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{SECT 0 {EXCHG {PARA 211 "" 0 "" {TEXT 206 151 "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 6 equation." }
{TEXT 206 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 206 118 "We first use t
he expression of the coefficients of the spectral curve in terms of th
e irregular times and monodromies." }{TEXT 206 0 "" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 9 "restart:\n" }{MPLTEXT 1 0 71 "CoherenceEquat
ion1 := tinfty10+tt10+tt20+t010+t020+tinfty20+t110+t120;\n" }{MPLTEXT 
1 0 38 "CoherenceEquation2 := Pt12+P012+P112;\n" }{MPLTEXT 1 0 83 "Coh
erenceEquation3 := -tinfty10*tinfty20+tt20*tt10+Pt12*t+t010*t020+P112+
t120*t110;" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I3CoherenceEquation1G6\",
2I)tinfty10GF$\"\"\"I%tt10GF$F'I%tt20GF$F'I%t010GF$F'I%t020GF$F'I)tinf
ty20GF$F'I%t110GF$F'I%t120GF$F'" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I3Co
herenceEquation2G6\",(I%Pt12GF$\"\"\"I%P012GF$F'I%P112GF$F'" }}{PARA 
11 "" 1 "" {XPPMATH 20 ">I3CoherenceEquation3G6\",.*&I%Pt12GF$\"\"\"I
\"tGF$F(F(*&I%t010GF$F(I%t020GF$F(F(*&I%t110GF$F(I%t120GF$F(F(*&I%tt10
GF$F(I%tt20GF$F(F(*&I)tinfty10GF$F(I)tinfty20GF$F(!\"\"I%P112GF$F(" }}
}{SECT 0 {PARA 3 "" 0 "" {TEXT 228 75 "Computation of the Lax matrix L
 using the asymptotics of the wave functions" }}{EXCHG {PARA 210 "" 0 
"" {TEXT 229 36 "Study of the asymptotics at infinity" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 182 "logPsi1Infty:=-tinfty10/h*ln(lambd
a)+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 0 195 "logPsi2Infty:=-tinfty20/h*ln(lambda)-1*l
n(lambda)+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)/l
ambda^(7-1)  ;\n" }{MPLTEXT 1 0 191 "Llogpsi1Infty:=-Ltinfty10/h*ln(la
mbda)+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 0 191 "Llogpsi2Infty:=-Ltinfty20/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 0 310 "Lpsi1Infty := exp(1/h*(-tin
fty10*ln(lambda)+h*A10-h*A12/lambda-1/2*h*A13/lambda^2-1/3*h*A14/lambd
a^3-1/4*h*A15/lambda^4-1/5*h*A16/lambda^5-1/6*h*A17/lambda^6))*1/h*(-L
tinfty10*ln(lambda)+h*LA10-h*LA12/lambda-1/2*h*LA13/lambda^2-1/3*h*LA1
4/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(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*(-Ltinft
y20*ln(lambda)+h*LA20-h*LA22/lambda-1/2*h*LA23/lambda^2-1/3*h*LA24/lam
bda^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 "
psi2Infty:=exp(logPsi2Infty);\n" }{MPLTEXT 1 0 43 "dpsi1dlambdaInfty:=
diff(psi1Infty,lambda):\n" }{MPLTEXT 1 0 43 "dpsi2dlambdaInfty:=diff(p
si2Infty,lambda):\n" }{MPLTEXT 1 0 47 "d2psi1dlambda2Infty:=diff(psi1I
nfty,lambda$2):\n" }{MPLTEXT 1 0 47 "d2psi2dlambda2Infty:=diff(psi2Inf
ty,lambda$2):\n" }{MPLTEXT 1 0 31 "Vinfty1:=+tinfty10*ln(lambda);\n" }
{MPLTEXT 1 0 31 "Vinfty2:=+tinfty20*ln(lambda);\n" }{MPLTEXT 1 0 1 "\n
" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 89 "WronskianLambdaInfty:=h*facto
r(psi1Infty*dpsi2dlambdaInfty-psi2Infty*dpsi1dlambdaInfty):\n" }
{MPLTEXT 1 0 132 "WronskianLambdabisInfty:=h*simplify(factor( (diff(lo
gPsi2Infty,lambda)-diff(logPsi1Infty,lambda))*exp(logPsi1Infty+logPsi2
Infty))):\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 116 "WronskianTildeLam
bdaInfty:=h^3*factor(dpsi2dlambdaInfty*d2psi1dlambda2Infty-dpsi1dlambd
aInfty*d2psi2dlambda2Infty):\n" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I-log
Psi1InftyG6\",2*(I)tinfty10GF$\"\"\"I\"hGF$!\"\"-I#lnG6$%*protectedGI(
_syslibGF$6#I'lambdaGF$F(F*I$A10GF$F(*&I$A12GF$F(F1F*F**&I$A13GF$F(F1!
\"##F*\"\"#*&I$A14GF$F(F1!\"$#F*\"\"$*&I$A15GF$F(F1!\"%#F*\"\"%*&I$A16
GF$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$A20GF$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%L
A10GF$F(*&I%LA12GF$F(F1F*F**&I%LA13GF$F(F1!\"##F*\"\"#*&I%LA14GF$F(F1!
\"$#F*\"\"$*&I%LA15GF$F(F1!\"%#F*\"\"%*&I%LA16GF$F(F1!\"&#F*\"\"&*&I%L
A17GF$F(F1!\"'#F*\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I.Llogpsi2Inf
tyG6\",2*(I*Ltinfty20GF$\"\"\"I\"hGF$!\"\"-I#lnG6$%*protectedGI(_sysli
bGF$6#I'lambdaGF$F(F*I%LA20GF$F(*&I%LA22GF$F(F1F*F**&I%LA23GF$F(F1!\"#
#F*\"\"#*&I%LA24GF$F(F1!\"$#F*\"\"$*&I%LA25GF$F(F1!\"%#F*\"\"%*&I%LA26
GF$F(F1!\"&#F*\"\"&*&I%LA27GF$F(F1!\"'#F*\"\"'" }}{PARA 11 "" 1 "" 
{XPPMATH 20 ">I+Lpsi1InftyG6\"*(-I$expG6$%*protectedGI(_syslibGF$6#*&I
\"hGF$!\"\",2*&I)tinfty10GF$\"\"\"-I#lnGF(6#I'lambdaGF$F2F.*&F-F2I$A10
GF$F2F2*(F-F2I$A12GF$F2F6F.F.*(F-F2I$A13GF$F2F6!\"##F.\"\"#*(F-F2I$A14
GF$F2F6!\"$#F.\"\"$*(F-F2I$A15GF$F2F6!\"%#F.\"\"%*(F-F2I$A16GF$F2F6!\"
&#F.\"\"&*(F-F2I$A17GF$F2F6!\"'#F.\"\"'F2F2F-F.,2*&I*Ltinfty10GF$F2F3F
2F.*&F-F2I%LA10GF$F2F2*(F-F2I%LA12GF$F2F6F.F.*(F-F2I%LA13GF$F2F6F=F>*(
F-F2I%LA14GF$F2F6FBFC*(F-F2I%LA15GF$F2F6FGFH*(F-F2I%LA16GF$F2F6FLFM*(F
-F2I%LA17GF$F2F6FQFRF2" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I+Lpsi2InftyG
6\"*(-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$F2F6!\"&#F.\"\"&*(F-F
2I$A27GF$F2F6!\"'#F.\"\"'F2F2F-F.,2*&I*Ltinfty20GF$F2F3F2F.*&F-F2I%LA2
0GF$F2F2*(F-F2I%LA22GF$F2F6F.F.*(F-F2I%LA23GF$F2F6F>F?*(F-F2I%LA24GF$F
2F6FCFD*(F-F2I%LA25GF$F2F6FHFI*(F-F2I%LA26GF$F2F6FMFN*(F-F2I%LA27GF$F2
F6FRFSF2" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I*psi1InftyG6\"-I$expG6$%*p
rotectedGI(_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*ps
i2InftyG6\"-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!\"%#F
0\"\"%*&I$A26GF$F.F4!\"&#F0\"\"&*&I$A27GF$F.F4!\"'#F0\"\"'" }}{PARA 11
 "" 1 "" {XPPMATH 20 ">I(Vinfty1G6\"*&I)tinfty10GF$\"\"\"-I#lnG6$%*pro
tectedGI(_syslibGF$6#I'lambdaGF$F'" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I
(Vinfty2G6\"*&I)tinfty20GF$\"\"\"-I#lnG6$%*protectedGI(_syslibGF$6#I'l
ambdaGF$F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "L21Infty:=fac
tor(simplify(WronskianTildeLambdaInfty/WronskianLambdabisInfty)):\n" }
{MPLTEXT 1 0 75 "L21InftyOrderlambda3:=factor(-residue(L21Infty/lambda
^4,lambda=infinity));\n" }{MPLTEXT 1 0 75 "L21InftyOrderlambda2:=facto
r(-residue(L21Infty/lambda^3,lambda=infinity));\n" }{MPLTEXT 1 0 75 "L
21InftyOrderlambda1:=factor(-residue(L21Infty/lambda^2,lambda=infinity
));\n" }{MPLTEXT 1 0 75 "L21InftyOrderlambda0:=factor(-residue(L21Inft
y/lambda^1,lambda=infinity));\n" }{MPLTEXT 1 0 80 "L21InftyOrderlambda
Minus1:=factor(-residue(L21Infty/lambda^0,lambda=infinity));\n" }
{MPLTEXT 1 0 59 "L21InftyNumer:=series(numer(L21Infty),lambda=infinity
,30):\n" }{MPLTEXT 1 0 59 "L21InftyDenom:=series(denom(L21Infty),lambd
a=infinity,30):\n" }{MPLTEXT 1 0 56 "series(L21InftyNumer/L21InftyDeno
m,lambda=infinity,12):\n" }{MPLTEXT 1 0 129 "L21InftyOrderlambdaMinus2
:=factor(-residue(series(L21InftyNumer/L21InftyDenom,lambda=infinity,1
2)/lambda^(-1),lambda=infinity));\n" }{MPLTEXT 1 0 129 "L21InftyOrderl
ambdaMinus3:=factor(-residue(series(L21InftyNumer/L21InftyDenom,lambda
=infinity,12)/lambda^(-2),lambda=infinity));\n" }{MPLTEXT 1 0 59 "L21I
nftyNumer:=series(numer(L21Infty),lambda=infinity,30):\n" }{MPLTEXT 1 
0 59 "L21InftyDenom:=series(denom(L21Infty),lambda=infinity,30):\n" }
{MPLTEXT 1 0 56 "series(L21InftyNumer/L21InftyDenom,lambda=infinity,10
):\n" }{MPLTEXT 1 0 128 "L21InftyOrderlambdaMinus2:=factor(-residue(se
ries(L21InftyNumer/L21InftyDenom,lambda=infinity,10)/lambda^(-1),lambd
a=infinity)):" }{MPLTEXT 1 0 1 "\n" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I
5L21InftyOrderlambda3G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I5L21
InftyOrderlambda2G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I5L21Inft
yOrderlambda1G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I5L21InftyOrd
erlambda0G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I:L21InftyOrderla
mbdaMinus1G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I:L21InftyOrderl
ambdaMinus2G6\",$*&,&I\"hGF$\"\"\"I)tinfty20GF$F)F)I)tinfty10GF$F)!\"
\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I:L21InftyOrderlambdaMinus3G6\",$
*(I\"hGF$\"\"\",0*(I$A12GF$F(F'F(I)tinfty10GF$F(F(*(F+F(F'F(I)tinfty20
GF$F(!\"\"*(F+F(F,F(F.F(F(*&F+F(F.\"\"#F/*(I$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 230 129 "We get that L21 behaves at infinity like -(h+tinfty20)
tinfty10/lambda^2= (-tinfty10tinfty20 -h*tinfty10)/lambda^2 +O(lambda^
\{-1\})" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "L22Infty:=factor
(h*simplify(diff(WronskianLambdabisInfty,lambda)/WronskianLambdabisInf
ty)):\n" }{MPLTEXT 1 0 75 "L22InftyOrderlambda3:=factor(-residue(L22In
fty/lambda^4,lambda=infinity));\n" }{MPLTEXT 1 0 75 "L22InftyOrderlamb
da2:=factor(-residue(L22Infty/lambda^3,lambda=infinity));\n" }{MPLTEXT
 1 0 75 "L22InftyOrderlambda1:=factor(-residue(L22Infty/lambda^2,lambd
a=infinity));\n" }{MPLTEXT 1 0 75 "L22InftyOrderlambda0:=factor(-resid
ue(L22Infty/lambda^1,lambda=infinity));\n" }{MPLTEXT 1 0 80 "L22InftyO
rderlambdaMinus1:=factor(-residue(L22Infty/lambda^0,lambda=infinity));
\n" }{MPLTEXT 1 0 83 "L22InftyOrderlambdaMinus2:=factor(-residue(L22In
fty/lambda^(-1),lambda=infinity));\n" }{MPLTEXT 1 0 82 "L22InftyOrderl
ambdaMinus3:=factor(-residue(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\"\"\"!" }}
{PARA 11 "" 1 "" {XPPMATH 20 ">I:L22InftyOrderlambdaMinus1G6\",(I\"hGF
$!\"#I)tinfty10GF$!\"\"I)tinfty20GF$F)" }}{PARA 11 "" 1 "" {XPPMATH 20
 ">I:L22InftyOrderlambdaMinus2G6\",$*(I\"hGF$\"\"\",,*&I$A12GF$F(I)tin
fty10GF$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 230 
20 "We get that L_\{2,2\} " }{TEXT 230 24 "behaves at infinity like" }
{TEXT 230 52 " -(tinfty10+tinfty20+2\\hbar)/lambda+h*O(1/lambda^2) " }
}}{EXCHG {PARA 209 "" 0 "" {TEXT 231 36 "Study of the asymptotics at l
ambda=0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 176 "logPsi1Zero:=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 0 176 "logPsi2Zero:=t020/h*ln(lamb
da)+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 0 185 "Llogpsi1Zero:=Lt010/h*ln(lambda)+LB10+L
B12/(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:=Lt020/h*ln(lambda)+LB20+LB22
/(2-1)*lambda^(2-1)+LB23/(3-1)*lambda^(3-1)+LB24/(4-1)*lambda^(4-1)+LB
25/(5-1)*lambda^(5-1)+LB26/(6-1)*lambda^(6-1)+LB27/(7-1)*lambda^(7-1) \+
 ;\n" }{MPLTEXT 1 0 180 "Lpsi1Zero := exp((t010/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+LB12/(2-1)*lambda^(2-1)+LB
13/(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)+B2
3/(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 "*(L
t020/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 28 "psi1Zero:=exp(logPsi
1Zero);\n" }{MPLTEXT 1 0 28 "psi2Zero:=exp(logPsi2Zero);\n" }{MPLTEXT 
1 0 41 "dpsi1dlambdaZero:=diff(psi1Zero,lambda):\n" }{MPLTEXT 1 0 41 "
dpsi2dlambdaZero:=diff(psi2Zero,lambda):\n" }{MPLTEXT 1 0 45 "d2psi1dl
ambda2Zero:=diff(psi1Zero,lambda$2):\n" }{MPLTEXT 1 0 45 "d2psi2dlambd
a2Zero:=diff(psi2Zero,lambda$2):\n" }{MPLTEXT 1 0 25 "VZero1:=t010*ln(
lambda);\n" }{MPLTEXT 1 0 25 "VZero2:=t020*ln(lambda);\n" }{MPLTEXT 1 
0 1 "\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 84 "WronskianLambdaZero:=h
*factor(psi1Zero*dpsi2dlambdaZero-psi2Zero*dpsi1dlambdaZero):\n" }
{MPLTEXT 1 0 127 "WronskianLambdabisZero:=h*simplify(factor( (diff(log
Psi2Zero,lambda)-diff(logPsi1Zero,lambda))*exp(logPsi1Zero+logPsi2Zero
))):\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 111 "WronskianTildeLambdaZe
ro:=h^3*factor(dpsi2dlambdaZero*d2psi1dlambda2Zero-dpsi1dlambdaZero*d2
psi2dlambda2Zero):\n" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I,logPsi1ZeroG6
\",2*(I%t010GF$\"\"\"I\"hGF$!\"\"-I#lnG6$%*protectedGI(_syslibGF$6#I'l
ambdaGF$F(F(I$B10GF$F(*&I$B12GF$F(F1F(F(*&I$B13GF$F(F1\"\"##F(F7*&I$B1
4GF$F(F1\"\"$#F(F;*&I$B15GF$F(F1\"\"%#F(F?*&I$B16GF$F(F1\"\"&#F(FC*&I$
B17GF$F(F1\"\"'#F(FG" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I,logPsi2ZeroG6
\",2*(I%t020GF$\"\"\"I\"hGF$!\"\"-I#lnG6$%*protectedGI(_syslibGF$6#I'l
ambdaGF$F(F(I$B20GF$F(*&I$B22GF$F(F1F(F(*&I$B23GF$F(F1\"\"##F(F7*&I$B2
4GF$F(F1\"\"$#F(F;*&I$B25GF$F(F1\"\"%#F(F?*&I$B26GF$F(F1\"\"&#F(FC*&I$
B27GF$F(F1\"\"'#F(FG" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I-Llogpsi1ZeroG
6\",2*(I&Lt010GF$\"\"\"I\"hGF$!\"\"-I#lnG6$%*protectedGI(_syslibGF$6#I
'lambdaGF$F(F(I%LB10GF$F(*&I%LB12GF$F(F1F(F(*&I%LB13GF$F(F1\"\"##F(F7*
&I%LB14GF$F(F1\"\"$#F(F;*&I%LB15GF$F(F1\"\"%#F(F?*&I%LB16GF$F(F1\"\"&#
F(FC*&I%LB17GF$F(F1\"\"'#F(FG" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I-Llog
psi2ZeroG6\",2*(I&Lt020GF$\"\"\"I\"hGF$!\"\"-I#lnG6$%*protectedGI(_sys
libGF$6#I'lambdaGF$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$F
(F1\"\"&#F(FC*&I%LB27GF$F(F1\"\"'#F(FG" }}{PARA 11 "" 1 "" {XPPMATH 20
 ">I*Lpsi1ZeroG6\"*&-I$expG6$%*protectedGI(_syslibGF$6#,2*(I%t010GF$\"
\"\"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$F/F5\"\"%#F/F
C*&I$B16GF$F/F5\"\"&#F/FG*&I$B17GF$F/F5\"\"'#F/FKF/,2*(I&Lt010GF$F/F0F
1F2F/F/I%LB10GF$F/*&I%LB12GF$F/F5F/F/*&I%LB13GF$F/F5F;F<*&I%LB14GF$F/F
5F?F@*&I%LB15GF$F/F5FCFD*&I%LB16GF$F/F5FGFH*&I%LB17GF$F/F5FKFLF/" }}
{PARA 11 "" 1 "" {XPPMATH 20 ">I*Lpsi2ZeroG6\"*&-I$expG6$%*protectedGI
(_syslibGF$6#,2*(I%t020GF$\"\"\"I\"hGF$!\"\"-I#lnGF(6#I'lambdaGF$F/F/I
$B20GF$F/*&I$B22GF$F/F5F/F/*&I$B23GF$F/F5\"\"##F/F;*&I$B24GF$F/F5\"\"$
#F/F?*&I$B25GF$F/F5\"\"%#F/FC*&I$B26GF$F/F5\"\"&FG*&I$B27GF$F/F5\"\"'F
JF/,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$F/F5FG#F/FG*&
I%LB27GF$F/F5FJ#F/FJF/" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I)psi1ZeroG6
\"-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$F.F4\"
\"&#F.FF*&I$B17GF$F.F4\"\"'#F.FJ" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I)p
si2ZeroG6\"-I$expG6$%*protectedGI(_syslibGF$6#,2*(I%t020GF$\"\"\"I\"hG
F$!\"\"-I#lnGF'6#I'lambdaGF$F.F.I$B20GF$F.*&I$B22GF$F.F4F.F.*&I$B23GF$
F.F4\"\"##F.F:*&I$B24GF$F.F4\"\"$#F.F>*&I$B25GF$F.F4\"\"%#F.FB*&I$B26G
F$F.F4\"\"&#F.FF*&I$B27GF$F.F4\"\"'#F.FJ" }}{PARA 11 "" 1 "" {XPPMATH 
20 ">I'VZero1G6\"*&I%t010GF$\"\"\"-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(Wro
nskianLambdabisZero,lambda)/WronskianLambdabisZero)):\n" }{MPLTEXT 1 
0 70 "L22ZeroOrderlambdaMinus3:=factor(residue(L22Zero*lambda^2,lambda
=0));\n" }{MPLTEXT 1 0 70 "L22ZeroOrderlambdaMinus2:=factor(residue(L2
2Zero*lambda^1,lambda=0));\n" }{MPLTEXT 1 0 70 "L22ZeroOrderlambdaMinu
s1:=factor(residue(L22Zero*lambda^0,lambda=0));\n" }{MPLTEXT 1 0 68 "L
22ZeroOrderlambda0:=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(resid
ue(L22Zero*lambda^(-3),lambda=0)):" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I
9L22ZeroOrderlambdaMinus3G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I
9L22ZeroOrderlambdaMinus2G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I
9L22ZeroOrderlambdaMinus1G6\",(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$B
22GF$F'F&F'F/*&F1F'F,F'F'*&F1F'F.F'F/F',&F,F'F.F/F/" }}}{EXCHG {PARA 0
 "" 0 "" {TEXT 230 71 "We get that L_\{2,2\} behaves at lambda=0 like \+
(t010+t020-h)/lambda +O(1)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
76 "L21Zero:=factor(simplify(WronskianTildeLambdaZero/WronskianLambdab
isZero)):\n" }{MPLTEXT 1 0 70 "L21ZeroOrderlambdaMinus5:=factor(residu
e(L21Zero*lambda^4,lambda=0));\n" }{MPLTEXT 1 0 70 "L21ZeroOrderlambda
Minus4:=factor(residue(L21Zero*lambda^3,lambda=0));\n" }{MPLTEXT 1 0 
70 "L21ZeroOrderlambdaMinus3:=factor(residue(L21Zero*lambda^2,lambda=0
));\n" }{MPLTEXT 1 0 70 "L21ZeroOrderlambdaMinus2:=factor(residue(L21Z
ero*lambda^1,lambda=0));\n" }{MPLTEXT 1 0 70 "L21ZeroOrderlambdaMinus1
:=factor(residue(L21Zero*lambda^0,lambda=0));\n" }{MPLTEXT 1 0 68 "L21
ZeroOrderlambda0:=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(residu
e(L21Zero*lambda^(-3),lambda=0)):" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I9
L21ZeroOrderlambdaMinus5G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I9
L21ZeroOrderlambdaMinus4G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I9
L21ZeroOrderlambdaMinus3G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I9
L21ZeroOrderlambdaMinus2G6\",$*&I%t010GF$\"\"\"I%t020GF$F(!\"\"" }}
{PARA 11 "" 1 "" {XPPMATH 20 ">I9L21ZeroOrderlambdaMinus1G6\",$*(I\"hG
F$\"\"\",.*(I$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$F'F&F+I%t0
10GF$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,\"\"$F0*(I$B23
GF$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 230 12 "We ge
t that " }{TEXT 230 8 "L_\{2,1\} " }{TEXT 230 24 "behaves at lambda=0 \+
like" }{TEXT 230 33 " -t010*t020/lambda^2+ O(1/lambda)" }}}{EXCHG 
{PARA 210 "" 0 "" {TEXT 229 36 "Study of the asymptotics at lambda=1" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 201 "logPsi1One:=t110/h*ln(la
mbda-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 0 201 "logPsi2One:
=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 
210 "Llogpsi1One:=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 0 210 "Llogpsi2One:=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 0 205 "Lpsi1One := exp((t110
/h*ln(lambda-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)*(l
ambda-1)^(6-1)+C17/(7-1)*(lambda-1)^(7-1)))\n" }{MPLTEXT 1 0 198 "*(Lt
110/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 
0 205 "Lpsi2One := exp((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 0 198 "*(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 26 "psi1One:=exp(logPsi1One);\n" }
{MPLTEXT 1 0 26 "psi2One:=exp(logPsi2One);\n" }{MPLTEXT 1 0 39 "dpsi1d
lambdaOne:=diff(psi1One,lambda):\n" }{MPLTEXT 1 0 39 "dpsi2dlambdaOne:
=diff(psi2One,lambda):\n" }{MPLTEXT 1 0 43 "d2psi1dlambda2One:=diff(ps
i1One,lambda$2):\n" }{MPLTEXT 1 0 43 "d2psi2dlambda2One:=diff(psi2One,
lambda$2):\n" }{MPLTEXT 1 0 26 "VOne1:=t110*ln(lambda-1);\n" }{MPLTEXT
 1 0 26 "VOne2:=t120*ln(lambda-1);\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 
1 0 1 "\n" }{MPLTEXT 1 0 79 "WronskianLambdaOne:=h*factor(psi1One*dpsi
2dlambdaOne-psi2One*dpsi1dlambdaOne):\n" }{MPLTEXT 1 0 122 "WronskianL
ambdabisOne:=h*simplify(factor( (diff(logPsi2One,lambda)-diff(logPsi1O
ne,lambda))*exp(logPsi1One+logPsi2One))):\n" }{MPLTEXT 1 0 1 "\n" }
{MPLTEXT 1 0 106 "WronskianTildeLambdaOne:=h^3*factor(dpsi2dlambdaOne*
d2psi1dlambda2One-dpsi1dlambdaOne*d2psi2dlambda2One):\n" }}{PARA 11 ""
 1 "" {XPPMATH 20 ">I+logPsi1OneG6\",2*(I%t110GF$\"\"\"I\"hGF$!\"\"-I#
lnG6$%*protectedGI(_syslibGF$6#,&I'lambdaGF$F(F*F(F(F(I$C10GF$F(*&I$C1
2GF$F(F1F(F(*&I$C13GF$F(F1\"\"##F(F8*&I$C14GF$F(F1\"\"$#F(F<*&I$C15GF$
F(F1\"\"%#F(F@*&I$C16GF$F(F1\"\"&#F(FD*&I$C17GF$F(F1\"\"'#F(FH" }}
{PARA 11 "" 1 "" {XPPMATH 20 ">I+logPsi2OneG6\",2*(I%t120GF$\"\"\"I\"h
GF$!\"\"-I#lnG6$%*protectedGI(_syslibGF$6#,&I'lambdaGF$F(F*F(F(F(I$C20
GF$F(*&I$C22GF$F(F1F(F(*&I$C23GF$F(F1\"\"##F(F8*&I$C24GF$F(F1\"\"$#F(F
<*&I$C25GF$F(F1\"\"%#F(F@*&I$C26GF$F(F1\"\"&#F(FD*&I$C27GF$F(F1\"\"'#F
(FH" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I,Llogpsi1OneG6\",2*(I&Lt110GF$
\"\"\"I\"hGF$!\"\"-I#lnG6$%*protectedGI(_syslibGF$6#,&I'lambdaGF$F(F*F
(F(F(I%LC10GF$F(*&I%LC12GF$F(F1F(F(*&I%LC13GF$F(F1\"\"##F(F8*&I%LC14GF
$F(F1\"\"$#F(F<*&I%LC15GF$F(F1\"\"%#F(F@*&I%LC16GF$F(F1\"\"&#F(FD*&I%L
C17GF$F(F1\"\"'#F(FH" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I,Llogpsi2OneG6
\",2*(I&Lt120GF$\"\"\"I\"hGF$!\"\"-I#lnG6$%*protectedGI(_syslibGF$6#,&
I'lambdaGF$F(F*F(F(F(I%LC20GF$F(*&I%LC22GF$F(F1F(F(*&I%LC23GF$F(F1\"\"
##F(F8*&I%LC24GF$F(F1\"\"$#F(F<*&I%LC25GF$F(F1\"\"%#F(F@*&I%LC26GF$F(F
1\"\"&#F(FD*&I%LC27GF$F(F1\"\"'#F(FH" }}{PARA 11 "" 1 "" {XPPMATH 20 "
>I)Lpsi1OneG6\"*&-I$expG6$%*protectedGI(_syslibGF$6#,2*(I%t110GF$\"\"
\"I\"hGF$!\"\"-I#lnGF(6#,&I'lambdaGF$F/F1F/F/F/I$C10GF$F/*&I$C12GF$F/F
5F/F/*&I$C13GF$F/F5\"\"##F/F<*&I$C14GF$F/F5\"\"$#F/F@*&I$C15GF$F/F5\"
\"%#F/FD*&I$C16GF$F/F5\"\"&#F/FH*&I$C17GF$F/F5\"\"'#F/FLF/,2*(I&Lt110G
F$F/F0F1F2F/F/I%LC10GF$F/*&I%LC12GF$F/F5F/F/*&I%LC13GF$F/F5F<F=*&I%LC1
4GF$F/F5F@FA*&I%LC15GF$F/F5FDFE*&I%LC16GF$F/F5FHFI*&I%LC17GF$F/F5FLFMF
/" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I)Lpsi2OneG6\"*&-I$expG6$%*protect
edGI(_syslibGF$6#,2*(I%t120GF$\"\"\"I\"hGF$!\"\"-I#lnGF(6#,&I'lambdaGF
$F/F1F/F/F/I$C20GF$F/*&I$C22GF$F/F5F/F/*&I$C23GF$F/F5\"\"##F/F<*&I$C24
GF$F/F5\"\"$#F/F@*&I$C25GF$F/F5\"\"%#F/FD*&I$C26GF$F/F5\"\"&#F/FH*&I$C
27GF$F/F5\"\"'#F/FLF/,2*(I&Lt120GF$F/F0F1F2F/F/I%LC20GF$F/*&I%LC22GF$F
/F5F/F/*&I%LC23GF$F/F5F<F=*&I%LC24GF$F/F5F@FA*&I%LC25GF$F/F5FDFE*&I%LC
26GF$F/F5FHFI*&I%LC27GF$F/F5FLFMF/" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I
(psi1OneG6\"-I$expG6$%*protectedGI(_syslibGF$6#,2*(I%t110GF$\"\"\"I\"h
GF$!\"\"-I#lnGF'6#,&I'lambdaGF$F.F0F.F.F.I$C10GF$F.*&I$C12GF$F.F4F.F.*
&I$C13GF$F.F4\"\"##F.F;*&I$C14GF$F.F4\"\"$#F.F?*&I$C15GF$F.F4\"\"%#F.F
C*&I$C16GF$F.F4\"\"&#F.FG*&I$C17GF$F.F4\"\"'#F.FK" }}{PARA 11 "" 1 "" 
{XPPMATH 20 ">I(psi2OneG6\"-I$expG6$%*protectedGI(_syslibGF$6#,2*(I%t1
20GF$\"\"\"I\"hGF$!\"\"-I#lnGF'6#,&I'lambdaGF$F.F0F.F.F.I$C20GF$F.*&I$
C22GF$F.F4F.F.*&I$C23GF$F.F4\"\"##F.F;*&I$C24GF$F.F4\"\"$#F.F?*&I$C25G
F$F.F4\"\"%#F.FC*&I$C26GF$F.F4\"\"&#F.FG*&I$C27GF$F.F4\"\"'#F.FK" }}
{PARA 11 "" 1 "" {XPPMATH 20 ">I&VOne1G6\"*&I%t110GF$\"\"\"-I#lnG6$%*p
rotectedGI(_syslibGF$6#,&I'lambdaGF$F'!\"\"F'F'" }}{PARA 11 "" 1 "" 
{XPPMATH 20 ">I&VOne2G6\"*&I%t120GF$\"\"\"-I#lnG6$%*protectedGI(_sysli
bGF$6#,&I'lambdaGF$F'!\"\"F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 86 "L22One:=factor(h*simplify(diff(WronskianLambdabisOne,lambda)/W
ronskianLambdabisOne)):\n" }{MPLTEXT 1 0 72 "L22OneOrderlambdaMinus3:=
factor(residue(L22One*(lambda-1)^2,lambda=1));\n" }{MPLTEXT 1 0 72 "L2
2OneOrderlambdaMinus2:=factor(residue(L22One*(lambda-1)^1,lambda=1));
\n" }{MPLTEXT 1 0 72 "L22OneOrderlambdaMinus1:=factor(residue(L22One*(
lambda-1)^0,lambda=1));\n" }{MPLTEXT 1 0 70 "L22OneOrderlambda0:=facto
r(residue(L22One*(lambda-1)^(-1),lambda=1));\n" }{MPLTEXT 1 0 70 "L22O
neOrderlambda1:=factor(residue(L22One*(lambda-1)^(-2),lambda=1)):\n" }
{MPLTEXT 1 0 69 "L22OneOrderlambda2:=factor(residue(L22One*(lambda-1)^
(-3),lambda=1)):" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I8L22OneOrderlambda
Minus3G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I8L22OneOrderlambdaM
inus2G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I8L22OneOrderlambdaMi
nus1G6\",(I%t110GF$\"\"\"I\"hGF$!\"\"I%t120GF$F'" }}{PARA 11 "" 1 "" 
{XPPMATH 20 ">I3L22OneOrderlambda0G6\"*(I\"hGF$\"\"\",.*&I$C12GF$F'F&F
'F'*&F*F'I%t110GF$F'F'*&F*F'I%t120GF$F'!\"\"*&I$C22GF$F'F&F'F/*&F1F'F,
F'F'*&F1F'F.F'F/F',&F,F'F.F/F/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 230 
75 "We get that L_\{2,2\} behaves at lambda=1 like (t110+t120-h)/(lamb
da-1) +O(1)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "L21One:=fact
or(simplify(WronskianTildeLambdaOne/WronskianLambdabisOne)):\n" }
{MPLTEXT 1 0 72 "L21OneOrderlambdaMinus5:=factor(residue(L21One*(lambd
a-1)^4,lambda=1));\n" }{MPLTEXT 1 0 72 "L21OneOrderlambdaMinus4:=facto
r(residue(L21One*(lambda-1)^3,lambda=1));\n" }{MPLTEXT 1 0 72 "L21OneO
rderlambdaMinus3:=factor(residue(L21One*(lambda-1)^2,lambda=1));\n" }
{MPLTEXT 1 0 72 "L21OneOrderlambdaMinus2:=factor(residue(L21One*(lambd
a-1)^1,lambda=1));\n" }{MPLTEXT 1 0 72 "L21OneOrderlambdaMinus1:=facto
r(residue(L21One*(lambda-1)^0,lambda=1));\n" }{MPLTEXT 1 0 70 "L21OneO
rderlambda0:=factor(residue(L21One*(lambda-1)^(-1),lambda=1));\n" }
{MPLTEXT 1 0 70 "L21OneOrderlambda1:=factor(residue(L21One*(lambda-1)^
(-2),lambda=1)):\n" }{MPLTEXT 1 0 69 "L21OneOrderlambda2:=factor(resid
ue(L21One*(lambda-1)^(-3),lambda=1)):" }}{PARA 11 "" 1 "" {XPPMATH 20 
">I8L21OneOrderlambdaMinus5G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "
>I8L21OneOrderlambdaMinus4G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">
I8L21OneOrderlambdaMinus3G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I
8L21OneOrderlambdaMinus2G6\",$*&I%t120GF$\"\"\"I%t110GF$F(!\"\"" }}
{PARA 11 "" 1 "" {XPPMATH 20 ">I8L21OneOrderlambdaMinus1G6\",$*(I\"hGF
$\"\"\",.*(I$C12GF$F(F'F(I%t120GF$F(F(*(F+F(I%t110GF$F(F,F(F(*&F+F(F,
\"\"#!\"\"*(I$C22GF$F(F'F(F.F(F1*&F3F(F.F0F(*(F3F(F.F(F,F(F1F(,&F.F(F,
F1F1F1" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I3L21OneOrderlambda0G6\"*(I\"
hGF$\"\"\",D*(I$C12GF$\"\"#F&F+I%t120GF$F'F'**F*F'I$C22GF$F'F&F+I%t110
GF$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$C13GF$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,\"\"$F0*(I$C23GF
$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 230 44 "We get \+
that L_\{2,1\} behaves at lambda=1 like" }{TEXT 230 41 " -t110*t120/(l
ambda-1)^2+ O(1/(lambda-1))" }}}{EXCHG {PARA 210 "" 0 "" {TEXT 229 87 
"Study of the asymtotics at lambda=t with the deformation operator giv
en by h*partial_t " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 199 "logP
si1T:=tt10/h*ln(lambda-t)+C10+C12/(2-1)*(lambda-t)^(2-1)+C13/(3-1)*(la
mbda-t)^(3-1)+C14/(4-1)*(lambda-t)^(4-1)+C15/(5-1)*(lambda-t)^(5-1)+C1
6/(6-1)*(lambda-t)^(6-1)+C17/(7-1)*(lambda-t)^(7-1)  ;\n" }{MPLTEXT 1 
0 199 "logPsi2T:=tt20/h*ln(lambda-t)+C20+C22/(2-1)*(lambda-t)^(2-1)+C2
3/(3-1)*(lambda-t)^(3-1)+C24/(4-1)*(lambda-t)^(4-1)+C25/(5-1)*(lambda-
t)^(5-1)+C26/(6-1)*(lambda-t)^(6-1)+C27/(7-1)*(lambda-t)^(7-1)  ;\n" }
{MPLTEXT 1 0 205 "Llogpsi1T:=Ltt10/h*ln(lambda-t)+LC10+LC12/(2-1)*(lam
bda-t)^(2-1)+LC13/(3-1)*(lambda-t)^(3-1)+LC14/(4-1)*(lambda-t)^(4-1)+L
C15/(5-1)*(lambda-t)^(5-1)+LC16/(6-1)*(lambda-t)^(6-1)+LC17/(7-1)*(lam
bda-t)^(7-1)\n" }{MPLTEXT 1 0 163 "+h*(-tt10/h/(lambda-t)-C12*(lambda-
t)^(2-1-1)-C13*(lambda-t)^(3-1-1)-C14*(lambda-t)^(4-1-1)-C15*(lambda-t
)^(5-1-1)+C16*(lambda-t)^(6-1-1)+C17*(lambda-t)^(7-1-1));\n" }{MPLTEXT
 1 0 205 "Llogpsi2T:=Ltt20/h*ln(lambda-t)+LC20+LC22/(2-1)*(lambda-t)^(
2-1)+LC23/(3-1)*(lambda-t)^(3-1)+LC24/(4-1)*(lambda-t)^(4-1)+LC25/(5-1
)*(lambda-t)^(5-1)+LC26/(6-1)*(lambda-t)^(6-1)+LC27/(7-1)*(lambda-t)^(
7-1)\n" }{MPLTEXT 1 0 163 "+h*(-tt20/h/(lambda-t)-C22*(lambda-t)^(2-1-
1)-C23*(lambda-t)^(3-1-1)-C24*(lambda-t)^(4-1-1)-C25*(lambda-t)^(5-1-1
)+C26*(lambda-t)^(6-1-1)+C27*(lambda-t)^(7-1-1));\n" }{MPLTEXT 1 0 
203 "Lpsi1T := exp((tt10/h*ln(lambda-t)+C10+C12/(2-1)*(lambda-t)^(2-1)
+C13/(3-1)*(lambda-t)^(3-1)+C14/(4-1)*(lambda-t)^(4-1)+C15/(5-1)*(lamb
da-t)^(5-1)+C16/(6-1)*(lambda-t)^(6-1)+C17/(7-1)*(lambda-t)^(7-1)))\n"
 }{MPLTEXT 1 0 196 "*(Ltt10/h*ln(lambda-t)+LC10+LC12/(2-1)*(lambda-t)^
(2-1)+LC13/(3-1)*(lambda-t)^(3-1)+LC14/(4-1)*(lambda-t)^(4-1)+LC15/(5-
1)*(lambda-t)^(5-1)+LC16/(6-1)*(lambda-t)^(6-1)+LC17/(7-1)*(lambda-t)^
(7-1)\n" }{MPLTEXT 1 0 162 "+h*(-tt10/h/(lambda-t)-C12*(lambda-t)^(2-1
-1)-C13*(lambda-t)^(3-1-1)-C14*(lambda-t)^(4-1-1)-C15*(lambda-t)^(5-1-
1)+C16*(lambda-t)^(6-1-1)+C17*(lambda-t)^(7-1-1))\n" }{MPLTEXT 1 0 3 "
);\n" }{MPLTEXT 1 0 203 "Lpsi2T := exp((tt20/h*ln(lambda-t)+C20+C22/(2
-1)*(lambda-t)^(2-1)+C23/(3-1)*(lambda-t)^(3-1)+C24/(4-1)*(lambda-t)^(
4-1)+C25/(5-1)*(lambda-t)^(5-1)+C26/(6-1)*(lambda-t)^(6-1)+C27/(7-1)*(
lambda-t)^(7-1)))\n" }{MPLTEXT 1 0 196 "*(Ltt20/h*ln(lambda-t)+LC20+LC
22/(2-1)*(lambda-t)^(2-1)+LC23/(3-1)*(lambda-t)^(3-1)+LC24/(4-1)*(lamb
da-t)^(4-1)+LC25/(5-1)*(lambda-t)^(5-1)+LC26/(6-1)*(lambda-t)^(6-1)+LC
27/(7-1)*(lambda-t)^(7-1)\n" }{MPLTEXT 1 0 164 "+h*(-tt20/h/(lambda-t)
-C22*(lambda-t)^(2-1-1)-C23*(lambda-t)^(3-1-1)-C24*(lambda-t)^(4-1-1)-
C25*(lambda-t)^(5-1-1)+C26*(lambda-t)^(6-1-1)+C27*(lambda-t)^(7-1-1)))
;\n" }{MPLTEXT 1 0 22 "psi1T:=exp(logPsi1T);\n" }{MPLTEXT 1 0 22 "psi2
T:=exp(logPsi2T);\n" }{MPLTEXT 1 0 35 "dpsi1dlambdaT:=diff(psi1T,lambd
a):\n" }{MPLTEXT 1 0 35 "dpsi2dlambdaT:=diff(psi2T,lambda):\n" }
{MPLTEXT 1 0 39 "d2psi1dlambda2T:=diff(psi1T,lambda$2):\n" }{MPLTEXT 
1 0 39 "d2psi2dlambda2T:=diff(psi2T,lambda$2):\n" }{MPLTEXT 1 0 24 "VT
1:=tt10*ln(lambda-t);\n" }{MPLTEXT 1 0 24 "VT2:=tt20*ln(lambda-t);\n" 
}{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 69 "WronskianL
ambdaT:=h*factor(psi1T*dpsi2dlambdaT-psi2T*dpsi1dlambdaT):\n" }
{MPLTEXT 1 0 112 "WronskianLambdabisT:=h*simplify(factor( (diff(logPsi
2T,lambda)-diff(logPsi1T,lambda))*exp(logPsi1T+logPsi2T))):\n" }
{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 96 "WronskianTildeLambdaT:=h^3*facto
r(dpsi2dlambdaT*d2psi1dlambda2T-dpsi1dlambdaT*d2psi2dlambda2T):\n" }}
{PARA 11 "" 1 "" {XPPMATH 20 ">I)logPsi1TG6\",2*(I%tt10GF$\"\"\"I\"hGF
$!\"\"-I#lnG6$%*protectedGI(_syslibGF$6#,&I'lambdaGF$F(I\"tGF$F*F(F(I$
C10GF$F(*&I$C12GF$F(F1F(F(*&I$C13GF$F(F1\"\"##F(F9*&I$C14GF$F(F1\"\"$#
F(F=*&I$C15GF$F(F1\"\"%#F(FA*&I$C16GF$F(F1\"\"&#F(FE*&I$C17GF$F(F1\"\"
'#F(FI" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I)logPsi2TG6\",2*(I%tt20GF$\"
\"\"I\"hGF$!\"\"-I#lnG6$%*protectedGI(_syslibGF$6#,&I'lambdaGF$F(I\"tG
F$F*F(F(I$C20GF$F(*&I$C22GF$F(F1F(F(*&I$C23GF$F(F1\"\"##F(F9*&I$C24GF$
F(F1\"\"$#F(F=*&I$C25GF$F(F1\"\"%#F(FA*&I$C26GF$F(F1\"\"&#F(FE*&I$C27G
F$F(F1\"\"'#F(FI" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I*Llogpsi1TG6\",4*(
I&Ltt10GF$\"\"\"I\"hGF$!\"\"-I#lnG6$%*protectedGI(_syslibGF$6#,&I'lamb
daGF$F(I\"tGF$F*F(F(I%LC10GF$F(*&I%LC12GF$F(F1F(F(*&I%LC13GF$F(F1\"\"#
#F(F9*&I%LC14GF$F(F1\"\"$#F(F=*&I%LC15GF$F(F1\"\"%#F(FA*&I%LC16GF$F(F1
\"\"&#F(FE*&I%LC17GF$F(F1\"\"'#F(FI*&F)F(,0*(I%tt10GF$F(F)F*F1F*F*I$C1
2GF$F**&I$C13GF$F(F1F(F**&I$C14GF$F(F1F9F**&I$C15GF$F(F1F=F**&I$C16GF$
F(F1FAF(*&I$C17GF$F(F1FEF(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I*Llo
gpsi2TG6\",4*(I&Ltt20GF$\"\"\"I\"hGF$!\"\"-I#lnG6$%*protectedGI(_sysli
bGF$6#,&I'lambdaGF$F(I\"tGF$F*F(F(I%LC20GF$F(*&I%LC22GF$F(F1F(F(*&I%LC
23GF$F(F1\"\"##F(F9*&I%LC24GF$F(F1\"\"$#F(F=*&I%LC25GF$F(F1\"\"%#F(FA*
&I%LC26GF$F(F1\"\"&#F(FE*&I%LC27GF$F(F1\"\"'#F(FI*&F)F(,0*(I%tt20GF$F(
F)F*F1F*F*I$C22GF$F**&I$C23GF$F(F1F(F**&I$C24GF$F(F1F9F**&I$C25GF$F(F1
F=F**&I$C26GF$F(F1FAF(*&I$C27GF$F(F1FEF(F(F(" }}{PARA 11 "" 1 "" 
{XPPMATH 20 ">I'Lpsi1TG6\"*&-I$expG6$%*protectedGI(_syslibGF$6#,2*(I%t
t10GF$\"\"\"I\"hGF$!\"\"-I#lnGF(6#,&I'lambdaGF$F/I\"tGF$F1F/F/I$C10GF$
F/*&I$C12GF$F/F5F/F/*&I$C13GF$F/F5\"\"##F/F=*&I$C14GF$F/F5\"\"$#F/FA*&
I$C15GF$F/F5\"\"%#F/FE*&I$C16GF$F/F5\"\"&#F/FI*&I$C17GF$F/F5\"\"'#F/FM
F/,4*(I&Ltt10GF$F/F0F1F2F/F/I%LC10GF$F/*&I%LC12GF$F/F5F/F/*&I%LC13GF$F
/F5F=F>*&I%LC14GF$F/F5FAFB*&I%LC15GF$F/F5FEFF*&I%LC16GF$F/F5FIFJ*&I%LC
17GF$F/F5FMFN*&F0F/,0*(F.F/F0F1F5F1F1F:F1*&F<F/F5F/F1*&F@F/F5F=F1*&FDF
/F5FAF1*&FHF/F5FEF/*&FLF/F5FIF/F/F/F/" }}{PARA 11 "" 1 "" {XPPMATH 20 
">I'Lpsi2TG6\"*&-I$expG6$%*protectedGI(_syslibGF$6#,2*(I%tt20GF$\"\"\"
I\"hGF$!\"\"-I#lnGF(6#,&I'lambdaGF$F/I\"tGF$F1F/F/I$C20GF$F/*&I$C22GF$
F/F5F/F/*&I$C23GF$F/F5\"\"##F/F=*&I$C24GF$F/F5\"\"$#F/FA*&I$C25GF$F/F5
\"\"%#F/FE*&I$C26GF$F/F5\"\"&#F/FI*&I$C27GF$F/F5\"\"'#F/FMF/,4*(I&Ltt2
0GF$F/F0F1F2F/F/I%LC20GF$F/*&I%LC22GF$F/F5F/F/*&I%LC23GF$F/F5F=F>*&I%L
C24GF$F/F5FAFB*&I%LC25GF$F/F5FEFF*&I%LC26GF$F/F5FIFJ*&I%LC27GF$F/F5FMF
N*&F0F/,0*(F.F/F0F1F5F1F1F:F1*&F<F/F5F/F1*&F@F/F5F=F1*&FDF/F5FAF1*&FHF
/F5FEF/*&FLF/F5FIF/F/F/F/" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I&psi1TG6
\"-I$expG6$%*protectedGI(_syslibGF$6#,2*(I%tt10GF$\"\"\"I\"hGF$!\"\"-I
#lnGF'6#,&I'lambdaGF$F.I\"tGF$F0F.F.I$C10GF$F.*&I$C12GF$F.F4F.F.*&I$C1
3GF$F.F4\"\"##F.F<*&I$C14GF$F.F4\"\"$#F.F@*&I$C15GF$F.F4\"\"%#F.FD*&I$
C16GF$F.F4\"\"&#F.FH*&I$C17GF$F.F4\"\"'#F.FL" }}{PARA 11 "" 1 "" 
{XPPMATH 20 ">I&psi2TG6\"-I$expG6$%*protectedGI(_syslibGF$6#,2*(I%tt20
GF$\"\"\"I\"hGF$!\"\"-I#lnGF'6#,&I'lambdaGF$F.I\"tGF$F0F.F.I$C20GF$F.*
&I$C22GF$F.F4F.F.*&I$C23GF$F.F4\"\"##F.F<*&I$C24GF$F.F4\"\"$#F.F@*&I$C
25GF$F.F4\"\"%#F.FD*&I$C26GF$F.F4\"\"&#F.FH*&I$C27GF$F.F4\"\"'#F.FL" }
}{PARA 11 "" 1 "" {XPPMATH 20 ">I$VT1G6\"*&I%tt10GF$\"\"\"-I#lnG6$%*pr
otectedGI(_syslibGF$6#,&I'lambdaGF$F'I\"tGF$!\"\"F'" }}{PARA 11 "" 1 "
" {XPPMATH 20 ">I$VT2G6\"*&I%tt20GF$\"\"\"-I#lnG6$%*protectedGI(_sysli
bGF$6#,&I'lambdaGF$F'I\"tGF$!\"\"F'" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 80 "L22T:=factor(h*simplify(diff(WronskianLambdabisT,lamb
da)/WronskianLambdabisT)):\n" }{MPLTEXT 1 0 68 "L22TOrderlambdaMinus3:
=factor(residue(L22T*(lambda-t)^2,lambda=t));\n" }{MPLTEXT 1 0 68 "L22
TOrderlambdaMinus2:=factor(residue(L22T*(lambda-t)^1,lambda=t));\n" }
{MPLTEXT 1 0 68 "L22TOrderlambdaMinus1:=factor(residue(L22T*(lambda-t)
^0,lambda=t));\n" }{MPLTEXT 1 0 66 "L22TOrderlambda0:=factor(residue(L
22T*(lambda-t)^(-1),lambda=t));\n" }{MPLTEXT 1 0 66 "L22TOrderlambda1:
=factor(residue(L22T*(lambda-t)^(-2),lambda=t)):\n" }{MPLTEXT 1 0 65 "
L22TOrderlambda2:=factor(residue(L22T*(lambda-t)^(-3),lambda=t)):" }}
{PARA 11 "" 1 "" {XPPMATH 20 ">I6L22TOrderlambdaMinus3G6\"\"\"!" }}
{PARA 11 "" 1 "" {XPPMATH 20 ">I6L22TOrderlambdaMinus2G6\"\"\"!" }}
{PARA 11 "" 1 "" {XPPMATH 20 ">I6L22TOrderlambdaMinus1G6\",(I%tt10GF$
\"\"\"I\"hGF$!\"\"I%tt20GF$F'" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I1L22T
Orderlambda0G6\"*(I\"hGF$\"\"\",.*&I$C12GF$F'F&F'F'*&F*F'I%tt10GF$F'F'
*&F*F'I%tt20GF$F'!\"\"*&I$C22GF$F'F&F'F/*&F1F'F,F'F'*&F1F'F.F'F/F',&F,
F'F.F/F/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 230 75 "We get that L_\{2,2
\} behaves at lambda=t like (tt10+tt20-h)/(lambda-t) +O(1)" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "L21T:=factor(simplify(WronskianTild
eLambdaT/WronskianLambdabisT)):\n" }{MPLTEXT 1 0 68 "L21TOrderlambdaMi
nus5:=factor(residue(L21T*(lambda-t)^4,lambda=t));\n" }{MPLTEXT 1 0 
68 "L21TOrderlambdaMinus4:=factor(residue(L21T*(lambda-t)^3,lambda=t))
;\n" }{MPLTEXT 1 0 68 "L21TOrderlambdaMinus3:=factor(residue(L21T*(lam
bda-t)^2,lambda=t));\n" }{MPLTEXT 1 0 68 "L21TOrderlambdaMinus2:=facto
r(residue(L21T*(lambda-t)^1,lambda=t));\n" }{MPLTEXT 1 0 68 "L21TOrder
lambdaMinus1:=factor(residue(L21T*(lambda-t)^0,lambda=t));\n" }
{MPLTEXT 1 0 66 "L21TOrderlambda0:=factor(residue(L21T*(lambda-t)^(-1)
,lambda=t));\n" }{MPLTEXT 1 0 66 "L21TOrderlambda1:=factor(residue(L21
T*(lambda-t)^(-2),lambda=t)):\n" }{MPLTEXT 1 0 65 "L21TOrderlambda2:=f
actor(residue(L21T*(lambda-t)^(-3),lambda=t)):" }}{PARA 11 "" 1 "" 
{XPPMATH 20 ">I6L21TOrderlambdaMinus5G6\"\"\"!" }}{PARA 11 "" 1 "" 
{XPPMATH 20 ">I6L21TOrderlambdaMinus4G6\"\"\"!" }}{PARA 11 "" 1 "" 
{XPPMATH 20 ">I6L21TOrderlambdaMinus3G6\"\"\"!" }}{PARA 11 "" 1 "" 
{XPPMATH 20 ">I6L21TOrderlambdaMinus2G6\",$*&I%tt10GF$\"\"\"I%tt20GF$F
(!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I6L21TOrderlambdaMinus1G6\",$
*(I\"hGF$\"\"\",.*(I$C12GF$F(F'F(I%tt20GF$F(F(*(F+F(I%tt10GF$F(F,F(F(*
&F+F(F,\"\"#!\"\"*(I$C22GF$F(F'F(F.F(F1*&F3F(F.F0F(*(F3F(F.F(F,F(F1F(,
&F.F(F,F1F1F1" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I1L21TOrderlambda0G6\"
*(I\"hGF$\"\"\",D*(I$C12GF$\"\"#F&F+I%tt20GF$F'F'**F*F'I$C22GF$F'F&F+I
%tt10GF$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$C13GF$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,\"\"$F0*(I$
C23GF$F'F&F'F/F+F+**F?F'F&F'F/F'F,F'F8*&F?F'F/F=F0*(F?F'F/F+F,F'F+*(F?
F'F/F'F,F+F0F',&F/F'F,F0F8" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 230 44 "We
 get that L_\{2,1\} behaves at lambda=t like" }{TEXT 230 41 " -tt10*tt
20/(lambda-t)^2+ O(1/(lambda-t))" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT
 1 0 99 "L22Form:=(t110+t120-h)/(lambda-1)+ (t010+t020-h)/lambda+ (tt1
0+tt20-h)/(lambda-t)   +h/(lambda-q);\n" }{MPLTEXT 1 0 55 "TermInLambd
aMinus1:=-residue(L22Form,lambda=infinity);\n" }{MPLTEXT 1 0 76 "simpl
ify(TermInLambdaMinus1- (-(tinfty10+tinfty20+2*h))-CoherenceEquation1)
;" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I(L22FormG6\",**&,(I%t110GF$\"\"\"
I\"hGF$!\"\"I%t120GF$F)F),&I'lambdaGF$F)F+F)F+F)*&,(I%t010GF$F)F*F+I%t
020GF$F)F)F.F+F)*&,(I%tt10GF$F)F*F+I%tt20GF$F)F),&F.F)I\"tGF$F+F+F)*&F
*F),&F.F)I\"qGF$F+F+F)" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I3TermInLambd
aMinus1G6\",0I%t110GF$\"\"\"I\"hGF$!\"#I%t120GF$F'I%t010GF$F'I%t020GF$
F'I%tt10GF$F'I%tt20GF$F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}
{EXCHG {PARA 209 "" 0 "" {TEXT 231 31 "Formulas for L_\{2,2\} et L_\{2
,1\}" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 230 36 "We have L_\{2,2\} behave
s at lambda=1 " }{TEXT 230 4 "like" }{TEXT 230 32 " (t110+t120-h)/(lam
bda-1) +O(1)\n" }{TEXT 230 23 "               L_\{2,2\} " }{TEXT 230 
10 "behaves at" }{TEXT 230 10 " lambda=0 " }{TEXT 230 4 "like" }{TEXT 
230 28 " (t010+t020-h)/lambda +O(1)\n" }{TEXT 230 23 "               L
_\{2,2\} " }{TEXT 230 10 "behaves at" }{TEXT 230 13 " en lambda=t " }
{TEXT 230 4 "like" }{TEXT 230 32 " (tt10+tt20-h)/(lambda-t) +O(1)\n" }
{TEXT 230 23 "               L_\{2,2\} " }{TEXT 230 10 "behaves at" }
{TEXT 230 74 " lambda=infinity like  -(tinfty10+tinfty20+2\\hbar)/lamb
da+h*O(1/lambda^2)\n" }{TEXT 230 5 "Thus," }{TEXT 214 1 " " }}{PARA 0 
"" 0 "" {TEXT 215 97 "L_\{2,2\}= (t110+t120-h)/(lambda-1)+ (t010+t020-
h)/lambda+ (tt10+tt20-h)/(lambda-t)   +h/(lambda-q)" }{TEXT 230 1 "\n"
 }{TEXT 230 63 "with the additional condition that (t110+t120-h)+(t010
+t020-h)+" }{TEXT 216 10 "(tt10+tt20" }{TEXT 217 1 "-" }{TEXT 218 2 "h
)" }{TEXT 230 23 "+h=t110+t120+t010+t020+" }{TEXT 219 9 "tt10+tt20" }
{TEXT 230 147 "-2h= -(tinfty10+tinfty20 +2h) which is equivalent to th
e vanishing of the sum of monodromies:  t110+t120+t010+t020+tt10+tt20+
tinfty10+tinfty20=0  \n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 230 16 "We ha
ve L_\{2,1\} " }{TEXT 230 11 "behaves at " }{TEXT 230 10 " lambda=1 " 
}{TEXT 230 4 "like" }{TEXT 230 42 " -t110*t120/(lambda-1)^2+ O(1/(lamb
da-1))\n" }{TEXT 230 23 "               L_\{2,1\} " }{TEXT 230 11 "beh
aves at " }{TEXT 230 10 " lambda=0 " }{TEXT 230 4 "like" }{TEXT 230 
34 " -t010*t020/lambda^2+ O(1/lambda)\n" }{TEXT 230 23 "              \+
 L_\{2,1\} " }{TEXT 230 11 "behaves at " }{TEXT 230 10 " lambda=t " }
{TEXT 230 4 "like" }{TEXT 230 42 " -tt10*tt20/(lambda-t)^2+ O(1/(lambd
a-t))\n" }{TEXT 230 23 "               L_\{2,1\} " }{TEXT 230 10 "beha
ves at" }{TEXT 230 22 " lambda=infinity like " }{TEXT 230 58 " (-tinft
y10tinfty20 -h*tinfty10)/lambda^2 +O(lambda^\{-1\})\n" }{TEXT 230 6 "T
hus, " }}{PARA 0 "" 0 "" {TEXT 220 7 "L_\{2,1\}" }{TEXT 221 2 "=-" }
{TEXT 222 129 "t110*t120/(lambda-1)^2 -a_1/(lambda-1)   -t010*t020/lam
bda^2 -a_0/lambda -tt10*tt20/(lambda-t)^2 -a_t/(lambda-t) - p*h/(lambd
a-q)" }{TEXT 223 1 "\n" }{TEXT 230 21 " with the conditions " }{TEXT 
224 18 "a_0+a_1+a_t+ h*p=0" }{TEXT 230 5 " and " }{TEXT 225 77 "a_1+t*
a_t+t010*t020+t110*t120+tt10*tt20-tinfty10*tinfty20+h*p*q -h*tinfty10=
0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 141 "L21Form:=-t110*t120/(
lambda-1)^2 -a_1/(lambda-1)   -t010*t020/lambda^2 -a_0/lambda -tt10*tt
20/(lambda-t)^2 -a_t/(lambda-t) - p*h/(lambda-q);\n" }{MPLTEXT 1 0 69 
"L21FormOrderLambdaMinus1:=factor(-residue(L21Form,lambda=infinity));
\n" }{MPLTEXT 1 0 76 "L21FormOrderLambdaMinus2:=factor(-residue(L21For
m*lambda,lambda=infinity));\n" }{MPLTEXT 1 0 75 "CoherenceEquation4:= \+
-(L21FormOrderLambdaMinus2-L21InftyOrderlambdaMinus2);" }}{PARA 11 "" 
1 "" {XPPMATH 20 ">I(L21FormG6\",0*(I%t110GF$\"\"\"I%t120GF$F(,&I'lamb
daGF$F(!\"\"F(!\"#F,*&I$a_1GF$F(F*F,F,*(I%t010GF$F(I%t020GF$F(F+F-F,*&
I$a_0GF$F(F+F,F,*(I%tt10GF$F(I%tt20GF$F(,&F+F(I\"tGF$F,F-F,*&I$a_tGF$F
(F8F,F,*(I\"pGF$F(I\"hGF$F(,&F+F(I\"qGF$F,F,F," }}{PARA 11 "" 1 "" 
{XPPMATH 20 ">I9L21FormOrderLambdaMinus1G6\",**&I\"hGF$\"\"\"I\"pGF$F(
!\"\"I$a_0GF$F*I$a_1GF$F*I$a_tGF$F*" }}{PARA 11 "" 1 "" {XPPMATH 20 ">
I9L21FormOrderLambdaMinus2G6\",.*(I\"hGF$\"\"\"I\"pGF$F(I\"qGF$F(!\"\"
*&I$a_tGF$F(I\"tGF$F(F+*&I%t010GF$F(I%t020GF$F(F+*&I%t110GF$F(I%t120GF
$F(F+*&I%tt10GF$F(I%tt20GF$F(F+I$a_1GF$F+" }}{PARA 11 "" 1 "" 
{XPPMATH 20 ">I3CoherenceEquation4G6\",0*(I\"pGF$\"\"\"I\"hGF$F(I\"qGF
$F(F(*&I$a_tGF$F(I\"tGF$F(F(*&I%t010GF$F(I%t020GF$F(F(*&I%t120GF$F(I%t
110GF$F(F(*&I%tt10GF$F(I%tt20GF$F(F(I$a_1GF$F(*&,&F)F(I)tinfty20GF$F(F
(I)tinfty10GF$F(!\"\"" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 228 85 "Compu
tation of the auxiliary Lax matrix A using the asymptotics of the wave
 functions" }{TEXT 228 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
68 "WronskianLInfty:=factor(psi1Infty*Lpsi2Infty-psi2Infty*Lpsi1Infty)
:\n" }{MPLTEXT 1 0 63 "WronskianLZero:=factor(psi1Zero*Lpsi2Zero-psi2Z
ero*Lpsi1Zero):\n" }{MPLTEXT 1 0 58 "WronskianLOne:=factor(psi1One*Lps
i2One-psi2One*Lpsi1One):\n" }{MPLTEXT 1 0 48 "WronskianLT:=factor(psi1
T*Lpsi2T-psi2T*Lpsi1T):\n" }{MPLTEXT 1 0 66 "A12Infty:=factor(simplify
(WronskianLInfty/WronskianLambdaInfty)):\n" }{MPLTEXT 1 0 63 "A12Zero:
=factor(simplify(WronskianLZero/WronskianLambdaZero)):\n" }{MPLTEXT 1 
0 60 "A12One:=factor(simplify(WronskianLOne/WronskianLambdaOne)):\n" }
{MPLTEXT 1 0 54 "A12T:=factor(simplify(WronskianLT/WronskianLambdaT)):
\n" }{MPLTEXT 1 0 48 "Y1Infty:=h*factor(dpsi1dlambdaInfty/psi1Infty):
\n" }{MPLTEXT 1 0 48 "Y2Infty:=h*factor(dpsi2dlambdaInfty/psi2Infty):
\n" }{MPLTEXT 1 0 45 "Y1Zero:=h*factor(dpsi1dlambdaZero/psi1Zero):\n" 
}{MPLTEXT 1 0 45 "Y2Zero:=h*factor(dpsi2dlambdaZero/psi2Zero):\n" }
{MPLTEXT 1 0 42 "Y1One:=h*factor(dpsi1dlambdaOne/psi1One):\n" }
{MPLTEXT 1 0 42 "Y2One:=h*factor(dpsi2dlambdaOne/psi2One):\n" }
{MPLTEXT 1 0 36 "Y1T:=h*factor(dpsi1dlambdaT/psi1T):\n" }{MPLTEXT 1 0 
36 "Y2T:=h*factor(dpsi2dlambdaT/psi2T):\n" }{MPLTEXT 1 0 39 "Z1Infty:=
factor(Lpsi1Infty/psi1Infty):\n" }{MPLTEXT 1 0 39 "Z2Infty:=factor(Lps
i2Infty/psi2Infty):\n" }{MPLTEXT 1 0 36 "Z1Zero:=factor(Lpsi1Zero/psi1
Zero):\n" }{MPLTEXT 1 0 36 "Z2Zero:=factor(Lpsi2Zero/psi2Zero):\n" }
{MPLTEXT 1 0 33 "Z1One:=factor(Lpsi1One/psi1One):\n" }{MPLTEXT 1 0 33 
"Z2One:=factor(Lpsi2One/psi2One):\n" }{MPLTEXT 1 0 27 "Z1T:=factor(Lps
i1T/psi1T):\n" }{MPLTEXT 1 0 27 "Z2T:=factor(Lpsi2T/psi2T):\n" }
{MPLTEXT 1 0 68 "A12bisInfty:=factor(simplify((Z2Infty-Z1Infty)/(Y2Inf
ty-Y1Infty))):\n" }{MPLTEXT 1 0 63 "A12bisZero:=factor(simplify((Z2Zer
o-Z1Zero)/(Y2Zero-Y1Zero))):\n" }{MPLTEXT 1 0 58 "A12bisOne:=factor(si
mplify((Z2One-Z1One)/(Y2One-Y1One))):\n" }{MPLTEXT 1 0 48 "A12bisT:=fa
ctor(simplify((Z2T-Z1T)/(Y2T-Y1T))):\n" }{MPLTEXT 1 0 83 "A11Infty:=fa
ctor(simplify( (Y2Infty*Z1Infty-Y1Infty*Z2Infty)/(Y2Infty-Y1Infty) )):
\n" }{MPLTEXT 1 0 76 "A11Zero:=factor(simplify( (Y2Zero*Z1Zero-Y1Zero*
Z2Zero)/(Y2Zero-Y1Zero) )):\n" }{MPLTEXT 1 0 69 "A11One:=factor(simpli
fy( (Y2One*Z1One-Y1One*Z2One)/(Y2One-Y1One) )):\n" }{MPLTEXT 1 0 55 "A
11T:=factor(simplify( (Y2T*Z1T-Y1T*Z2T)/(Y2T-Y1T) )):\n" }{MPLTEXT 1 
0 40 "factor(simplify(A12bisInfty-A12Infty));\n" }{MPLTEXT 1 0 38 "fac
tor(simplify(A12bisZero-A12Zero));\n" }{MPLTEXT 1 0 37 "factor(simplif
y(A12bisZero-A12Zero));" }{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 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "Lt020:=Lt010:\n" }{MPLTEXT 
1 0 14 "Lt120:=Lt110:\n" }{MPLTEXT 1 0 14 "Ltt20:=Ltt10:\n" }{MPLTEXT 
1 0 10 "Lt010:=0:\n" }{MPLTEXT 1 0 10 "Lt110:=0:\n" }{MPLTEXT 1 0 10 "
Ltt10:=0:\n" }{MPLTEXT 1 0 22 "Ltinfty20:=Ltinfty10:\n" }{MPLTEXT 1 0 
14 "Ltinfty10:=0:\n" }}}{EXCHG {PARA 212 "" 0 "" {TEXT 206 16 "Study o
f A_\{1,2\}" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "A12InftyLamb
da3:=factor(-residue(A12Infty/lambda^4,lambda=infinity));\n" }{MPLTEXT
 1 0 70 "A12InftyLambda2:=factor(-residue(A12Infty/lambda^3,lambda=inf
inity));\n" }{MPLTEXT 1 0 70 "A12InftyLambda1:=factor(-residue(A12Inft
y/lambda^2,lambda=infinity));\n" }{MPLTEXT 1 0 70 "A12InftyLambda0:=fa
ctor(-residue(A12Infty/lambda^1,lambda=infinity));\n" }{MPLTEXT 1 0 
75 "A12InftyLambdaMinus1:=factor(-residue(A12Infty/lambda^0,lambda=inf
inity)):\n" }}{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$F*F,F*F/*&F:F*F6F*F**&F:F*F8F*F/F*,(F,F*F6F/
F8F*!\"#F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "A12ZeroLambda
Minus3:=factor(residue(A12Zero*lambda^2,lambda=0));\n" }{MPLTEXT 1 0 
65 "A12ZeroLambdaMinus2:=factor(residue(A12Zero*lambda^1,lambda=0));\n
" }{MPLTEXT 1 0 65 "A12ZeroLambdaMinus1:=factor(residue(A12Zero*lambda
^0,lambda=0));\n" }{MPLTEXT 1 0 63 "A12ZeroLambda0:=factor(residue(A12
Zero*lambda^(-1),lambda=0));\n" }{MPLTEXT 1 0 63 "A12ZeroLambda1:=fact
or(residue(A12Zero*lambda^(-2),lambda=0));\n" }{MPLTEXT 1 0 1 "\n" }}
{PARA 11 "" 1 "" {XPPMATH 20 ">I4A12ZeroLambdaMinus3G6\"\"\"!" }}
{PARA 11 "" 1 "" {XPPMATH 20 ">I4A12ZeroLambdaMinus2G6\"\"\"!" }}
{PARA 11 "" 1 "" {XPPMATH 20 ">I4A12ZeroLambdaMinus1G6\"\"\"!" }}
{PARA 11 "" 1 "" {XPPMATH 20 ">I/A12ZeroLambda0G6\"\"\"!" }}{PARA 11 "
" 1 "" {XPPMATH 20 ">I/A12ZeroLambda1G6\"*&,&I%LB20GF$!\"\"I%LB10GF$\"
\"\"F*,&I%t010GF$F*I%t020GF$F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT
 1 0 67 "A12OneLambdaMinus3:=factor(residue(A12One*(lambda-1)^2,lambda
=1));\n" }{MPLTEXT 1 0 67 "A12OneLambdaMinus2:=factor(residue(A12One*(
lambda-1)^1,lambda=1));\n" }{MPLTEXT 1 0 67 "A12OneLambdaMinus1:=facto
r(residue(A12One*(lambda-1)^0,lambda=1));\n" }{MPLTEXT 1 0 65 "A12OneL
ambda0:=factor(residue(A12One*(lambda-1)^(-1),lambda=1));\n" }{MPLTEXT
 1 0 64 "A12OneLambda1:=factor(residue(A12One*(lambda-1)^(-2),lambda=1
));" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I3A12OneLambdaMinus3G6\"\"\"!" }
}{PARA 11 "" 1 "" {XPPMATH 20 ">I3A12OneLambdaMinus2G6\"\"\"!" }}
{PARA 11 "" 1 "" {XPPMATH 20 ">I3A12OneLambdaMinus1G6\"\"\"!" }}{PARA 
11 "" 1 "" {XPPMATH 20 ">I.A12OneLambda0G6\"\"\"!" }}{PARA 11 "" 1 "" 
{XPPMATH 20 ">I.A12OneLambda1G6\"*&,&I%LC10GF$\"\"\"I%LC20GF$!\"\"F(,&
I%t110GF$F(I%t120GF$F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 
"A12TLambdaMinus3:=factor(residue(A12T*(lambda-t)^2,lambda=t));\n" }
{MPLTEXT 1 0 63 "A12TLambdaMinus2:=factor(residue(A12T*(lambda-t)^1,la
mbda=t));\n" }{MPLTEXT 1 0 63 "A12TLambdaMinus1:=factor(residue(A12T*(
lambda-t)^0,lambda=t));\n" }{MPLTEXT 1 0 61 "A12TLambda0:=factor(resid
ue(A12T*(lambda-t)^(-1),lambda=t));\n" }{MPLTEXT 1 0 60 "A12TLambda1:=
factor(residue(A12T*(lambda-t)^(-2),lambda=t));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 ">I1A12TLambdaMinus3G6\"\"\"!" }}{PARA 11 "" 1 "" 
{XPPMATH 20 ">I1A12TLambdaMinus2G6\"\"\"!" }}{PARA 11 "" 1 "" 
{XPPMATH 20 ">I1A12TLambdaMinus1G6\"\"\"!" }}{PARA 11 "" 1 "" 
{XPPMATH 20 ">I,A12TLambda0G6\"!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "
>I,A12TLambda1G6\"*&,&I%LC10GF$\"\"\"I%LC20GF$!\"\"F(,&I%tt10GF$F(I%tt
20GF$F*F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 230 20 "We thus obtain that
 " }{TEXT 226 44 "A_\{1,2\}= a*(lambda-t)+b-1+(q-t)*b/(lambda-q)" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "A12Form:=a*(lambda-t)+b-1+ (
q-t)*b/(lambda-q);\n" }{MPLTEXT 1 0 52 "factor( residue(A12Form/(lambd
a-t),lambda=t)-(-1));\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 55 "A12For
m:=(a*(lambda-t)^2+b*(lambda-t)+q-t)/(lambda-q);\n" }{MPLTEXT 1 0 51 "
factor( residue(A12Form/(lambda-t),lambda=t)-(-1));" }}{PARA 11 "" 1 "
" {XPPMATH 20 ">I(A12FormG6\",**&I\"aGF$\"\"\",&I'lambdaGF$F(I\"tGF$!
\"\"F(F(I\"bGF$F(F,F(*(,&I\"qGF$F(F+F,F(F-F(,&F*F(F0F,F,F(" }}{PARA 11
 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I(A12For
mG6\"*&,**&I\"aGF$\"\"\",&I'lambdaGF$F)I\"tGF$!\"\"\"\"#F)*&I\"bGF$F)F
*F)F)I\"qGF$F)F,F-F),&F+F)F1F-F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"
!" }}}{EXCHG {PARA 213 "" 0 "" {TEXT 206 16 "Study of A_\{1,1\}" }}}
{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 67 "A11O
neLambdaMinus3:=factor(residue(A11One*(lambda-1)^2,lambda=1));\n" }
{MPLTEXT 1 0 67 "A11OneLambdaMinus2:=factor(residue(A11One*(lambda-1)^
1,lambda=1));\n" }{MPLTEXT 1 0 67 "A11OneLambdaMinus1:=factor(residue(
A11One*(lambda-1)^0,lambda=1));\n" }{MPLTEXT 1 0 65 "A11OneLambda0:=fa
ctor(residue(A11One*(lambda-1)^(-1),lambda=1));\n" }{MPLTEXT 1 0 65 "A
11OneLambda1:=factor(residue(A11One*(lambda-1)^(-2),lambda=1)):\n" }
{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 63 "A11TLambdaMinus3:=factor(residue
(A11T*(lambda-t)^2,lambda=t));\n" }{MPLTEXT 1 0 63 "A11TLambdaMinus2:=
factor(residue(A11T*(lambda-t)^1,lambda=t));\n" }{MPLTEXT 1 0 63 "A11T
LambdaMinus1:=factor(residue(A11T*(lambda-t)^0,lambda=t));\n" }
{MPLTEXT 1 0 61 "A11TLambda0:=factor(residue(A11T*(lambda-t)^(-1),lamb
da=t));\n" }{MPLTEXT 1 0 61 "A11TLambda1:=factor(residue(A11T*(lambda-
t)^(-2),lambda=t)):\n" }{MPLTEXT 1 0 1 "\n" }}{PARA 11 "" 1 "" 
{XPPMATH 20 ">I0A11InftyLambda3G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 
20 ">I0A11InftyLambda2G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I0A1
1InftyLambda1G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I0A11InftyLam
bda0G6\"*&,(*&I%LA10GF$\"\"\"I\"hGF$F)F)*&F(F)I)tinfty20GF$F)F)*&I%LA2
0GF$F)I)tinfty10GF$F)!\"\"F),(F*F)F/F0F,F)F0" }}{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$F*!\"\"F*,&F.F*F+F/F/F/" }}{PARA 11 "" 1 "" 
{XPPMATH 20 ">I3A11OneLambdaMinus3G6\"\"\"!" }}{PARA 11 "" 1 "" 
{XPPMATH 20 ">I3A11OneLambdaMinus2G6\"\"\"!" }}{PARA 11 "" 1 "" 
{XPPMATH 20 ">I3A11OneLambdaMinus1G6\"\"\"!" }}{PARA 11 "" 1 "" 
{XPPMATH 20 ">I.A11OneLambda0G6\",$*&,&*&I%LC10GF$\"\"\"I%t120GF$F*F**
&I%LC20GF$F*I%t110GF$F*!\"\"F*,&F.F*F+F/F/F/" }}{PARA 11 "" 1 "" 
{XPPMATH 20 ">I1A11TLambdaMinus3G6\"\"\"!" }}{PARA 11 "" 1 "" 
{XPPMATH 20 ">I1A11TLambdaMinus2G6\"\"\"!" }}{PARA 11 "" 1 "" 
{XPPMATH 20 ">I1A11TLambdaMinus1G6\"\"\"!" }}{PARA 11 "" 1 "" 
{XPPMATH 20 ">I,A11TLambda0G6\",$*&,&*&I%LC10GF$\"\"\"I%tt20GF$F*F**&I
%LC20GF$F*I%tt10GF$F*!\"\"F*,&F.F*F+F/F/F/" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT 230 20 "We thus obtain that " }{TEXT 227 23 "A_\{1,1\}=C+mu/(lam
bda-q)" }}}{EXCHG {PARA 214 "" 0 "" {TEXT 206 64 "In the end the first
 line of the auxiliary matrix A is given by\n" }{TEXT 206 25 "A_\{1,1
\}=C+ mu/(lambda-q)\n" }{TEXT 206 44 "A_\{1,2\}= a*(lambda-t)+b-1+(q-t
)*b/(lambda-q)" }}}}{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 }