 1 "" {XPPMATH 20 ">I)EQinfty6G6\",.*&I)tinfty23GF$\"\"\"I)
tinfty21GF$F(!\"#*$I)tinfty22GF$\"\"#!\"\"*&,&I)tinfty13GF$F.F'F.F(F)F
(F.*&,&I)tinfty12GF$F.F,F.F(F,F(F.*&,&I)tinfty11GF$F.F)F.F(F'F(F.I)Pin
fty22GF$F." }}{PARA 11 "" 1 "" {XPPMATH 20 ">I)EQinfty7G6\",.*&I)tinft
y13GF$\"\"\"I)tinfty10GF$F(!\"#*&I)tinfty12GF$F(I)tinfty11GF$F(F**&,&F
'!\"\"I)tinfty23GF$F0F(F)F(F0*&,&F,F0I)tinfty22GF$F0F(F-F(F0*&,&F-F0I)
tinfty21GF$F0F(F,F(F0I)Pinfty12GF$F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "
>I)EQinfty8G6\",.*&I)tinfty10GF$\"\"\"I)tinfty23GF$F(\"\"#*&I)tinfty22
GF$F(I)tinfty21GF$F(!\"#*&,&I)tinfty13GF$!\"\"F)F2F(F'F(F(*&,&I)tinfty
12GF$F2F,F2F(F-F(F2*&,&I)tinfty11GF$F2F-F2F(F,F(F2I)Pinfty12GF$F2" }}
{PARA 11 "" 1 "" {XPPMATH 20 ">I)EQinfty9G6\",0*&I)tinfty13GF$\"\"\"I(
UnknownGF$F(\"\"#*&I)tinfty12GF$F(I)tinfty10GF$F(!\"#*$I)tinfty11GF$F*
!\"\"*&,&F'F1I)tinfty23GF$F1F(F)F(F(*&,&F,F1I)tinfty22GF$F1F(F-F(F1*&,
&F0F1I)tinfty21GF$F1F(F0F(F1I)Pinfty02GF$F1" }}{PARA 11 "" 1 "" 
{XPPMATH 20 ">I*EQinfty10G6\",0*&I)tinfty23GF$\"\"\"I)Unknown2GF$F(\"
\"#*&I)tinfty10GF$F(I)tinfty22GF$F(F**$I)tinfty21GF$F*!\"\"*&,&I)tinft
y13GF$F0F'F0F(F)F(F(*&,&I)tinfty12GF$F0F-F0F(F,F(F(*&,&I)tinfty11GF$F0
F/F0F(F/F(F0I)Pinfty02GF$F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
44 "Pinfty42:=factor(solve(EQinfty1,Pinfty42));\n" }{MPLTEXT 1 0 44 "P
infty32:=factor(solve(EQinfty3,Pinfty32));\n" }{MPLTEXT 1 0 44 "Pinfty
22:=factor(solve(EQinfty5,Pinfty22));\n" }{MPLTEXT 1 0 44 "Pinfty12:=f
actor(solve(EQinfty7,Pinfty12));\n" }{MPLTEXT 1 0 20 "simplify(EQinfty
8);\n" }{MPLTEXT 1 0 19 "simplify(EQinfty9);" }}{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$F(F(" }}{PARA 11 "" 1 "" {XPPMATH 
20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ",**&,&I)tinfty12G6\"!\"\"I)
tinfty22GF&\"\"\"F)I)tinfty10GF&F)F)*&,&I)tinfty13GF&F)I)tinfty23GF&F'
F)I(UnknownGF&F)F)*&I)tinfty11GF&F)I)tinfty21GF&F)F)I)Pinfty02GF&F'" }
}}{EXCHG {PARA 0 "" 0 "" {TEXT 210 27 "Summary of the coefficients" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "Pinfty01:=Pinfty01;\n" }
{MPLTEXT 1 0 20 "Pinfty11:=Pinfty11;\n" }{MPLTEXT 1 0 20 "Pinfty21:=Pi
nfty21;\n" }{MPLTEXT 1 0 20 "Pinfty02:=Pinfty02;\n" }{MPLTEXT 1 0 20 "
Pinfty12:=Pinfty12;\n" }{MPLTEXT 1 0 20 "Pinfty22:=Pinfty22;\n" }
{MPLTEXT 1 0 20 "Pinfty32:=Pinfty32;\n" }{MPLTEXT 1 0 20 "Pinfty42:=Pi
nfty42;\n" }{MPLTEXT 1 0 39 "CoherenceEquation1:=tinfty10+tinfty20;\n"
 }{MPLTEXT 1 0 1 "\n" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I)Pinfty01G6\",
&I)tinfty11GF$!\"\"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)tinfty23GF$F'" }}
{PARA 11 "" 1 "" {XPPMATH 20 ">I)Pinfty02G6\"F#" }}{PARA 11 "" 1 "" 
{XPPMATH 20 ">I)Pinfty12G6\",**&I)tinfty10GF$\"\"\"I)tinfty13GF$F(!\"
\"*&F'F(I)tinfty23GF$F(F(*&I)tinfty11GF$F(I)tinfty22GF$F(F(*&I)tinfty1
2GF$F(I)tinfty21GF$F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I)Pinfty22G6
\",(*&I)tinfty11GF$\"\"\"I)tinfty23GF$F(F(*&I)tinfty12GF$F(I)tinfty22G
F$F(F(*&I)tinfty13GF$F(I)tinfty21GF$F(F(" }}{PARA 11 "" 1 "" {XPPMATH 
20 ">I)Pinfty32G6\",&*&I)tinfty12GF$\"\"\"I)tinfty23GF$F(F(*&I)tinfty1
3GF$F(I)tinfty22GF$F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I)Pinfty42G6
\"*&I)tinfty13GF$\"\"\"I)tinfty23GF$F'" }}{PARA 11 "" 1 "" {XPPMATH 20
 ">I3CoherenceEquation1G6\"\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 210 
132 "We have one undetermined coefficient Pinfty02, all the other coef
ficients are expressed in terms of irregular times and monodromies." }
}}{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 }