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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT 205 151 "In this Maple sheet, we
 compute the Lax matrices using the asymptotics of the wave functions \+
and the local diagonalization for the Painlev\351 3 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 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*tinfty22;\n" }{MPLTEXT 1 0 32 "Pinfty01 := \+
-tinfty11-tinfty21;\n" }{MPLTEXT 1 0 32 "Pinfty11 := -tinfty12-tinfty2
2;\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+Pinfty22*x^2+Pinfty32*x^3+Pinfty42*x^4:" 
}{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 50 "CoherenceEquation1 :=tinfty10+t
infty20+t010+t020;\n" }}{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)tinft
y23GF$F(F(*&I)tinfty11GF$F(I)tinfty22GF$F(F(*&I)tinfty12GF$F(I)tinfty2
1GF$F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I)Pinfty01G6\",&I)tinfty11G
F$!\"\"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$F'" }}{PARA 11 "" 1 "
" {XPPMATH 20 ">I3CoherenceEquation1G6\",&I%t010GF$\"\"\"I%t020GF$F'" 
}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 222 30 "Expression of the Lax matrix \+
L" }}{EXCHG {PARA 210 "" 0 "" {TEXT 223 36 "Study of the asymptotics a
t infinity" }{TEXT 223 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
200 "logPsi1Infty:=-tinfty11/h*lambda-tinfty10/h*ln(lambda)+A10-A12/(2
-1)/lambda^(2-1)-A13/(3-1)/lambda^(3-1)-A14/(4-1)/lambda^(4-1)-A15/(5-
1)/lambda^(5-1)-A16/(6-1)/lambda^(6-1)-A17/(7-1)/lambda^(7-1)  ;\n" }
{MPLTEXT 1 0 200 "logPsi2Infty:=-tinfty21/h*lambda-tinfty20/h*ln(lambd
a)+A20-A22/(2-1)/lambda^(2-1)-A23/(3-1)/lambda^(3-1)-A24/(4-1)/lambda^
(4-1)-A25/(5-1)/lambda^(5-1)-A26/(6-1)/lambda^(6-1)-A27/(7-1)/lambda^(
7-1)  ;\n" }{MPLTEXT 1 0 210 "Llogpsi1Infty:=-Ltinfty11/h*lambda-Ltinf
ty10/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 0 210 "Llogpsi2Infty:=-Lt
infty21/h*lambda-Ltinfty20/h*ln(lambda)+LA20-LA22/(2-1)/lambda^(2-1)-L
A23/(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 
343 "Lpsi1Infty := exp(1/h*(-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*(-Ltinfty11*lambda-Ltinft
y10*ln(lambda)+h*LA10-h*LA12/lambda-1/2*h*LA13/lambda^2-1/3*h*LA14/lam
bda^3-1/4*h*LA15/lambda^4-1/5*h*LA16/lambda^5-1/6*h*LA17/lambda^6);\n"
 }{MPLTEXT 1 0 343 "Lpsi2Infty := exp(1/h*(-tinfty21*lambda-tinfty20*l
n(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/lambda^5-1/6*h*A27/lambda^6))*1/h*(-Ltinfty2
1*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/lambda^5-1/6*h*LA2
7/lambda^6);\n" }{MPLTEXT 1 0 30 "psi1Infty:=exp(logPsi1Infty);\n" }
{MPLTEXT 1 0 30 "psi2Infty:=exp(logPsi2Infty);\n" }{MPLTEXT 1 0 43 "dp
si1dlambdaInfty:=diff(psi1Infty,lambda):\n" }{MPLTEXT 1 0 43 "dpsi2dla
mbdaInfty:=diff(psi2Infty,lambda):\n" }{MPLTEXT 1 0 47 "d2psi1dlambda2
Infty:=diff(psi1Infty,lambda$2):\n" }{MPLTEXT 1 0 47 "d2psi2dlambda2In
fty:=diff(psi2Infty,lambda$2):\n" }{MPLTEXT 1 0 46 "Vinfty1:=tinfty11*
lambda+tinfty10*ln(lambda);\n" }{MPLTEXT 1 0 46 "Vinfty2:=tinfty21*lam
bda+tinfty20*ln(lambda);\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" 
}{MPLTEXT 1 0 89 "WronskianLambdaInfty:=h*factor(psi1Infty*dpsi2dlambd
aInfty-psi2Infty*dpsi1dlambdaInfty):\n" }{MPLTEXT 1 0 132 "WronskianLa
mbdabisInfty:=h*simplify(factor( (diff(logPsi2Infty,lambda)-diff(logPs
i1Infty,lambda))*exp(logPsi1Infty+logPsi2Infty))):\n" }{MPLTEXT 1 0 1 
"\n" }{MPLTEXT 1 0 116 "WronskianTildeLambdaInfty:=h^3*factor(dpsi2dla
mbdaInfty*d2psi1dlambda2Infty-dpsi1dlambdaInfty*d2psi2dlambda2Infty):
\n" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I-logPsi1InftyG6\",4*(I)tinfty11G
F$\"\"\"I\"hGF$!\"\"I'lambdaGF$F(F**(I)tinfty10GF$F(F)F*-I#lnG6$%*prot
ectedGI(_syslibGF$6#F+F(F*I$A10GF$F(*&I$A12GF$F(F+F*F**&I$A13GF$F(F+!
\"##F*\"\"#*&I$A14GF$F(F+!\"$#F*\"\"$*&I$A15GF$F(F+!\"%#F*\"\"%*&I$A16
GF$F(F+!\"&#F*\"\"&*&I$A17GF$F(F+!\"'#F*\"\"'" }}{PARA 11 "" 1 "" 
{XPPMATH 20 ">I-logPsi2InftyG6\",4*(I)tinfty21GF$\"\"\"I\"hGF$!\"\"I'l
ambdaGF$F(F**(I)tinfty10GF$F(F)F*-I#lnG6$%*protectedGI(_syslibGF$6#F+F
(F(I$A20GF$F(*&I$A22GF$F(F+F*F**&I$A23GF$F(F+!\"##F*\"\"#*&I$A24GF$F(F
+!\"$#F*\"\"$*&I$A25GF$F(F+!\"%#F*\"\"%*&I$A26GF$F(F+!\"&#F*\"\"&*&I$A
27GF$F(F+!\"'#F*\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I.Llogpsi1Inft
yG6\",4*(I*Ltinfty11GF$\"\"\"I\"hGF$!\"\"I'lambdaGF$F(F**(I*Ltinfty10G
F$F(F)F*-I#lnG6$%*protectedGI(_syslibGF$6#F+F(F*I%LA10GF$F(*&I%LA12GF$
F(F+F*F**&I%LA13GF$F(F+!\"##F*\"\"#*&I%LA14GF$F(F+!\"$#F*\"\"$*&I%LA15
GF$F(F+!\"%#F*\"\"%*&I%LA16GF$F(F+!\"&#F*\"\"&*&I%LA17GF$F(F+!\"'#F*\"
\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I.Llogpsi2InftyG6\",4*(I*Ltinfty
21GF$\"\"\"I\"hGF$!\"\"I'lambdaGF$F(F**(I*Ltinfty20GF$F(F)F*-I#lnG6$%*
protectedGI(_syslibGF$6#F+F(F*I%LA20GF$F(*&I%LA22GF$F(F+F*F**&I%LA23GF
$F(F+!\"##F*\"\"#*&I%LA24GF$F(F+!\"$#F*\"\"$*&I%LA25GF$F(F+!\"%#F*\"\"
%*&I%LA26GF$F(F+!\"&#F*\"\"&*&I%LA27GF$F(F+!\"'#F*\"\"'" }}{PARA 11 ""
 1 "" {XPPMATH 20 ">I+Lpsi1InftyG6\"*(-I$expG6$%*protectedGI(_syslibGF
$6#*&I\"hGF$!\"\",4*&I)tinfty11GF$\"\"\"I'lambdaGF$F2F.*&I)tinfty10GF$
F2-I#lnGF(6#F3F2F.*&F-F2I$A10GF$F2F2*(F-F2I$A12GF$F2F3F.F.*(F-F2I$A13G
F$F2F3!\"##F.\"\"#*(F-F2I$A14GF$F2F3!\"$#F.\"\"$*(F-F2I$A15GF$F2F3!\"%
#F.\"\"%*(F-F2I$A16GF$F2F3!\"&#F.\"\"&*(F-F2I$A17GF$F2F3!\"'#F.\"\"'F2
F2F-F.,4*&I*Ltinfty11GF$F2F3F2F.*&I*Ltinfty10GF$F2F6F2F.*&F-F2I%LA10GF
$F2F2*(F-F2I%LA12GF$F2F3F.F.*(F-F2I%LA13GF$F2F3F?F@*(F-F2I%LA14GF$F2F3
FDFE*(F-F2I%LA15GF$F2F3FIFJ*(F-F2I%LA16GF$F2F3FNFO*(F-F2I%LA17GF$F2F3F
SFTF2" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I+Lpsi2InftyG6\"*(-I$expG6$%*p
rotectedGI(_syslibGF$6#*&I\"hGF$!\"\",4*&I)tinfty21GF$\"\"\"I'lambdaGF
$F2F.*&I)tinfty10GF$F2-I#lnGF(6#F3F2F2*&F-F2I$A20GF$F2F2*(F-F2I$A22GF$
F2F3F.F.*(F-F2I$A23GF$F2F3!\"##F.\"\"#*(F-F2I$A24GF$F2F3!\"$#F.\"\"$*(
F-F2I$A25GF$F2F3!\"%#F.\"\"%*(F-F2I$A26GF$F2F3!\"&#F.\"\"&*(F-F2I$A27G
F$F2F3!\"'#F.\"\"'F2F2F-F.,4*&I*Ltinfty21GF$F2F3F2F.*&I*Ltinfty20GF$F2
F6F2F.*&F-F2I%LA20GF$F2F2*(F-F2I%LA22GF$F2F3F.F.*(F-F2I%LA23GF$F2F3F?F
@*(F-F2I%LA24GF$F2F3FDFE*(F-F2I%LA25GF$F2F3FIFJ*(F-F2I%LA26GF$F2F3FNFO
*(F-F2I%LA27GF$F2F3FSFTF2" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I*psi1Inft
yG6\"-I$expG6$%*protectedGI(_syslibGF$6#,4*(I)tinfty11GF$\"\"\"I\"hGF$
!\"\"I'lambdaGF$F.F0*(I)tinfty10GF$F.F/F0-I#lnGF'6#F1F.F0I$A10GF$F.*&I
$A12GF$F.F1F0F0*&I$A13GF$F.F1!\"##F0\"\"#*&I$A14GF$F.F1!\"$#F0\"\"$*&I
$A15GF$F.F1!\"%#F0\"\"%*&I$A16GF$F.F1!\"&#F0\"\"&*&I$A17GF$F.F1!\"'#F0
\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I*psi2InftyG6\"-I$expG6$%*prot
ectedGI(_syslibGF$6#,4*(I)tinfty21GF$\"\"\"I\"hGF$!\"\"I'lambdaGF$F.F0
*(I)tinfty10GF$F.F/F0-I#lnGF'6#F1F.F.I$A20GF$F.*&I$A22GF$F.F1F0F0*&I$A
23GF$F.F1!\"##F0\"\"#*&I$A24GF$F.F1!\"$#F0\"\"$*&I$A25GF$F.F1!\"%#F0\"
\"%*&I$A26GF$F.F1!\"&#F0\"\"&*&I$A27GF$F.F1!\"'#F0\"\"'" }}{PARA 11 ""
 1 "" {XPPMATH 20 ">I(Vinfty1G6\",&*&I)tinfty11GF$\"\"\"I'lambdaGF$F(F
(*&I)tinfty10GF$F(-I#lnG6$%*protectedGI(_syslibGF$6#F)F(F(" }}{PARA 11
 "" 1 "" {XPPMATH 20 ">I(Vinfty2G6\",&*&I)tinfty21GF$\"\"\"I'lambdaGF$
F(F(*&I)tinfty10GF$F(-I#lnG6$%*protectedGI(_syslibGF$6#F)F(!\"\"" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "L21Infty:=factor(simplify(Wr
onskianTildeLambdaInfty/WronskianLambdabisInfty)):\n" }{MPLTEXT 1 0 
75 "L21InftyOrderlambda3:=factor(-residue(L21Infty/lambda^4,lambda=inf
inity));\n" }{MPLTEXT 1 0 75 "L21InftyOrderlambda2:=factor(-residue(L2
1Infty/lambda^3,lambda=infinity));\n" }{MPLTEXT 1 0 75 "L21InftyOrderl
ambda1:=factor(-residue(L21Infty/lambda^2,lambda=infinity));\n" }
{MPLTEXT 1 0 75 "L21InftyOrderlambda0:=factor(-residue(L21Infty/lambda
^1,lambda=infinity));\n" }{MPLTEXT 1 0 80 "L21InftyOrderlambdaMinus1:=
factor(-residue(L21Infty/lambda^0,lambda=infinity));\n" }{MPLTEXT 1 0 
129 "L21InftyOrderlambdaMinus2:=factor(-residue(series(L21InftyNumer/L
21InftyDenom,lambda=infinity,12)/lambda^(-1),lambda=infinity));\n" }
{MPLTEXT 1 0 129 "L21InftyOrderlambdaMinus3:=factor(-residue(series(L2
1InftyNumer/L21InftyDenom,lambda=infinity,12)/lambda^(-2),lambda=infin
ity));\n" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I5L21InftyOrderlambda3G6\"
\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I5L21InftyOrderlambda2G6\"\"\"
!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I5L21InftyOrderlambda1G6\"\"\"!" }
}{PARA 11 "" 1 "" {XPPMATH 20 ">I5L21InftyOrderlambda0G6\",$*&I)tinfty
11GF$\"\"\"I)tinfty21GF$F(!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I:L2
1InftyOrderlambdaMinus1G6\"*&,&I)tinfty11GF$\"\"\"I)tinfty21GF$!\"\"F(
I)tinfty10GF$F(" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I:L21InftyOrderlambd
aMinus2G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I:L21InftyOrderlamb
daMinus3G6\"\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 224 127 "We deduce \+
that L_\{2,1\} behaves at infinity like  -tinfty11*tinfty21-(tinfty11*
tinfty20+tinfty21*tinfty10)/lambda +O(1/lambda^2)" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 92 "L22Infty:=factor(h*simplify(diff(WronskianLa
mbdabisInfty,lambda)/WronskianLambdabisInfty)):\n" }{MPLTEXT 1 0 75 "L
22InftyOrderlambda3:=factor(-residue(L22Infty/lambda^4,lambda=infinity
));\n" }{MPLTEXT 1 0 75 "L22InftyOrderlambda2:=factor(-residue(L22Inft
y/lambda^3,lambda=infinity));\n" }{MPLTEXT 1 0 75 "L22InftyOrderlambda
1:=factor(-residue(L22Infty/lambda^2,lambda=infinity));\n" }{MPLTEXT 
1 0 75 "L22InftyOrderlambda0:=factor(-residue(L22Infty/lambda^1,lambda
=infinity));\n" }{MPLTEXT 1 0 80 "L22InftyOrderlambdaMinus1:=factor(-r
esidue(L22Infty/lambda^0,lambda=infinity));\n" }{MPLTEXT 1 0 83 "L22In
ftyOrderlambdaMinus2:=factor(-residue(L22Infty/lambda^(-1),lambda=infi
nity));\n" }{MPLTEXT 1 0 82 "L22InftyOrderlambdaMinus3:=factor(-residu
e(L22Infty/lambda^(-2),lambda=infinity)):" }}{PARA 11 "" 1 "" 
{XPPMATH 20 ">I5L22InftyOrderlambda3G6\"\"\"!" }}{PARA 11 "" 1 "" 
{XPPMATH 20 ">I5L22InftyOrderlambda2G6\"\"\"!" }}{PARA 11 "" 1 "" 
{XPPMATH 20 ">I5L22InftyOrderlambda1G6\"\"\"!" }}{PARA 11 "" 1 "" 
{XPPMATH 20 ">I5L22InftyOrderlambda0G6\",&I)tinfty11GF$!\"\"I)tinfty21
GF$F'" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I:L22InftyOrderlambdaMinus1G6
\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I:L22InftyOrderlambdaMinus2G
6\"*(I\"hGF$\"\"\",,*&I$A12GF$F'I)tinfty11GF$F'F'*&F*F'I)tinfty21GF$F'
!\"\"*&I$A22GF$F'F+F'F'*&F0F'F-F'F.I)tinfty10GF$!\"#F',&F+F'F-F.F." }}
}{EXCHG {PARA 0 "" 0 "" {TEXT 224 113 "We deduce that L_\{2,2\} behave
s at infinity like  -(tinfty11+tinfty21)-(tinfty10+tinfty20)/lambda+h*
O(1/lambda^2) " }}}{EXCHG {PARA 209 "" 0 "" {TEXT 225 17 "Study at lam
bda=0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 191 "logPsi1Zero:=-t01
1/h/lambda+t010/h*ln(lambda)+B10+B12/(2-1)*lambda^(2-1)+B13/(3-1)*lamb
da^(3-1)+B14/(4-1)*lambda^(4-1)+B15/(5-1)*lambda^(5-1)+B16/(6-1)*lambd
a^(6-1)+B17/(7-1)*lambda^(7-1)  ;\n" }{MPLTEXT 1 0 191 "logPsi2Zero:=-
t021/h/lambda+t020/h*ln(lambda)+B20+B22/(2-1)*lambda^(2-1)+B23/(3-1)*l
ambda^(3-1)+B24/(4-1)*lambda^(4-1)+B25/(5-1)*lambda^(5-1)+B26/(6-1)*la
mbda^(6-1)+B27/(7-1)*lambda^(7-1)  ;\n" }{MPLTEXT 1 0 201 "Llogpsi1Zer
o:=-Lt011/h/lambda+Lt010/h*ln(lambda)+LB10+LB12/(2-1)*lambda^(2-1)+LB1
3/(3-1)*lambda^(3-1)+LB14/(4-1)*lambda^(4-1)+LB15/(5-1)*lambda^(5-1)+L
B16/(6-1)*lambda^(6-1)+LB17/(7-1)*lambda^(7-1)  ;\n" }{MPLTEXT 1 0 
201 "Llogpsi2Zero:=-Lt021/h/lambda+Lt020/h*ln(lambda)+LB20+LB22/(2-1)*
lambda^(2-1)+LB23/(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 0 195 "Lpsi1Zero := exp((-t011/h/lambda+t010/h*ln(lambda)+B
10+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 0 188 "*(-Lt011/h/lambda+Lt010/h*ln(lambda)+LB10+LB1
2/(2-1)*lambda^(2-1)+LB13/(3-1)*lambda^(3-1)+LB14/(4-1)*lambda^(4-1)+L
B15/(5-1)*lambda^(5-1)+LB16/(6-1)*lambda^(6-1)+LB17/(7-1)*lambda^(7-1)
);\n" }{MPLTEXT 1 0 195 "Lpsi2Zero := exp((-t021/h/lambda+t020/h*ln(la
mbda)+B20+B22/(2-1)*lambda^(2-1)+B23/(3-1)*lambda^(3-1)+B24/(4-1)*lamb
da^(4-1)+B25/(5-1)*lambda^(5-1)+B26*(6-1)*lambda^(6-1)+B27*(7-1)*lambd
a^(7-1)))\n" }{MPLTEXT 1 0 188 "*(-Lt021/h/lambda+Lt020/h*ln(lambda)+L
B20+LB22/(2-1)*lambda^(2-1)+LB23/(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)*lambd
a^(7-1));\n" }{MPLTEXT 1 0 28 "psi1Zero:=exp(logPsi1Zero);\n" }
{MPLTEXT 1 0 28 "psi2Zero:=exp(logPsi2Zero);\n" }{MPLTEXT 1 0 41 "dpsi
1dlambdaZero:=diff(psi1Zero,lambda):\n" }{MPLTEXT 1 0 41 "dpsi2dlambda
Zero:=diff(psi2Zero,lambda):\n" }{MPLTEXT 1 0 45 "d2psi1dlambda2Zero:=
diff(psi1Zero,lambda$2):\n" }{MPLTEXT 1 0 45 "d2psi2dlambda2Zero:=diff
(psi2Zero,lambda$2):\n" }{MPLTEXT 1 0 25 "VZero1:=t010*ln(lambda);\n" 
}{MPLTEXT 1 0 25 "VZero2:=t020*ln(lambda);\n" }{MPLTEXT 1 0 1 "\n" }
{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 84 "WronskianLambdaZero:=h*factor(ps
i1Zero*dpsi2dlambdaZero-psi2Zero*dpsi1dlambdaZero):\n" }{MPLTEXT 1 0 
127 "WronskianLambdabisZero:=h*simplify(factor( (diff(logPsi2Zero,lamb
da)-diff(logPsi1Zero,lambda))*exp(logPsi1Zero+logPsi2Zero))):\n" }
{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 111 "WronskianTildeLambdaZero:=h^3*f
actor(dpsi2dlambdaZero*d2psi1dlambda2Zero-dpsi1dlambdaZero*d2psi2dlamb
da2Zero):\n" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I,logPsi1ZeroG6\",4*(I%t
011GF$\"\"\"I\"hGF$!\"\"I'lambdaGF$F*F**(I%t010GF$F(F)F*-I#lnG6$%*prot
ectedGI(_syslibGF$6#F+F(F(I$B10GF$F(*&I$B12GF$F(F+F(F(*&I$B13GF$F(F+\"
\"##F(F9*&I$B14GF$F(F+\"\"$#F(F=*&I$B15GF$F(F+\"\"%#F(FA*&I$B16GF$F(F+
\"\"&#F(FE*&I$B17GF$F(F+\"\"'#F(FI" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I
,logPsi2ZeroG6\",4*(I%t021GF$\"\"\"I\"hGF$!\"\"I'lambdaGF$F*F**(I%t020
GF$F(F)F*-I#lnG6$%*protectedGI(_syslibGF$6#F+F(F(I$B20GF$F(*&I$B22GF$F
(F+F(F(*&I$B23GF$F(F+\"\"##F(F9*&I$B24GF$F(F+\"\"$#F(F=*&I$B25GF$F(F+
\"\"%#F(FA*&I$B26GF$F(F+\"\"&#F(FE*&I$B27GF$F(F+\"\"'#F(FI" }}{PARA 11
 "" 1 "" {XPPMATH 20 ">I-Llogpsi1ZeroG6\",4*(I&Lt011GF$\"\"\"I\"hGF$!
\"\"I'lambdaGF$F*F**(I&Lt010GF$F(F)F*-I#lnG6$%*protectedGI(_syslibGF$6
#F+F(F(I%LB10GF$F(*&I%LB12GF$F(F+F(F(*&I%LB13GF$F(F+\"\"##F(F9*&I%LB14
GF$F(F+\"\"$#F(F=*&I%LB15GF$F(F+\"\"%#F(FA*&I%LB16GF$F(F+\"\"&#F(FE*&I
%LB17GF$F(F+\"\"'#F(FI" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I-Llogpsi2Zer
oG6\",4*(I&Lt021GF$\"\"\"I\"hGF$!\"\"I'lambdaGF$F*F**(I&Lt020GF$F(F)F*
-I#lnG6$%*protectedGI(_syslibGF$6#F+F(F(I%LB20GF$F(*&I%LB22GF$F(F+F(F(
*&I%LB23GF$F(F+\"\"##F(F9*&I%LB24GF$F(F+\"\"$#F(F=*&I%LB25GF$F(F+\"\"%
#F(FA*&I%LB26GF$F(F+\"\"&#F(FE*&I%LB27GF$F(F+\"\"'#F(FI" }}{PARA 11 ""
 1 "" {XPPMATH 20 ">I*Lpsi1ZeroG6\"*&-I$expG6$%*protectedGI(_syslibGF$
6#,4*(I%t011GF$\"\"\"I\"hGF$!\"\"I'lambdaGF$F1F1*(I%t010GF$F/F0F1-I#ln
GF(6#F2F/F/I$B10GF$F/*&I$B12GF$F/F2F/F/*&I$B13GF$F/F2\"\"##F/F=*&I$B14
GF$F/F2\"\"$#F/FA*&I$B15GF$F/F2\"\"%#F/FE*&I$B16GF$F/F2\"\"&#F/FI*&I$B
17GF$F/F2\"\"'#F/FMF/,4*(I&Lt011GF$F/F0F1F2F1F1*(I&Lt010GF$F/F0F1F5F/F
/I%LB10GF$F/*&I%LB12GF$F/F2F/F/*&I%LB13GF$F/F2F=F>*&I%LB14GF$F/F2FAFB*
&I%LB15GF$F/F2FEFF*&I%LB16GF$F/F2FIFJ*&I%LB17GF$F/F2FMFNF/" }}{PARA 11
 "" 1 "" {XPPMATH 20 ">I*Lpsi2ZeroG6\"*&-I$expG6$%*protectedGI(_syslib
GF$6#,4*(I%t021GF$\"\"\"I\"hGF$!\"\"I'lambdaGF$F1F1*(I%t020GF$F/F0F1-I
#lnGF(6#F2F/F/I$B20GF$F/*&I$B22GF$F/F2F/F/*&I$B23GF$F/F2\"\"##F/F=*&I$
B24GF$F/F2\"\"$#F/FA*&I$B25GF$F/F2\"\"%#F/FE*&I$B26GF$F/F2\"\"&FI*&I$B
27GF$F/F2\"\"'FLF/,4*(I&Lt021GF$F/F0F1F2F1F1*(I&Lt020GF$F/F0F1F5F/F/I%
LB20GF$F/*&I%LB22GF$F/F2F/F/*&I%LB23GF$F/F2F=F>*&I%LB24GF$F/F2FAFB*&I%
LB25GF$F/F2FEFF*&I%LB26GF$F/F2FI#F/FI*&I%LB27GF$F/F2FL#F/FLF/" }}
{PARA 11 "" 1 "" {XPPMATH 20 ">I)psi1ZeroG6\"-I$expG6$%*protectedGI(_s
yslibGF$6#,4*(I%t011GF$\"\"\"I\"hGF$!\"\"I'lambdaGF$F0F0*(I%t010GF$F.F
/F0-I#lnGF'6#F1F.F.I$B10GF$F.*&I$B12GF$F.F1F.F.*&I$B13GF$F.F1\"\"##F.F
<*&I$B14GF$F.F1\"\"$#F.F@*&I$B15GF$F.F1\"\"%#F.FD*&I$B16GF$F.F1\"\"&#F
.FH*&I$B17GF$F.F1\"\"'#F.FL" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I)psi2Ze
roG6\"-I$expG6$%*protectedGI(_syslibGF$6#,4*(I%t021GF$\"\"\"I\"hGF$!\"
\"I'lambdaGF$F0F0*(I%t020GF$F.F/F0-I#lnGF'6#F1F.F.I$B20GF$F.*&I$B22GF$
F.F1F.F.*&I$B23GF$F.F1\"\"##F.F<*&I$B24GF$F.F1\"\"$#F.F@*&I$B25GF$F.F1
\"\"%#F.FD*&I$B26GF$F.F1\"\"&#F.FH*&I$B27GF$F.F1\"\"'#F.FL" }}{PARA 11
 "" 1 "" {XPPMATH 20 ">I'VZero1G6\"*&I%t010GF$\"\"\"-I#lnG6$%*protecte
dGI(_syslibGF$6#I'lambdaGF$F'" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I'VZer
o2G6\"*&I%t020GF$\"\"\"-I#lnG6$%*protectedGI(_syslibGF$6#I'lambdaGF$F'
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "L22Zero:=factor(h*simpl
ify(diff(WronskianLambdabisZero,lambda)/WronskianLambdabisZero)):\n" }
{MPLTEXT 1 0 70 "L22ZeroOrderlambdaMinus3:=factor(residue(L22Zero*lamb
da^2,lambda=0));\n" }{MPLTEXT 1 0 70 "L22ZeroOrderlambdaMinus2:=factor
(residue(L22Zero*lambda^1,lambda=0));\n" }{MPLTEXT 1 0 70 "L22ZeroOrde
rlambdaMinus1:=factor(residue(L22Zero*lambda^0,lambda=0));\n" }
{MPLTEXT 1 0 68 "L22ZeroOrderlambda0:=factor(residue(L22Zero*lambda^(-
1),lambda=0));\n" }{MPLTEXT 1 0 68 "L22ZeroOrderlambda1:=factor(residu
e(L22Zero*lambda^(-2),lambda=0)):\n" }{MPLTEXT 1 0 67 "L22ZeroOrderlam
bda2:=factor(residue(L22Zero*lambda^(-3),lambda=0)):" }}{PARA 11 "" 1 
"" {XPPMATH 20 ">I9L22ZeroOrderlambdaMinus3G6\"\"\"!" }}{PARA 11 "" 1 
"" {XPPMATH 20 ">I9L22ZeroOrderlambdaMinus2G6\",&I%t011GF$\"\"\"I%t021
GF$F'" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I9L22ZeroOrderlambdaMinus1G6\"
,(I\"hGF$!\"#I%t010GF$\"\"\"I%t020GF$F)" }}{PARA 11 "" 1 "" {XPPMATH 
20 ">I4L22ZeroOrderlambda0G6\"*(I\"hGF$\"\"\",.*&I$B12GF$F'I%t011GF$F'
F'*&F*F'I%t021GF$F'!\"\"*&I$B22GF$F'F+F'F'*&F0F'F-F'F.I%t010GF$F'I%t02
0GF$F.F',&F+F'F-F.F." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 224 96 "We deduc
e that L_\{2,2\} behaves at lambda=0 like (t011+t021)/lambda^2+(t010+t
020-2h)/lambda +O(1)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "L21
Zero:=factor(simplify(WronskianTildeLambdaZero/WronskianLambdabisZero)
):\n" }{MPLTEXT 1 0 70 "L21ZeroOrderlambdaMinus5:=factor(residue(L21Ze
ro*lambda^4,lambda=0));\n" }{MPLTEXT 1 0 70 "L21ZeroOrderlambdaMinus4:
=factor(residue(L21Zero*lambda^3,lambda=0));\n" }{MPLTEXT 1 0 70 "L21Z
eroOrderlambdaMinus3:=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\",$*&I%t011GF$\"\"\"I%t021GF$F(!\"\"" }}{PARA 11 "" 
1 "" {XPPMATH 20 ">I9L21ZeroOrderlambdaMinus3G6\",&*&I%t010GF$\"\"\"I%
t021GF$F(!\"\"*&I%t011GF$F(I%t020GF$F(F*" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT 224 0 "" }}}{EXCHG {PARA 209 "" 0 "" {TEXT 225 31 "Formulas for \+
L_\{2,2\} et L_\{2,1\}" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 224 105 "We ha
ve L_\{2,2\} behaves at infinity like -(tinfty11+tinfty21)-(tinfty10+t
infty20)/lambda+h*O(1/lambda^2)\n" }{TEXT 224 92 "        L_\{2,2\}  b
ehaves at lambda=0 like (t011+t021)/lambda^2+(t010+t020-2h)/lambda +O(
1) \n" }{TEXT 224 6 "Thus, " }{TEXT 214 54 "L_\{2,2\}=  (t011+t021)/la
mbda^2+ (t010+t020-2*h)/lambda" }{TEXT 215 1 " " }{TEXT 216 20 "-(tinf
ty11+tinfty21)" }{TEXT 217 1 " " }{TEXT 218 15 "  +h/(lambda-q)" }
{TEXT 224 1 "\n" }{TEXT 224 49 "with the condition t010+t020+tinfty10+
tinfty20=h\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 224 132 "We have L_\{2,1
\} behaves at infinity like -tinfty11*tinfty21-(tinfty11*tinfty20+tinf
ty21*tinfty10 +h*tinfty11)/lambda +O(1/lambda^2)\n" }{TEXT 224 22 "   \+
       and L_\{2,1\} " }{TEXT 224 12 "behaves at  " }{TEXT 224 81 "lam
bda=0 like -t011*t021/lambda^4-(t010*t021+t020*t011)/lambda^3+ O(1/lam
bda^2) \n" }{TEXT 224 6 "Thus, " }{TEXT 219 7 "L_\{2,1\}" }{TEXT 220 
2 "=-" }{TEXT 221 160 "t011*t021/lambda^4  -(t010*t021+t020*t011)/lamb
da^3- tinfty11*tinfty21- H/lambda^2 - (tinfty11*tinfty20+tinfty21*tinf
ty10+h*tinfty11-h*p)/lambda- p*h/(lambda-q)" }}}}{EXCHG {PARA 0 "> " 0
 "" {MPLTEXT 1 0 171 "L21Form:=-t011*t021/lambda^4  -(t010*t021+t020*t
011)/lambda^3-H/lambda^2- tinfty11*tinfty21 - (tinfty11*tinfty20+tinft
y21*tinfty10+h*tinfty11-h*p)/lambda- p*h/(lambda-q);\n" }{MPLTEXT 1 0 
69 "L21FormOrderLambdaMinus1:=factor(-residue(L21Form,lambda=infinity)
);\n" }{MPLTEXT 1 0 75 "L21FormOrderLambdaMinus2:=factor(-residue(L21F
orm*lambda,lambda=infinity)):" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I(L21F
ormG6\",.*(I%t011GF$\"\"\"I%t021GF$F(I'lambdaGF$!\"%!\"\"*&,&*&I%t010G
F$F(F)F(F(*&F'F(I%t020GF$F(F(F(F*!\"$F,*&I\"HGF$F(F*!\"#F,*&I)tinfty11
GF$F(I)tinfty21GF$F(F,*&,**&I\"hGF$F(I\"pGF$F(F,*&F=F(F8F(F(*&I)tinfty
10GF$F(F8F(F,*&FAF(F9F(F(F(F*F,F,*(F>F(F=F(,&F*F(I\"qGF$F,F,F," }}
{PARA 11 "" 1 "" {XPPMATH 20 ">I9L21FormOrderLambdaMinus1G6\",(*&I\"hG
F$\"\"\"I)tinfty11GF$F(!\"\"*&I)tinfty10GF$F(F)F(F(*&F,F(I)tinfty21GF$
F(F*" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 222 36 "Expression of the auxil
iary matrix A" }}{EXCHG {PARA 0 "" 0 "" {TEXT 205 194 "The deformation
 operator is \\mathcal\{L\}=\\hbar (alphainf11\\partial_\{t_\{\\infty^
\{(1)\},1\} +alphainf21\\partial_\{t_\{\\infty^\{(2)\},1\}+alpha011\\p
artial_\{t_\{0^\{(1)\},1\} +alpha021\\partial_\{t_\{0^\{(2)\},1\})) )"
 }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "WronskianLInfty:=factor(
psi1Infty*Lpsi2Infty-psi2Infty*Lpsi1Infty):\n" }{MPLTEXT 1 0 63 "Wrons
kianLZero:=factor(psi1Zero*Lpsi2Zero-psi2Zero*Lpsi1Zero):\n" }{MPLTEXT
 1 0 66 "A12Infty:=factor(simplify(WronskianLInfty/WronskianLambdaInft
y)):\n" }{MPLTEXT 1 0 63 "A12Zero:=factor(simplify(WronskianLZero/Wron
skianLambdaZero)):\n" }{MPLTEXT 1 0 48 "Y1Infty:=h*factor(dpsi1dlambda
Infty/psi1Infty):\n" }{MPLTEXT 1 0 48 "Y2Infty:=h*factor(dpsi2dlambdaI
nfty/psi2Infty):\n" }{MPLTEXT 1 0 45 "Y1Zero:=h*factor(dpsi1dlambdaZer
o/psi1Zero):\n" }{MPLTEXT 1 0 45 "Y2Zero:=h*factor(dpsi2dlambdaZero/ps
i2Zero):\n" }{MPLTEXT 1 0 39 "Z1Infty:=factor(Lpsi1Infty/psi1Infty):\n
" }{MPLTEXT 1 0 39 "Z2Infty:=factor(Lpsi2Infty/psi2Infty):\n" }
{MPLTEXT 1 0 36 "Z1Zero:=factor(Lpsi1Zero/psi1Zero):\n" }{MPLTEXT 1 0 
36 "Z2Zero:=factor(Lpsi2Zero/psi2Zero):\n" }{MPLTEXT 1 0 68 "A12bisInf
ty:=factor(simplify((Z2Infty-Z1Infty)/(Y2Infty-Y1Infty))):\n" }
{MPLTEXT 1 0 63 "A12bisZero:=factor(simplify((Z2Zero-Z1Zero)/(Y2Zero-Y
1Zero))):\n" }{MPLTEXT 1 0 83 "A11Infty:=factor(simplify( (Y2Infty*Z1I
nfty-Y1Infty*Z2Infty)/(Y2Infty-Y1Infty) )):\n" }{MPLTEXT 1 0 76 "A11Ze
ro:=factor(simplify( (Y2Zero*Z1Zero-Y1Zero*Z2Zero)/(Y2Zero-Y1Zero) )):
\n" }{MPLTEXT 1 0 40 "factor(simplify(A12bisInfty-A12Infty));\n" }
{MPLTEXT 1 0 38 "factor(simplify(A12bisZero-A12Zero));\n" }{MPLTEXT 1 
0 1 "\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "Lt011:=h*alpha011:\n" }{MPLTEXT 1 
0 9 "Lt021:=h*" }{MPLTEXT 1 0 8 "alpha021" }{MPLTEXT 1 0 2 ":\n" }
{MPLTEXT 1 0 10 "Lt010:=0:\n" }{MPLTEXT 1 0 10 "Lt020:=0:\n" }{MPLTEXT
 1 0 25 "Ltinfty11:=h*alphainf11:\n" }{MPLTEXT 1 0 25 "Ltinfty21:=h*al
phainf21:\n" }{MPLTEXT 1 0 14 "Ltinfty10:=0:\n" }{MPLTEXT 1 0 14 "Ltin
fty20:=0:\n" }{MPLTEXT 1 0 1 "\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT
 1 0 70 "A12InftyLambda3:=factor(-residue(A12Infty/lambda^4,lambda=inf
inity));\n" }{MPLTEXT 1 0 70 "A12InftyLambda2:=factor(-residue(A12Inft
y/lambda^3,lambda=infinity));\n" }{MPLTEXT 1 0 70 "A12InftyLambda1:=fa
ctor(-residue(A12Infty/lambda^2,lambda=infinity));\n" }{MPLTEXT 1 0 
70 "A12InftyLambda0:=factor(-residue(A12Infty/lambda^1,lambda=infinity
));\n" }{MPLTEXT 1 0 75 "A12InftyLambdaMinus1:=factor(-residue(A12Inft
y/lambda^0,lambda=infinity)):\n" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I0A1
2InftyLambda3G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I0A12InftyLam
bda2G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I0A12InftyLambda1G6\"*
&,&I+alphainf11GF$\"\"\"I+alphainf21GF$!\"\"F(,&I)tinfty11GF$F(I)tinft
y21GF$F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I0A12InftyLambda0G6\",$*&
,.*&I%LA10GF$\"\"\"I)tinfty11GF$F*F**&F)F*I)tinfty21GF$F*!\"\"*&I%LA20
GF$F*F+F*F.*&F0F*F-F*F**&I+alphainf11GF$F*I)tinfty10GF$F*\"\"#*&I+alph
ainf21GF$F*F4F*!\"#F*,&F+F*F-F.F8F." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 68 "A12ZeroLambdaMinus3:=factor(residue(A12bisZero*lambda
^2,lambda=0));\n" }{MPLTEXT 1 0 68 "A12ZeroLambdaMinus2:=factor(residu
e(A12bisZero*lambda^1,lambda=0));\n" }{MPLTEXT 1 0 68 "A12ZeroLambdaMi
nus1:=factor(residue(A12bisZero*lambda^0,lambda=0));\n" }{MPLTEXT 1 0 
66 "A12ZeroLambda0:=factor(residue(A12bisZero*lambda^(-1),lambda=0));
\n" }{MPLTEXT 1 0 66 "A12ZeroLambda1:=factor(residue(A12bisZero*lambda
^(-2),lambda=0));\n" }{MPLTEXT 1 0 66 "A12ZeroLambda2:=factor(residue(
A12bisZero*lambda^(-3),lambda=0));\n" }{MPLTEXT 1 0 1 "\n" }}{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\",$*&,&I)alpha021GF$!\"\"I)alpha011GF$\"\"\"F+
,&I%t011GF$F+I%t021GF$F)F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I/A12Ze
roLambda2G6\"*&,2*&I%LB10GF$\"\"\"I%t011GF$F)F)*&F(F)I%t021GF$F)!\"\"*
&I%LB20GF$F)F*F)F-*&F/F)F,F)F)*&I%t010GF$F)I)alpha011GF$F)F)*&F2F)I)al
pha021GF$F)F-*&I%t020GF$F)F3F)F-*&F7F)F5F)F)F),&F*F)F,F-!\"#" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT 205 87 "We get that A_\{1,2\}=(alphainf11
-alphainf21)/(tinfty11-tinfty21)*lambda+nu+mu/(lambda-q)" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "A12Form:=(alphainf11-alphainf21)/(-
tinfty21+tinfty11)*lambda+nu+ mu/(lambda-q);\n" }{MPLTEXT 1 0 34 "fact
or(series(A12Form,lambda=0));\n" }{MPLTEXT 1 0 132 "solve(\{factor(res
idue(A12Form/lambda,lambda=0))=A12ZeroLambda0,factor(residue(A12Form/l
ambda^2,lambda=0))=A12ZeroLambda1\},\{mu,nu\});\n" }{MPLTEXT 1 0 1 "\n
" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I(A12FormG6\",(*(,&I+alphainf11GF$
\"\"\"I+alphainf21GF$!\"\"F),&I)tinfty11GF$F)I)tinfty21GF$F+F+I'lambda
GF$F)F)I#nuGF$F)*&I#muGF$F),&F/F)I\"qGF$F+F+F)" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "+1I'lambdaG6\",$*&,&*&I#nuGF$\"\"\"I\"qGF$F*!\"\"I#muGF$F
*F*F+F,F,\"\"!,$*(,**&F+\"\"#I+alphainf11GF$F*F,*&F+F3I+alphainf21GF$F
*F**&F-F*I)tinfty11GF$F*F**&F-F*I)tinfty21GF$F*F,F*,&F8F*F:F,F,F+!\"#F
,F*,$*&F-F*F+!\"$F,F3,$*&F-F*F+!\"%F,\"\"$,$*&F-F*F+!\"&F,\"\"%,$*&F-F
*F+!\"'F,\"\"&-I\"OG6$%*protectedGI(_syslibGF$6#F*\"\"'" }}{PARA 11 ""
 1 "" {XPPMATH 20 "<$/I#muG6\"*(I\"qGF%\"\"#,2*&I%t011GF%\"\"\"I+alpha
inf11GF%F,F,*&F+F,I+alphainf21GF%F,!\"\"*&I%t021GF%F,F-F,F0*&F2F,F/F,F
,*&I)alpha011GF%F,I)tinfty11GF%F,F,*&F5F,I)tinfty21GF%F,F0*&I)alpha021
GF%F,F6F,F0*&F:F,F8F,F,F,,**&F+F,F6F,F,*&F+F,F8F,F0*&F2F,F6F,F0*&F2F,F
8F,F,F0/I#nuGF%*(F'F,F)F,F<F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 207 "mu := q^2*(t011*alphainf11-t011*alphainf21-t021*alphainf11+t021
*alphainf21+alpha011*tinfty11-alpha011*tinfty21-alpha021*tinfty11+alph
a021*tinfty21)/(t011*tinfty11-t011*tinfty21-t021*tinfty11+t021*tinfty2
1);\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 204 "nu := q*(t011*alphainf1
1-t011*alphainf21-t021*alphainf11+t021*alphainf21+alpha011*tinfty11-al
pha011*tinfty21-alpha021*tinfty11+alpha021*tinfty21)/(t011*tinfty11-t0
11*tinfty21-t021*tinfty11+t021*tinfty21);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 ">I#muG6\"*(I\"qGF$\"\"#,2*&I%t011GF$\"\"\"I+alphainf11GF$
F+F+*&F*F+I+alphainf21GF$F+!\"\"*&I%t021GF$F+F,F+F/*&F1F+F.F+F+*&I)alp
ha011GF$F+I)tinfty11GF$F+F+*&F4F+I)tinfty21GF$F+F/*&I)alpha021GF$F+F5F
+F/*&F9F+F7F+F+F+,**&F*F+F5F+F+*&F*F+F7F+F/*&F1F+F5F+F/*&F1F+F7F+F+F/"
 }}{PARA 11 "" 1 "" {XPPMATH 20 ">I#nuG6\"*(I\"qGF$\"\"\",2*&I%t011GF$
F'I+alphainf11GF$F'F'*&F*F'I+alphainf21GF$F'!\"\"*&I%t021GF$F'F+F'F.*&
F0F'F-F'F'*&I)alpha011GF$F'I)tinfty11GF$F'F'*&F3F'I)tinfty21GF$F'F.*&I
)alpha021GF$F'F4F'F.*&F8F'F6F'F'F',**&F*F'F4F'F'*&F*F'F6F'F.*&F0F'F4F'
F.*&F0F'F6F'F'F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "A11Inft
yLambda3:=factor(-residue(A11Infty/lambda^4,lambda=infinity));\n" }
{MPLTEXT 1 0 70 "A11InftyLambda2:=factor(-residue(A11Infty/lambda^3,la
mbda=infinity));\n" }{MPLTEXT 1 0 70 "A11InftyLambda1:=factor(-residue
(A11Infty/lambda^2,lambda=infinity));\n" }{MPLTEXT 1 0 70 "A11InftyLam
bda0:=factor(-residue(A11Infty/lambda^1,lambda=infinity));\n" }
{MPLTEXT 1 0 75 "A11InftyLambdaMinus1:=factor(-residue(A11Infty/lambda
^0,lambda=infinity)):\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 65 "A11Zer
oLambdaMinus3:=factor(residue(A11Zero*lambda^2,lambda=0));\n" }
{MPLTEXT 1 0 65 "A11ZeroLambdaMinus2:=factor(residue(A11Zero*lambda^1,
lambda=0));\n" }{MPLTEXT 1 0 65 "A11ZeroLambdaMinus1:=factor(residue(A
11Zero*lambda^0,lambda=0));\n" }{MPLTEXT 1 0 63 "A11ZeroLambda0:=facto
r(residue(A11Zero*lambda^(-1),lambda=0));\n" }{MPLTEXT 1 0 63 "A11Zero
Lambda1:=factor(residue(A11Zero*lambda^(-2),lambda=0)):\n" }}{PARA 11 
"" 1 "" {XPPMATH 20 ">I0A11InftyLambda3G6\"\"\"!" }}{PARA 11 "" 1 "" 
{XPPMATH 20 ">I0A11InftyLambda2G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 
20 ">I0A11InftyLambda1G6\"*&,&*&I+alphainf11GF$\"\"\"I)tinfty21GF$F)F)
*&I+alphainf21GF$F)I)tinfty11GF$F)!\"\"F),&F-F)F*F.F." }}{PARA 11 "" 1
 "" {XPPMATH 20 ">I0A11InftyLambda0G6\",$*&,2*(I%LA10GF$\"\"\"I)tinfty
11GF$F*I)tinfty21GF$F*F**&F)F*F,\"\"#!\"\"*&I%LA20GF$F*F+F.F/*(F1F*F+F
*F,F*F**(I+alphainf11GF$F*I)tinfty10GF$F*F+F*F**(F4F*F5F*F,F*F**(I+alp
hainf21GF$F*F5F*F+F*F/*(F8F*F5F*F,F*F/F*,&F+F*F,F/!\"#F/" }}{PARA 11 "
" 1 "" {XPPMATH 20 ">I4A11ZeroLambdaMinus3G6\"\"\"!" }}{PARA 11 "" 1 "
" {XPPMATH 20 ">I4A11ZeroLambdaMinus2G6\"\"\"!" }}{PARA 11 "" 1 "" 
{XPPMATH 20 ">I4A11ZeroLambdaMinus1G6\",$*&,&*&I%t011GF$\"\"\"I)alpha0
21GF$F*F**&I%t021GF$F*I)alpha011GF$F*!\"\"F*,&F)F*F-F/F/F/" }}{PARA 11
 "" 1 "" {XPPMATH 20 ">I/A11ZeroLambda0G6\",$*&,2*(I%LB10GF$\"\"\"I%t0
11GF$F*I%t021GF$F*F**&F)F*F,\"\"#!\"\"*&I%LB20GF$F*F+F.F/*(F1F*F+F*F,F
*F**(I%t010GF$F*F,F*I)alpha011GF$F*F**(F4F*F,F*I)alpha021GF$F*F/*(F+F*
I%t020GF$F*F5F*F/*(F+F*F9F*F7F*F*F*,&F+F*F,F/!\"#F/" }}}{EXCHG {PARA 
211 "" 0 "" {TEXT 205 7 "A_\{1,1\}" }{TEXT 205 1 "=" }{TEXT 205 68 "(a
lphainf11*tinfty21-alphainf21*tinfty11)/(tinfty11-tinfty21)*lambda" }
{TEXT 205 1 "+" }{TEXT 205 16 "C-rho/(lambda-q)" }{TEXT 205 49 "-(t011
*alpha021-t021*alpha011)/(t011-t021)/lambda" }}}{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 }