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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT 208 135 "In this Maple file, we \+
compute the coefficients of the polynomials P_1 and P_2 in terms of th
e irregular times for the Painlev\351 1 case." }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 9 "restart:\n" }{MPLTEXT 1 0 29 "P1:=x-> Pinfty01+Pi
nfty11*x;\n" }{MPLTEXT 1 0 54 "P2:=x-> Pinfty02+Pinfty12*x+Pinfty22*x^
2+Pinfty32*x^3;" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I#P1G6\"f*6#I\"xGF$F
$6$I)operatorGF$I&arrowGF$F$,&I)Pinfty01GF$\"\"\"*&I)Pinfty11GF$F-9$F-
F-F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I#P2G6\"f*6#I\"xGF$F$6$I)op
eratorGF$I&arrowGF$F$,*I)Pinfty02GF$\"\"\"*&I)Pinfty12GF$F-9$F-F-*&I)P
infty22GF$F-F0\"\"#F-*&I)Pinfty32GF$F-F0\"\"$F-F$F$F$" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "ClassicalSpectralCurve:=unapply(y^2
-P1(x)*y+P2(x),y);" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I7ClassicalSpectr
alCurveG6\"f*6#I\"yGF$F$6$I)operatorGF$I&arrowGF$F$,.*$9$\"\"#\"\"\"*&
,&*&I\"xGF$F/I)Pinfty11GF$F/F/I)Pinfty01GF$F/F/F-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 29 "Yinfty:=-1/2*t
infty15*x^(3/2)" }{MPLTEXT 1 0 15 "-1/2*tinfty14*x" }{MPLTEXT 1 0 21 "
-1/2*tinfty13*x^(1/2)" }{MPLTEXT 1 0 13 "-1/2*tinfty12" }{MPLTEXT 1 0 
59 "-1/2*tinfty11*x^(-1/2)-1/2*tinfty10*x^(-1)+ Unknown/x^(3/2)" }
{MPLTEXT 1 0 2 ";\n" }{MPLTEXT 1 0 11 "Yinftybis:=" }{MPLTEXT 1 0 21 "
-1/2*tinfty25*x^(3/2)" }{MPLTEXT 1 0 109 "-1/2*tinfty24*x-1/2*tinfty23
*x^(1/2)-1/2*tinfty22-1/2*tinfty21*x^(-1/2)-1/2*tinfty20*x^(-1)+ Unkno
wnn/x^(3/2)" }{MPLTEXT 1 0 1 ";" }{MPLTEXT 1 0 1 "\n" }}{PARA 11 "" 1 
"" {XPPMATH 20 ">I'YinftyG6\",0*&I)tinfty15GF$\"\"\"I\"xGF$#\"\"$\"\"#
#!\"\"F,*&I)tinfty14GF$F(F)F(F-*&I)tinfty13GF$F(F)#F(F,F-I)tinfty12GF$
F-*&I)tinfty11GF$F(F)F-F-*&I)tinfty10GF$F(F)F.F-*&I(UnknownGF$F(F)#!\"
$F,F(" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I*YinftybisG6\",0*&I)tinfty25G
F$\"\"\"I\"xGF$#\"\"$\"\"##!\"\"F,*&I)tinfty24GF$F(F)F(F-*&I)tinfty23G
F$F(F)#F(F,F-I)tinfty22GF$F-*&I)tinfty21GF$F(F)F-F-*&I)tinfty20GF$F(F)
F.F-*&I)UnknownnGF$F(F)#!\"$F,F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 213 
40 "Expression of P1 in terms of both sheets" }}}{EXCHG {PARA 0 "> " 0
 "" {MPLTEXT 1 0 42 "series(Yinfty+Yinftybis-P1(x),x=infinity);" }}
{PARA 11 "" 1 "" {XPPMATH 20 ",4*&,&I)tinfty15G6\"#!\"\"\"\"#I)tinfty2
5GF&F'\"\"\"*$I\"xGF&F(#!\"$F)F+*&,(I)tinfty14GF&F'I)tinfty24GF&F'I)Pi
nfty11GF&F(F+F-F+F+*&,&I)tinfty13GF&F'I)tinfty23GF&F'F+F,F'F+I)tinfty1
2GF&F'I)tinfty22GF&F'I)Pinfty01GF&F(*&,&I)tinfty11GF&F'I)tinfty21GF&F'
F+F,#F+F)F+*&,&I)tinfty10GF&F'I)tinfty20GF&F'F+F-F(F+*&,&I(UnknownGF&F
+I)UnknownnGF&F+F+F,#\"\"$F)F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 21 "tinfty25:=-tinfty15:\n" }{MPLTEXT 1 0 21 "tinfty23:=-tinfty13:
\n" }{MPLTEXT 1 0 21 "tinfty21:=-tinfty11:\n" }{MPLTEXT 1 0 21 "tinfty
20:=-tinfty10:\n" }{MPLTEXT 1 0 34 "Pinfty11:=-(tinfty14+tinfty24)/2;
\n" }{MPLTEXT 1 0 34 "Pinfty01:=-(tinfty12+tinfty22)/2;\n" }{MPLTEXT 
1 0 20 "Unknownn:=-Unknown:\n" }{MPLTEXT 1 0 20 "Unknownn2:=Unknown2:"
 }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 42 "series(Yinfty+Yinftybis-P1(x),
x=infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I)Pinfty11G6\",&I)tinft
y14GF$#!\"\"\"\"#I)tinfty24GF$F'" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I)P
infty01G6\",&I)tinfty12GF$#!\"\"\"\"#I)tinfty22GF$F'" }}{PARA 11 "" 1 
"" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 208 92 "This im
plies that coefficients in the second sheet are related to the one in \+
the first sheet" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "series(s
implify(ClassicalSpectralCurve(Yinfty)),x=infinity);" }{MPLTEXT 1 0 0 
"" }}{PARA 11 "" 1 "" {XPPMATH 20 ",D*&,&*$I)tinfty15G6\"\"\"##\"\"\"
\"\"%I)Pinfty32GF'F*F*I\"xGF'\"\"$F**(F&F*,&I)tinfty14GF'F*I)tinfty24G
F'!\"\"F**$F-F3#!\"&F(F)*&,(*&I)tinfty13GF'F*F&F*#F*F(*&F1F*F2F*#F3F+I
)Pinfty22GF'F*F*F-F(F**&,&*&F0F*F:F*F)*&F&F*,&I)tinfty22GF'F*I)tinfty1
2GF'F3F*F=F*F4#!\"$F(F**&,,*&I)tinfty11GF'F*F&F*F;*&FEF*F2F*F=*$F:F(F)
*&F1F*FDF*F=I)Pinfty12GF'F*F*F-F*F**&,(*&F&F*I)tinfty10GF'F*F;*&F0F*FK
F*F)*&F:F*FCF*F=F*F4#F3F(F**&F0F*FSF*F)*&F:F*FKF*F;*&F&F*I(UnknownGF'F
*F3*&FEF*FDF*F=I)Pinfty02GF'F**&,(*&F:F*FSF*F;*&,&FDF3FEF*F*FKF*F)*&FZ
F*F0F*FVF*F4F;F**&,(*&F[oF*FSF*F)*&F:F*FZF*F3*$FKF(F)F*F-F3F**&,&*&FSF
*FKF*F;*&FZF*FCF*F;F*F4#F.F(F**&,&*&FZF*FKF*F3*$FSF(F)F*F-!\"#F**(FZF*
FSF*F4#\"\"&F(F3*&FZF(F-FGF*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 20 "tinfty24:=tinfty14:\n" }{MPLTEXT 1 0 20 "tinfty22:=tinfty12:\n" 
}{MPLTEXT 1 0 13 "tinfty10:=0:\n" }{MPLTEXT 1 0 60 "series(simplify(Cl
assicalSpectralCurve(Yinfty)),x=infinity);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 ",6*&,&*$I)tinfty15G6\"\"\"##\"\"\"\"\"%I)Pinfty32GF'F*F*I
\"xGF'\"\"$F**&,(*&I)tinfty13GF'F*F&F*#F*F(*$I)tinfty14GF'F(#!\"\"F+I)
Pinfty22GF'F*F*F-F(F**&,**&I)tinfty11GF'F*F&F*F3*&I)tinfty12GF'F*F5F*#
F7F(*$F2F(F)I)Pinfty12GF'F*F*F-F*F**&F&F*I(UnknownGF'F*F7*&F2F*F<F*F3*
$F>F(F6I)Pinfty02GF'F**&,&*&F2F*FCF*F7*$F<F(F)F*F-F7F**(FCF*F<F*F-!\"#
F7*&FCF(F-!\"$F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 213 17 "Study at inf
inity" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "series(ClassicalSp
ectralCurve(Yinfty),x=infinity,6):\n" }{MPLTEXT 1 0 56 "series(Classic
alSpectralCurve(Yinftybis),x=infinity,6):\n" }{MPLTEXT 1 0 18 "EQinfty
1:=residue(" }{MPLTEXT 1 0 9 "simplify(" }{MPLTEXT 1 0 52 "x^(-5)*Clas
sicalSpectralCurve(Yinfty)),x=infinity);\n" }{MPLTEXT 1 0 82 "EQinfty2
:=residue(simplify(x^(-5)*ClassicalSpectralCurve(Yinftybis)),x=infinit
y);\n" }{MPLTEXT 1 0 18 "EQinfty3:=residue(" }{MPLTEXT 1 0 9 "simplify
(" }{MPLTEXT 1 0 52 "x^(-4)*ClassicalSpectralCurve(Yinfty)),x=infinity
);\n" }{MPLTEXT 1 0 18 "EQinfty4:=residue(" }{MPLTEXT 1 0 9 "simplify(
" }{MPLTEXT 1 0 55 "x^(-4)*ClassicalSpectralCurve(Yinftybis)),x=infini
ty);\n" }{MPLTEXT 1 0 18 "EQinfty5:=residue(" }{MPLTEXT 1 0 9 "simplif
y(" }{MPLTEXT 1 0 52 "x^(-3)*ClassicalSpectralCurve(Yinfty)),x=infinit
y);\n" }{MPLTEXT 1 0 18 "EQinfty6:=residue(" }{MPLTEXT 1 0 9 "simplify
(" }{MPLTEXT 1 0 55 "x^(-3)*ClassicalSpectralCurve(Yinftybis)),x=infin
ity);\n" }{MPLTEXT 1 0 18 "EQinfty7:=residue(" }{MPLTEXT 1 0 9 "simpli
fy(" }{MPLTEXT 1 0 52 "x^(-2)*ClassicalSpectralCurve(Yinfty)),x=infini
ty);\n" }{MPLTEXT 1 0 18 "EQinfty8:=residue(" }{MPLTEXT 1 0 9 "simplif
y(" }{MPLTEXT 1 0 55 "x^(-2)*ClassicalSpectralCurve(Yinftybis)),x=infi
nity);\n" }{MPLTEXT 1 0 18 "EQinfty9:=residue(" }{MPLTEXT 1 0 9 "simpl
ify(" }{MPLTEXT 1 0 52 "x^(-1)*ClassicalSpectralCurve(Yinfty)),x=infin
ity);\n" }{MPLTEXT 1 0 19 "EQinfty10:=residue(" }{MPLTEXT 1 0 9 "simpl
ify(" }{MPLTEXT 1 0 55 "x^(-1)*ClassicalSpectralCurve(Yinftybis)),x=in
finity);\n" }{MPLTEXT 1 0 1 "\n" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I)EQ
infty1G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I)EQinfty2G6\"\"\"!"
 }}{PARA 11 "" 1 "" {XPPMATH 20 ">I)EQinfty3G6\",&*$I)tinfty15GF$\"\"#
#!\"\"\"\"%I)Pinfty32GF$F*" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I)EQinfty
4G6\",&*$I)tinfty15GF$\"\"##!\"\"\"\"%I)Pinfty32GF$F*" }}{PARA 11 "" 1
 "" {XPPMATH 20 ">I)EQinfty5G6\",(*&I)tinfty13GF$\"\"\"I)tinfty15GF$F(
#!\"\"\"\"#*$I)tinfty14GF$F,#F(\"\"%I)Pinfty22GF$F+" }}{PARA 11 "" 1 "
" {XPPMATH 20 ">I)EQinfty6G6\",(*&I)tinfty13GF$\"\"\"I)tinfty15GF$F(#!
\"\"\"\"#*$I)tinfty14GF$F,#F(\"\"%I)Pinfty22GF$F+" }}{PARA 11 "" 1 "" 
{XPPMATH 20 ">I)EQinfty7G6\",**&I)tinfty11GF$\"\"\"I)tinfty15GF$F(#!\"
\"\"\"#*&I)tinfty12GF$F(I)tinfty14GF$F(#F(F,*$I)tinfty13GF$F,#F+\"\"%I
)Pinfty12GF$F+" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I)EQinfty8G6\",**&I)t
infty11GF$\"\"\"I)tinfty15GF$F(#!\"\"\"\"#*&I)tinfty12GF$F(I)tinfty14G
F$F(#F(F,*$I)tinfty13GF$F,#F+\"\"%I)Pinfty12GF$F+" }}{PARA 11 "" 1 "" 
{XPPMATH 20 ">I)EQinfty9G6\",**&I)tinfty15GF$\"\"\"I(UnknownGF$F(F(*&I
)tinfty13GF$F(I)tinfty11GF$F(#!\"\"\"\"#*$I)tinfty12GF$F/#F(\"\"%I)Pin
fty02GF$F." }}{PARA 11 "" 1 "" {XPPMATH 20 ">I*EQinfty10G6\",**&I)tinf
ty15GF$\"\"\"I(UnknownGF$F(F(*&I)tinfty13GF$F(I)tinfty11GF$F(#!\"\"\"
\"#*$I)tinfty12GF$F/#F(\"\"%I)Pinfty02GF$F." }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 44 "Pinfty32:=factor(solve(EQinfty3,Pinfty32));\n" }
{MPLTEXT 1 0 44 "Pinfty22:=factor(solve(EQinfty5,Pinfty22));\n" }
{MPLTEXT 1 0 44 "Pinfty12:=factor(solve(EQinfty7,Pinfty12));\n" }
{MPLTEXT 1 0 19 "simplify(EQinfty4);" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 
1 0 20 "simplify(EQinfty6);\n" }{MPLTEXT 1 0 19 "simplify(EQinfty8);" 
}}{PARA 11 "" 1 "" {XPPMATH 20 ">I)Pinfty32G6\",$*$I)tinfty15GF$\"\"##
!\"\"\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I)Pinfty22G6\",&*&I)tinft
y13GF$\"\"\"I)tinfty15GF$F(#!\"\"\"\"#*$I)tinfty14GF$F,#F(\"\"%" }}
{PARA 11 "" 1 "" {XPPMATH 20 ">I)Pinfty12G6\",(*&I)tinfty11GF$\"\"\"I)
tinfty15GF$F(#!\"\"\"\"#*&I)tinfty12GF$F(I)tinfty14GF$F(#F(F,*$I)tinft
y13GF$F,#F+\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 ""
 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT 213 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 1 "\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 "Pinfty3
2:=Pinfty32;\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 
0 1 "\n" }{MPLTEXT 1 0 1 "\n" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I)Pinft
y01G6\",$I)tinfty12GF$!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I)Pinfty
11G6\",$I)tinfty14GF$!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I)Pinfty0
2G6\"F#" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I)Pinfty12G6\",(*&I)tinfty11
GF$\"\"\"I)tinfty15GF$F(#!\"\"\"\"#*&I)tinfty12GF$F(I)tinfty14GF$F(#F(
F,*$I)tinfty13GF$F,#F+\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I)Pinfty
22G6\",&*&I)tinfty13GF$\"\"\"I)tinfty15GF$F(#!\"\"\"\"#*$I)tinfty14GF$
F,#F(\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I)Pinfty32G6\",$*$I)tinft
y15GF$\"\"##!\"\"\"\"%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 213 47 "We hav
e one undetermined coefficients: Pinfty02" }}}{EXCHG {PARA 0 "> " 0 ""
 {MPLTEXT 1 0 23 "P1(lambda);\nP2(lambda);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 ",&*&I'lambdaG6\"\"\"\"I)tinfty14GF%F&!\"\"I)tinfty12GF%F(
" }}{PARA 11 "" 1 "" {XPPMATH 20 ",*I)Pinfty02G6\"\"\"\"*&,(*&I)tinfty
11GF$F%I)tinfty15GF$F%#!\"\"\"\"#*&I)tinfty12GF$F%I)tinfty14GF$F%#F%F-
*$I)tinfty13GF$F-#F,\"\"%F%I'lambdaGF$F%F%*&,&*&F3F%F*F%F+*$F0F-#F%F5F
%F6F-F%*&F*F-F6\"\"$F4" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0
 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 ""
 {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 ""
 {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 ""
 {MPLTEXT 1 0 0 "" }}}{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 }