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2 2 2 0 0 0 1 }{CSTYLE "" -1 222 "Times " 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 223 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 224 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{PSTYLE "" -1 209 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "" -1 210 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "" -1 211 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 } {PSTYLE "" -1 212 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT 225 240 "In this Maple file, we \+ compute the evolution equations for the Painlev\351 6 equations using \+ the compatibility equation of the Lax system. We also obtain the expre ssion of the Lax matrices in the geometric gauge without apparent sing ularities." }{TEXT 225 1 "\n" }{TEXT 225 1 "T" }{TEXT 222 79 "he auxil iary operator is \\hbar*\\partial_t where t is the position of one pol e.\n" }{TEXT 223 104 "The symplectic reduction has already been done: \+ the simple poles are located at \\lambda= 0,1,infinity, t" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 226 56 "Lax matrices in the oper gauge from prev ious Maple files" }{TEXT 226 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT 215 102 "Summary of previous files: We have the expression for some coeff icients of the Lax matrix L and of A." }{TEXT 215 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "restart:\n" }{MPLTEXT 1 0 21 "with(L inearAlgebra):\n" }{MPLTEXT 1 0 19 "P011 := t010+t020;\n" }{MPLTEXT 1 0 19 "P111 := t110+t120;\n" }{MPLTEXT 1 0 19 "Pt11 := tt10+tt20;\n" } {MPLTEXT 1 0 19 "P022 := t010*t020;\n" }{MPLTEXT 1 0 19 "P122 := t120* t110;\n" }{MPLTEXT 1 0 19 "Pt22 := tt20*tt10;\n" }{MPLTEXT 1 0 71 "Coh erenceEquation1 := tinfty10+tt10+tt20+t010+t020+tinfty20+t110+t120;\n" }{MPLTEXT 1 0 38 "CoherenceEquation2 := Pt12+P012+P112;\n" }{MPLTEXT 1 0 84 "CoherenceEquation3 := -tinfty10*tinfty20+tt20*tt10+Pt12*t+t010 *t020+P112+t120*t110;\n" }{MPLTEXT 1 0 37 "CoherenceEquation4 := a0+a1 +at+ h*p;\n" }{MPLTEXT 1 0 97 "CoherenceEquation5 := a1+t*at+t010*t020 +t110*t120+tt10*tt20-tinfty10*tinfty20+h*p*q -h*tinfty10;\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 57 "P1:=lambda->P011/lambda+P111/(lambda-1)+ Pt11/(lambda-t);\n" }{MPLTEXT 1 0 53 "dP1dlambda:=unapply(diff(P1(lamb da),lambda),lambda):\n" }{MPLTEXT 1 0 107 "P2:=lambda->P022/lambda^2+P 012/lambda+P122/(lambda-1)^2+P112/(lambda-1)+Pt22/(lambda-t)^2+Pt12/(l ambda-t);\n" }{MPLTEXT 1 0 53 "dP2dlambda:=unapply(diff(P2(lambda),lam bda),lambda):\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 52 "tdP2:=unapply( P2(lambda)-P012/lambda-P112/(lambda-1)" }{MPLTEXT 1 0 16 "-Pt12/(lambd a-t)" }{MPLTEXT 1 0 9 ",lambda);" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 56 "dtdP2dlambda:=unapply(diff(tdP2(lambda),lambda),lambda):" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 30 "mubis:=q*(q -1)*(q-t)/t/(t-1);\n" }{MPLTEXT 1 0 28 "nuMinus1bis:=mubis/q/(q-1);\n" }{MPLTEXT 1 0 16 "nu0bis:=mubis/q;" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 18 "L:=Matrix(2,2,0):\n" } {MPLTEXT 1 0 11 "L[1,1]:=0:\n" }{MPLTEXT 1 0 11 "L[1,2]:=1:\n" } {MPLTEXT 1 0 137 "L[2,1]:=-t110*t120/(lambda-1)^2 -a1/(lambda-1) -t0 10*t020/lambda^2 -a0/lambda -tt10*tt20/(lambda-t)^2 -at/(lambda-t) - p *h/(lambda-q):\n" }{MPLTEXT 1 0 98 "L[2,2]:=(t110+t120-h)/(lambda-1)+ \+ (t010+t020-h)/lambda+ (tt10+tt20-h)/(lambda-t) +h/(lambda-q):\n" } {MPLTEXT 1 0 61 "P1(lambda)-h/lambda-h/(lambda-1)-h/(lambda-t)+ h/(lam bda-q);\n" }{MPLTEXT 1 0 29 "-P2(lambda) -p*h/(lambda-q);\n" }{MPLTEXT 1 0 10 "C01:=-a0:\n" }{MPLTEXT 1 0 10 "C11:=-a1:\n" }{MPLTEXT 1 0 10 "Ct1:=-at:\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 30 "rho:=q*p*(q-1)*(q -t)/t/(t-1);\n" }{MPLTEXT 1 0 18 "A:=Matrix(2,2,0):\n" }{MPLTEXT 1 0 26 "A[1,1]:=C-rho/(lambda-q):\n" }{MPLTEXT 1 0 54 "A[1,2]:=(a*(lambda- t)^2+b*(lambda-t)+q-t)/(lambda-q):\n" }{MPLTEXT 1 0 22 "A[2,1]:=AA21(l ambda):\n" }{MPLTEXT 1 0 22 "A[2,2]:=AA22(lambda):\n" }{MPLTEXT 1 0 26 "dAdlambda:=Matrix(2,2,0):\n" }{MPLTEXT 1 0 88 "for i from 1 to 2 d o for j from 1 to 2 do dAdlambda[i,j]:=diff(A[i,j],lambda): od: od:\n " }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 3 "L;\n" }{MPLTEXT 1 0 3 "A;\n" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 61 "nuMinus1:=factor(-residue(A[1,2] /lambda^2,lambda=infinity));\n" }{MPLTEXT 1 0 54 "nu0:=factor(-residue (A[1,2]/lambda,lambda=infinity));\n" }{MPLTEXT 1 0 37 "mu:=factor(resi due(A[1,2],lambda=q));" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 42 "Q2:=unapply(-p*(q-0)*(q-1)*(q-t) ,lambda):\n" }{MPLTEXT 1 0 18 "J:=Matrix(2,2,0):\n" }{MPLTEXT 1 0 11 " J[1,1]:=1:\n" }{MPLTEXT 1 0 11 "J[1,2]:=0:\n" }{MPLTEXT 1 0 31 "J[2,1] :=Q2(lambda)/(lambda-q):\n" }{MPLTEXT 1 0 58 "J[2,2]:=(lambda-0)^1*(la mbda-1)^1*(lambda-t)^1/(lambda-q):" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 26 "dJdlambda:=Matrix(2,2,0):\n" }{MPLTEXT 1 0 87 "for i from 1 to 2 do for j from 1 to 2 do dJdlambda[i,j]:=diff(J[i,j],lambda): od: od: \n" }{MPLTEXT 1 0 3 "J:\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 19 "LJ:= Matrix(2,2,0):\n" }{MPLTEXT 1 0 12 "LJ[1,1]:=0:\n" }{MPLTEXT 1 0 12 "L J[1,2]:=0:\n" }{MPLTEXT 1 0 63 "LJ[2,2]:=diff(J[2,2],q)*Lq+diff(J[2,2] ,p)*Lp+h*diff(J[2,2],t):\n" }{MPLTEXT 1 0 63 "LJ[2,1]:=diff(J[2,1],q)* Lq+diff(J[2,1],p)*Lp+h*diff(J[2,1],t):\n" }{MPLTEXT 1 0 4 "LJ:\n" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" } {MPLTEXT 1 0 79 "checkL:=simplify(Multiply(Multiply(J,L),J^(-1))+h*Mul tiply(dJdlambda,J^(-1))):\n" }{MPLTEXT 1 0 70 "checkA:=simplify(Multip ly(Multiply(J,A),J^(-1))+Multiply(LJ,J^(-1))):\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 18 "V:=Matrix(3,3,0):\n" }{MPLTEXT 1 0 11 "V[1,1]:=1:\n" }{MPLTEXT 1 0 11 "V[1,2]:=1:\n" }{MPLTEXT 1 0 11 "V[1,3]:=1:\n" }{MPLTEXT 1 0 11 "V[2,1]:=0:\n" }{MPLTEXT 1 0 11 "V[ 2,2]:=1:\n" }{MPLTEXT 1 0 11 "V[2,3]:=t:\n" }{MPLTEXT 1 0 13 "V[3,1]:= 1/q:\n" }{MPLTEXT 1 0 17 "V[3,2]:=1/(q-1):\n" }{MPLTEXT 1 0 17 "V[3,3] :=1/(q-t):\n" }{MPLTEXT 1 0 3 "V;\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 18 "B:=Matrix(3,1,0):\n" }{MPLTEXT 1 0 13 "B[1,1]:=h*p:\n" }{MPLTEXT 1 0 46 "B[2,1]:=h*p*q+t1+t2+t3-tinfty10*(h+tinfty20):\n" }{MPLTEXT 1 0 82 "B[3,1]:=p^2 -p*((T1-h)/q+(T2-h)/(q-1)+(T3-h)/(q-t))+t1/q^2+t2/(q -1)^2+t3/(q-t)^2:\n" }{MPLTEXT 1 0 3 "B;\n" }{MPLTEXT 1 0 15 "t1:=t010 *t020;\n" }{MPLTEXT 1 0 15 "t2:=t110*t120;\n" }{MPLTEXT 1 0 15 "t3:=tt 10*tt20;\n" }{MPLTEXT 1 0 15 "T1:=t010+t020;\n" }{MPLTEXT 1 0 15 "T2:= t110+t120;\n" }{MPLTEXT 1 0 1 "T" }{MPLTEXT 1 0 13 "3:=tt10+tt20;" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 29 "CMatrix:=Mu ltiply(V^(-1),B):\n" }{MPLTEXT 1 0 22 "C01bis:=CMatrix[1,1]:\n" } {MPLTEXT 1 0 22 "C11bis:=CMatrix[2,1]:\n" }{MPLTEXT 1 0 21 "Ct1bis:=CM atrix[3,1]:" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I%P011G6\",&I%t010GF$\"\"\"I%t 020GF$F'" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I%P111G6\",&I%t110GF$\"\"\" I%t120GF$F'" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I%Pt11G6\",&I%tt10GF$\" \"\"I%tt20GF$F'" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I%P022G6\"*&I%t010GF $\"\"\"I%t020GF$F'" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I%P122G6\"*&I%t12 0GF$\"\"\"I%t110GF$F'" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I%Pt22G6\"*&I% tt20GF$\"\"\"I%tt10GF$F'" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I3Coherence Equation1G6\",2I)tinfty10GF$\"\"\"I%tt10GF$F'I%tt20GF$F'I%t010GF$F'I%t 020GF$F'I)tinfty20GF$F'I%t110GF$F'I%t120GF$F'" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I3CoherenceEquation2G6\",(I%Pt12GF$\"\"\"I%P012GF$F'I%P1 12GF$F'" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I3CoherenceEquation3G6\",.*& I%Pt12GF$\"\"\"I\"tGF$F(F(*&I%t010GF$F(I%t020GF$F(F(*&I%t110GF$F(I%t12 0GF$F(F(*&I%tt10GF$F(I%tt20GF$F(F(*&I)tinfty10GF$F(I)tinfty20GF$F(!\" \"I%P112GF$F(" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I3CoherenceEquation4G6 \",**&I\"hGF$\"\"\"I\"pGF$F(F(I#a0GF$F(I#a1GF$F(I#atGF$F(" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I3CoherenceEquation5G6\",2*(I\"hGF$\"\"\"I\"pGF$ F(I\"qGF$F(F(*&I#atGF$F(I\"tGF$F(F(*&F'F(I)tinfty10GF$F(!\"\"*&I%t010G F$F(I%t020GF$F(F(*&I%t110GF$F(I%t120GF$F(F(*&I%tt10GF$F(I%tt20GF$F(F(* &F/F(I)tinfty20GF$F(F0I#a1GF$F(" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I#P1 G6\"f*6#I'lambdaGF$F$6$I)operatorGF$I&arrowGF$F$,(*&I%P011GF$\"\"\"9$! \"\"F.*&I%P111GF$F.,&F/F.F0F.F0F.*&I%Pt11GF$F.,&F/F.I\"tGF$F0F0F.F$F$F $" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I#P2G6\"f*6#I'lambdaGF$F$6$I)opera torGF$I&arrowGF$F$,.*&I%P022GF$\"\"\"9$!\"#F.*&I%P012GF$F.F/!\"\"F.*&I %P122GF$F.,&F/F.F3F.F0F.*&I%P112GF$F.F6F3F.*&I%Pt22GF$F.,&F/F.I\"tGF$F 3F0F.*&I%Pt12GF$F.F;F3F.F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I%tdP 2G6\"f*6#I'lambdaGF$F$6$I)operatorGF$I&arrowGF$F$,(*(I%t010GF$\"\"\"I% t020GF$F.9$!\"#F.*(I%t120GF$F.I%t110GF$F.,&F0F.!\"\"F.F1F.*(I%tt20GF$F .I%tt10GF$F.,&F0F.I\"tGF$F6F1F.F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I&mubisG6\"*,I\"qGF$\"\"\",&F&F'!\"\"F'F',&F&F'I\"tGF$F)F'F+F),&F+F' F)F'F)" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I,nuMinus1bisG6\"*(,&I\"qGF$ \"\"\"I\"tGF$!\"\"F(F)F*,&F)F(F*F(F*" }}{PARA 11 "" 1 "" {XPPMATH 20 " >I'nu0bisG6\"**,&I\"qGF$\"\"\"!\"\"F(F(,&F'F(I\"tGF$F)F(F+F),&F+F(F)F( F)" }}{PARA 11 "" 1 "" {XPPMATH 20 ",0*&,&I%t010G6\"\"\"\"I%t020GF&F'F 'I'lambdaGF&!\"\"F'*&,&I%t110GF&F'I%t120GF&F'F',&F)F'F*F'F*F'*&,&I%tt1 0GF&F'I%tt20GF&F'F',&F)F'I\"tGF&F*F*F'*&I\"hGF&F'F)F*F**&F7F'F/F*F**&F 7F'F4F*F**&F7F',&F)F'I\"qGF&F*F*F'" }}{PARA 11 "" 1 "" {XPPMATH 20 ",0 *(I%t010G6\"\"\"\"I%t020GF%F&I'lambdaGF%!\"#!\"\"*&I%P012GF%F&F(F*F**( I%t120GF%F&I%t110GF%F&,&F(F&F*F&F)F**&I%P112GF%F&F0F*F**(I%tt20GF%F&I% tt10GF%F&,&F(F&I\"tGF%F*F)F**&I%Pt12GF%F&F6F*F**(I\"pGF%F&I\"hGF%F&,&F (F&I\"qGF%F*F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I$rhoG6\"*.I\"qGF$ \"\"\"I\"pGF$F',&F&F'!\"\"F'F',&F&F'I\"tGF$F*F'F,F*,&F,F'F*F'F*" }} {PARA 11 "" 1 "" {XPPMATH 20 "-I(mfencedG6#/I+modulenameG6\"I,Typesett ingGI(_syslibGF'6'-I%mrowGF$6#-I'mtableGF$66-I$mtrGF$6'-I$mtdGF$6(-I#m nGF$6%Q\"0F'/%+foregroundGQ([0,0,0]F'/%,mathvariantGQ'normalF'/%)rowal ignGQ!F'/%,columnalignGFC/%+groupalignGFC/%(rowspanGQ\"1F'/%+columnspa nGFJ-F56(-F86%FJF;F>FAFDFFFHFKFAFDFF-F26'-F56(-F,60-I#moGF$6.Q*&uminus 0;F'F;F>/%&fenceGQ&falseF'/%*separatorGFgn/%)stretchyGFgn/%*symmetricG Fgn/%(largeopGFgn/%.movablelimitsGFgn/%'accentGFgn/%'lspaceGQ,0.222222 2emF'/%'rspaceGFfo-I&mfracGF$6)-F,6%-I#miGF$6&Q%t120F'/%'italicGQ%true F'F;/F?Q'italicF'-FX6-Q1⁢F'F>FenFhnFjnF\\oF^oF`oFbo/Feo Q&0.0emF'/FhoF[q-F_p6&Q%t110F'FbpF;Fep-F,6#-I%msupGF$6%-F#6%-F,6%-F_p6 &Q)λF'/FcpFgnF;F>-FX6.Q(−F'F;F>FenFhnFjnF\\oF^oF`oFboFdoF goFOF;F>-F86%Q\"2F'F;F>/%1superscriptshiftGF:/%.linethicknessGFJ/%+den omalignGQ'centerF'/%)numalignGFir/%)bevelledGFgnF;F]r-Fjo6)-F,6#-F_p6& Q#a1F'FbpF;Fep-F,6#FgqFerFgrFjrF\\sF;F]r-Fjo6)-F,6%-F_p6&Q%t010F'FbpF; 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labelspacingGF^zF@FJ/%%openGQ\"[F'/%&closeGQ\"]F'" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I#t1G6\"*&I%t010GF$\"\"\"I%t020GF$F'" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I#t2G6\"*&I%t120GF$\"\"\"I%t110GF$F'" }}{PARA 11 "" 1 " " {XPPMATH 20 ">I#t3G6\"*&I%tt20GF$\"\"\"I%tt10GF$F'" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I#T1G6\",&I%t010GF$\"\"\"I%t020GF$F'" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I#T2G6\",&I%t110GF$\"\"\"I%t120GF$F'" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I#T3G6\",&I%tt10GF$\"\"\"I%tt20GF$F'" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 226 73 "Solving the compatibility equations to o btain the Hamiltonian evolutions." }{TEXT 226 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT 225 69 "The compatibility equation is \\mathcal\{L\}L=h\\p artial_\\lambda A+[A,L]\n" }{TEXT 225 111 "Since the first line of L i s trivial, we may easily obtain A[2,1] et A[2,2] to obtain the full ex pression for A" }{TEXT 225 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "LL:=h*dAdlambda+(Multiply(A,L)-Multiply(L,A)):" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 18 "Entry11:=LL[1,1]:\n" }{MPLTEXT 1 0 17 "Entry1 2:=LL[1,2]:" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 53 "AA21:=unapply(solv e(Entry11=0,AA21(lambda)),lambda):\n" }{MPLTEXT 1 0 40 "AA21bis:=h*dAd lambda[1,1]+A[1,2]*L[2,1]:" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 32 "sim plify(AA21(lambda)-AA21bis);\n" }{MPLTEXT 1 0 53 "AA22:=unapply(solve( Entry12=0,AA22(lambda)),lambda):\n" }{MPLTEXT 1 0 47 "AA22bis:=h*dAdla mbda[1,2]+A[1,1]+A[1,2]*L[2,2]:" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 32 "simplify(AA22(lambda)-AA22bis);\n" }{MPLTEXT 1 0 19 "simplify(Entr y11);\n" }{MPLTEXT 1 0 19 "simplify(Entry12);\n" }{MPLTEXT 1 0 47 "LL: =h*dAdlambda+(Multiply(A,L)-Multiply(L,A)):\n" }{MPLTEXT 1 0 1 "\n" }} {PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 " \"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 225 95 "We now com pute the action of \\mathcal\{L\} on L[2,2] et L[2,1] to obtain the e volution equations" }{TEXT 225 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 215 26 "Evolution of entry L_\{2,2\}" }{TEXT 215 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "Entry22:=simplify(LL[2,2]):\n" }{MPLTEXT 1 0 77 "Entry22TermLambdaMinusqSquare:=factor(residue(Entry22*(lambda- q),lambda=q)):\n" }{MPLTEXT 1 0 60 "Entry22TermLambdaMinusq:=factor(re sidue(Entry22,lambda=q));\n" }{MPLTEXT 1 0 73 "Entry22TermLambdaZeroMi nus4:=factor(residue(Entry22*lambda^3,lambda=0));\n" }{MPLTEXT 1 0 73 "Entry22TermLambdaZeroMinus3:=factor(residue(Entry22*lambda^2,lambda=0 ));\n" }{MPLTEXT 1 0 71 "Entry22TermLambdaZeroMinus2:=factor(residue(E ntry22*lambda,lambda=0));\n" }{MPLTEXT 1 0 64 "Entry22TermLambdaZeroMi nus1:=factor(residue(Entry22,lambda=0));\n" }{MPLTEXT 1 0 77 "Entry22T ermLambdaInfty2:=factor(-residue(Entry22/lambda^3,lambda=infinity));\n " }{MPLTEXT 1 0 77 "Entry22TermLambdaInfty1:=factor(-residue(Entry22/l ambda^2,lambda=infinity));\n" }{MPLTEXT 1 0 75 "Entry22TermLambdaInfty 0:=factor(-residue(Entry22/lambda,lambda=infinity));\n" }{MPLTEXT 1 0 76 "Entry22TermLambdaOneMinus4:=factor(residue(Entry22*(lambda-1)^3,la mbda=1));\n" }{MPLTEXT 1 0 76 "Entry22TermLambdaOneMinus3:=factor(resi due(Entry22*(lambda-1)^2,lambda=1));\n" }{MPLTEXT 1 0 74 "Entry22TermL ambdaOneMinus2:=factor(residue(Entry22*(lambda-1),lambda=1));\n" } {MPLTEXT 1 0 63 "Entry22TermLambdaOneMinus1:=factor(residue(Entry22,la mbda=1));\n" }{MPLTEXT 1 0 74 "Entry22TermLambdaTMinus4:=factor(residu e(Entry22*(lambda-t)^3,lambda=t));\n" }{MPLTEXT 1 0 74 "Entry22TermLam bdaTMinus3:=factor(residue(Entry22*(lambda-t)^2,lambda=t));\n" } {MPLTEXT 1 0 72 "Entry22TermLambdaTMinus2:=factor(residue(Entry22*(lam bda-t),lambda=t));\n" }{MPLTEXT 1 0 61 "Entry22TermLambdaTMinus1:=fact or(residue(Entry22,lambda=t));\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 97 "simplify( Entry22-(Entry22TermLambdaMinusqSquare/(lambda-q)^2+Entr y22TermLambdaMinusq/(lambda-q)\n" }{MPLTEXT 1 0 147 "+Entry22TermLambd aZeroMinus4/lambda^4+Entry22TermLambdaZeroMinus3/lambda^3+Entry22TermL ambdaZeroMinus2/lambda^2+Entry22TermLambdaZeroMinus1/lambda\n" } {MPLTEXT 1 0 159 "+Entry22TermLambdaOneMinus4/(lambda-1)^4+Entry22Term LambdaOneMinus3/(lambda-1)^3+Entry22TermLambdaOneMinus2/(lambda-1)^2+E ntry22TermLambdaOneMinus1/(lambda-1)\n" }{MPLTEXT 1 0 151 "+Entry22Ter mLambdaTMinus4/(lambda-t)^4+Entry22TermLambdaTMinus3/(lambda-t)^3+Entr y22TermLambdaTMinus2/(lambda-t)^2+Entry22TermLambdaTMinus1/(lambda-t) \n" }{MPLTEXT 1 0 93 "+Entry22TermLambdaInfty0+Entry22TermLambdaInfty1 *lambda+Entry22TermLambdaInfty2*lambda^2) );\n" }{MPLTEXT 1 0 8 "L[2,2 ];\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I8Entry22TermLambdaMinusqG6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">IIIII8En try22TermLambdaInfty2G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I8Ent ry22TermLambdaInfty1G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I8Entr y22TermLambdaInfty0G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I;Entry 22TermLambdaOneMinus4G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I;Ent ry22TermLambdaOneMinus3G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I;E ntry22TermLambdaOneMinus2G6\",$**,(I%t120GF$!\"\"I\"hGF$\"\"\"I%t110GF $F)F+,0*&I\"aGF$F+I\"tGF$\"\"#F+*&F/F+F0F+!\"#*&I\"bGF$F+F0F+F)F/F+F5F +I\"qGF$F+F0F)F+F*F+,&F6F+F)F+F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I ;Entry22TermLambdaOneMinus1G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 " >I9Entry22TermLambdaTMinus4G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 " >I9Entry22TermLambdaTMinus3G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 " >I9Entry22TermLambdaTMinus2G6\",$*&,(I%tt10GF$!\"\"I%tt20GF$F)I\"hGF$ \"\"\"F,F+F,F)" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I9Entry22TermLambdaTM inus1G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ",**&,(I%t110G6\"\"\"\"I%t120GF&F'I\"hGF&!\"\"F',&I'la mbdaGF&F'F*F'F*F'*&,(I%t010GF&F'I%t020GF&F'F)F*F'F,F*F'*&,(I%tt10GF&F' I%tt20GF&F'F)F*F',&F,F'I\"tGF&F*F*F'*&F)F',&F,F'I\"qGF&F*F*F'" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 225 201 "Since the deformation operator i s \\hbar*partial_t we cannot have double poles at lambda=1 or lambda=0 but only a double pole at lambda=t with a numerator equal to tt10+tt 20-h which is indeed the case." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "a:=(q-t)/t/(t-1);\n" }{MPLTEXT 1 0 26 "b:=(q-t)*(2*t-1)/t/(t-1 );\n" }{MPLTEXT 1 0 39 "simplify(Entry22TermLambdaZeroMinus2);\n" } {MPLTEXT 1 0 38 "simplify(Entry22TermLambdaOneMinus2);\n" }{MPLTEXT 1 0 16 "factor(A[1,2]);\n" }{MPLTEXT 1 0 18 "factor(nuMinus1);\n" } {MPLTEXT 1 0 13 "factor(nu0);\n" }{MPLTEXT 1 0 12 "factor(mu);\n" } {MPLTEXT 1 0 20 "simplify(mu-mubis);\n" }{MPLTEXT 1 0 32 "simplify(nuM inus1-nuMinus1bis);\n" }{MPLTEXT 1 0 21 "simplify(nu0-nu0bis);" }} {PARA 11 "" 1 "" {XPPMATH 20 ">I\"aG6\"*(,&I\"qGF$\"\"\"I\"tGF$!\"\"F( F)F*,&F)F(F*F(F*" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I\"bG6\"**,&I\"qGF$ \"\"\"I\"tGF$!\"\"F(,&F)\"\"#F*F(F(F)F*,&F)F(F*F(F*" }}{PARA 11 "" 1 " " {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "*.I'lambdaG6\"\"\"\",&I\"qGF$F%I\"tGF$!\"\"F%, &F#F%F)F%F%,&F(F%F)F%F)F(F),&F#F%F'F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "*(,&I\"qG6\"\"\"\"I\"tGF%!\"\"F&F'F(,&F'F&F(F&F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "**,&I\"qG6\"\"\"\"!\"\"F&F&,&F$F&I\"tGF%F'F&F)F',&F)F& F'F&F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "*,I\"qG6\"\"\"\",&F#F%!\"\"F%F %,&F#F%I\"tGF$F'F%F)F',&F)F%F'F%F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "\" \"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "Lq:=f actor(Entry22TermLambdaMinusqSquare/h):\n" }{MPLTEXT 1 0 80 "Lqbis:=2* q*p*(q-1)*(q-t)/t/(t-1)-q*(q-1)*(q-t)/t/(t-1)*P1(q)+q*(q-1)/t/(t-1)*h; \n" }{MPLTEXT 1 0 19 "simplify(Lq-Lqbis);" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I&LqbisG6\",(*.I\"qGF$\"\"\"I\"pGF$F(,&F'F(!\"\"F(F(,&F' F(I\"tGF$F+F(F-F+,&F-F(F+F(F+\"\"#*.F'F(F*F(F,F(F-F+F.F+,(*&,&I%t010GF $F(I%t020GF$F(F(F'F+F(*&,&I%t110GF$F(I%t120GF$F(F(F*F+F(*&,&I%tt10GF$F (I%tt20GF$F(F(F,F+F(F(F+*,F'F(F*F(F-F+F.F+I\"hGF$F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 225 34 "Let us look at \\mathcal\{L\}[L[2,1]]" }{TEXT 225 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "Entry21:=simplify(LL[2,1]):\n" }{MPLTEXT 1 0 77 "Entry21TermLambdaMinusqCube:=factor(residue(Entry21*(lambda-q)^2,l ambda=q));\n" }{MPLTEXT 1 0 77 "Entry21TermLambdaMinusqSquare:=factor( residue(Entry21*(lambda-q),lambda=q));\n" }{MPLTEXT 1 0 60 "Entry21Ter mLambdaMinusq:=factor(residue(Entry21,lambda=q));\n" }{MPLTEXT 1 0 73 "Entry21TermLambdaZeroMinus5:=factor(residue(Entry21*lambda^4,lambda=0 ));\n" }{MPLTEXT 1 0 73 "Entry21TermLambdaZeroMinus4:=factor(residue(E ntry21*lambda^3,lambda=0));\n" }{MPLTEXT 1 0 73 "Entry21TermLambdaZero Minus3:=factor(residue(Entry21*lambda^2,lambda=0));\n" }{MPLTEXT 1 0 71 "Entry21TermLambdaZeroMinus2:=factor(residue(Entry21*lambda,lambda= 0));\n" }{MPLTEXT 1 0 64 "Entry21TermLambdaZeroMinus1:=factor(residue( Entry21,lambda=0));\n" }{MPLTEXT 1 0 77 "Entry21TermLambdaInfty3:=fact or(-residue(Entry21/lambda^4,lambda=infinity));\n" }{MPLTEXT 1 0 77 "E ntry21TermLambdaInfty2:=factor(-residue(Entry21/lambda^3,lambda=infini ty));\n" }{MPLTEXT 1 0 77 "Entry21TermLambdaInfty1:=factor(-residue(En try21/lambda^2,lambda=infinity));\n" }{MPLTEXT 1 0 75 "Entry21TermLamb daInfty0:=factor(-residue(Entry21/lambda,lambda=infinity));\n" } {MPLTEXT 1 0 76 "Entry21TermLambdaOneMinus5:=factor(residue(Entry21*(l ambda-1)^4,lambda=1));\n" }{MPLTEXT 1 0 76 "Entry21TermLambdaOneMinus4 :=factor(residue(Entry21*(lambda-1)^3,lambda=1));\n" }{MPLTEXT 1 0 76 "Entry21TermLambdaOneMinus3:=factor(residue(Entry21*(lambda-1)^2,lambd a=1));\n" }{MPLTEXT 1 0 74 "Entry21TermLambdaOneMinus2:=factor(residue (Entry21*(lambda-1),lambda=1));\n" }{MPLTEXT 1 0 63 "Entry21TermLambda OneMinus1:=factor(residue(Entry21,lambda=1));\n" }{MPLTEXT 1 0 74 "Ent ry21TermLambdaTMinus5:=factor(residue(Entry21*(lambda-t)^4,lambda=t)); \n" }{MPLTEXT 1 0 74 "Entry21TermLambdaTMinus4:=factor(residue(Entry21 *(lambda-t)^3,lambda=t));\n" }{MPLTEXT 1 0 74 "Entry21TermLambdaTMinus 3:=factor(residue(Entry21*(lambda-t)^2,lambda=t));\n" }{MPLTEXT 1 0 72 "Entry21TermLambdaTMinus2:=factor(residue(Entry21*(lambda-t),lambda =t));\n" }{MPLTEXT 1 0 61 "Entry21TermLambdaTMinus1:=factor(residue(En try21,lambda=t));\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 138 "simplify( Entry21-(Entry21TermLambdaMinusqCube/(lambda-q)^3+Entry21TermLambdaMi nusqSquare/(lambda-q)^2+Entry21TermLambdaMinusq/(lambda-q)\n" } {MPLTEXT 1 0 185 "+Entry21TermLambdaZeroMinus5/lambda^5+Entry21TermLam bdaZeroMinus4/lambda^4+Entry21TermLambdaZeroMinus3/lambda^3+ Entry21Te rmLambdaZeroMinus2/lambda^2+Entry21TermLambdaZeroMinus1/lambda\n" } {MPLTEXT 1 0 200 "+Entry21TermLambdaOneMinus5/(lambda-1)^5+Entry21Term LambdaOneMinus4/(lambda-1)^4+Entry21TermLambdaOneMinus3/(lambda-1)^3+ \+ Entry21TermLambdaOneMinus2/(lambda-1)^2+Entry21TermLambdaOneMinus1/(la mbda-1)\n" }{MPLTEXT 1 0 190 "+Entry21TermLambdaTMinus5/(lambda-t)^5+E ntry21TermLambdaTMinus4/(lambda-t)^4+Entry21TermLambdaTMinus3/(lambda- t)^3+ Entry21TermLambdaTMinus2/(lambda-t)^2+Entry21TermLambdaTMinus1/( lambda-t)\n" }{MPLTEXT 1 0 126 "+Entry21TermLambdaInfty0+Entry21TermLa mbdaInfty1*lambda+Entry21TermLambdaInfty2*lambda^2+Entry21TermLambdaIn fty3*lambda^3) );\n" }{MPLTEXT 1 0 8 "L[2,1];\n" }{MPLTEXT 1 0 1 "\n" }}{PARA 11 "" 1 "" {XPPMATH 20 ">II>Entry21TermLambdaMinusqSquareG6 \"*0,ju*(I\"hGF$\"\"\"I\"pGF$F)I\"qGF$\"\"&F,**F(F)F*F)F+\"\"%I\"tGF$F )!\"***F(F)F*F)F+\"\"$F/\"\"#F.*(F*F)F+F,I%t010GF$F)!\"\"*(F*F)F+F,I%t 020GF$F)F6*(F*F)F+F,I%t110GF$F)F6*(F*F)F+F,I%t120GF$F)F6*(F*F)F+F,I%tt 10GF$F)F6*(F*F)F+F,I%tt20GF$F)F6**F*F)F+F.F/F)F5F)F3**F*F)F+F.F/F)F8F) F3**F*F)F+F.F/F)F:F)F3**F*F)F+F.F/F)FF)F)**F*F)F+ F.F/F)F@F)F)**F*F)F+F2F/F3F5F)F6**F*F)F+F2F/F3F8F)F6**F*F)F+F2F/F3F:F) F6**F*F)F+F2F/F3FF)F3*(F*F)F+F.F@F)F3**F*F)F+F2F/F)F5F)FN**F*F)F+F2F/F)F 8F)FN**F*F)F+F2F/F)F:F)FW**F*F)F+F2F/F)FF)FW**F*F )F+F2F/F)F@F)FW**F*F)F+F3F/F3F5F)F3**F*F)F+F3F/F3F8F)F3**F*F)F+F3F/F3F :F)F)**F*F)F+F3F/F3FF)F@F )F3**F+F2F/F)F5F)F8F)FN**F+F2F/F)F:F)FF)F6*(F*F)F+F2F@F)F6**F*F)F+F3F/F)F5F)F3**F*F) F+F3F/F)F8F)F3**F*F)F+F3F/F)F>F)F)**F*F)F+F3F/F)F@F)F)**F*F)F+F)F/F3F5 F)F6**F*F)F+F)F/F3F8F)F6*(F+F2F5F)F8F)FN*(F+F2F>F)F@F)FN**F+F3F/F)F5F) F8F)Fap**F+F)F/F3F5F)F8F)FN*&FLF)F+F2F3*(FLF)F+F3F/F)FN*(FLF)F+F)F/F3F 3*&FUF)F+F2F3*(FUF)F+F3F/F)FW*(F+F3F5F)F8F)F3*(F+F3F>F)F@F)F3**F+F)F/F )F5F)F8F)FN*(F/F3F5F)F8F)F3F)F(F)F/F6,&F/F)F6F)F6,&F+F)F/F6F6F+F6,&F+F )F6F)F6" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I8Entry21TermLambdaMinusqG6 \",$*0I\"hGF$\"\"\",\\\\l*(F'F(I\"pGF$F(I\"qGF$\"\"(\"\"%**F'F(F+F(F, \"\"'I\"tGF$F(!#5**F'F(F+F(F,\"\"&F1\"\"#\"\"***F'F(F+F(F,F.F1\"\"$!\" $*(F+F(F,F-I%t010GF$F(!\"\"*(F+F(F,F-I%t020GF$F(F<*(F+F(F,F-I%t110GF$F (F<*(F+F(F,F-I%t120GF$F(F<*(F+F(F,F-I%tt10GF$F(F<*(F+F(F,F-I%tt20GF$F( F<**F+F(F,F0F1F(F;F(F8**F+F(F,F0F1F(F>F(F8**F+F(F,F0F1F(F@F(F8**F+F(F, F0F1F(FBF(F8**F+F(F,F0F1F(FDF(F(**F+F(F,F0F1F(FFF(F(**F+F(F,F4F1F5F;F( F9**F+F(F,F4F1F5F>F(F9**F+F(F,F4F1F5F@F(F9**F+F(F,F4F1F5FBF(F9**F+F(F, F.F1F8F;F(F(**F+F(F,F.F1F8F>F(F(**F+F(F,F.F1F8F@F(F(**F+F(F,F.F1F8FBF( F(*&I#a0GF$F(F,F-F(*(FVF(F,F0F1F(F9*(FVF(F,F4F1F5F8*(FVF(F,F.F1F8F<*&I #a1GF$F(F,F-F(*(FenF(F,F0F1F(F9*(FenF(F,F4F1F5F8*(FenF(F,F.F1F8F<*&I#a tGF$F(F,F-F(*(FjnF(F,F0F1F(F<*(F'F(F+F(F,F0!\"***F'F(F+F(F,F4F1F(\"#@* *F'F(F+F(F,F.F1F5!#=**F'F(F+F(F,F8F1F8F0*(F+F(F,F0F;F(F8*(F+F(F,F0F>F( F8*(F+F(F,F0F@F(F(*(F+F(F,F0FBF(F(*(F+F(F,F0FDF(F8*(F+F(F,F0FFF(F8**F+ F(F,F4F1F(F;F(F]o**F+F(F,F4F1F(F>F(F]o**F+F(F,F4F1F(F@F(F9**F+F(F,F4F1 F(FBF(F9**F+F(F,F4F1F(FDF(F9**F+F(F,F4F1F(FFF(F9**F+F(F,F.F1F5F;F(F6** F+F(F,F.F1F5F>F(F6**F+F(F,F.F1F5F@F(F8**F+F(F,F.F1F5FBF(F8**F+F(F,F8F1 F8F;F(F9**F+F(F,F8F1F8F>F(F9**F+F(F,F8F1F8F@F(F<**F+F(F,F8F1F8FBF(F<*( F,F0F;F(F>F(F5*(F,F0F@F(FBF(F5*(F,F0FDF(FFF(F5**F,F4F1F(F;F(F>F(!\"'** F,F4F1F(F@F(FBF(F[q**F,F.F1F5F;F(F>F(F0**F,F.F1F5F@F(FBF(F0**F,F8F1F8F ;F(F>F(!\"#**F,F8F1F8F@F(FBF(F`q*&FVF(F,F0F9*(FVF(F,F4F1F(F6*(FVF(F,F. F1F5F]o*(FVF(F,F8F1F8F8*&FenF(F,F0F<*(FenF(F,F4F1F(F8*(FenF(F,F.F1F5F9 *(FenF(F,F8F1F8F(*&FjnF(F,F0F9*(FjnF(F,F4F1F(F8*(F'F(F+F(F,F4F-**F'F(F +F(F,F.F1F(!#:**F'F(F+F(F,F8F1F5\"#7**F'F(F+F(F,F5F1F8!\"%*(F+F(F,F4F; F(F9*(F+F(F,F4F>F(F9*(F+F(F,F4FDF(F9*(F+F(F,F4FFF(F9**F+F(F,F.F1F(F;F( F6**F+F(F,F.F1F(F>F(F6**F+F(F,F.F1F(FDF(F8**F+F(F,F.F1F(FFF(F8**F+F(F, F8F1F5F;F(F]o**F+F(F,F8F1F5F>F(F]o**F+F(F,F5F1F8F;F(F8**F+F(F,F5F1F8F> F(F8*(F,F4F;F(F>F(F[q*(F,F4FDF(FFF(F[q**F,F.F1F(F;F(F>F(\"#=**F,F8F1F5 F;F(F>F(Fao**F,F5F1F8F;F(F>F(F0*&FVF(F,F4F8*(FVF(F,F.F1F(F]o*(FVF(F,F8 F1F5F6*(FVF(F,F5F1F8F9*&FjnF(F,F4F8*(FjnF(F,F.F1F(F9*(F'F(F+F(F,F.F`q* *F'F(F+F(F,F8F1F(F.**F'F(F+F(F,F5F1F5F9**F'F(F+F(F,F(F1F8F(*(F+F(F,F.F ;F(F(*(F+F(F,F.F>F(F(*(F+F(F,F.FDF(F(*(F+F(F,F.FFF(F(**F+F(F,F8F1F(F;F (F9**F+F(F,F8F1F(F>F(F9**F+F(F,F8F1F(FDF(F<**F+F(F,F8F1F(FFF(F<**F+F(F ,F5F1F5F;F(F8**F+F(F,F5F1F5F>F(F8**F+F(F,F(F1F8F;F(F<**F+F(F,F(F1F8F>F (F<*(F,F.F;F(F>F(F0*(F,F.FDF(FFF(F0**F,F8F1F(F;F(F>F(Fao**F,F5F1F5F;F( F>F(Fbs**F,F(F1F8F;F(F>F(F[q*&FVF(F,F.F<*(FVF(F,F8F1F(F8*(FVF(F,F5F1F5 F9*(FVF(F,F(F1F8F(*&FjnF(F,F.F<*(FjnF(F,F8F1F(F(*(F,F8F;F(F>F(F`q*(F,F 8FDF(FFF(F`q**F,F5F1F(F;F(F>F(F0**F,F(F1F5F;F(F>F(F[q*(F1F8F;F(F>F(F5F (F1F<,&F1F(FIIIIII8Entry21TermLamb daInfty3G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I8Entry21TermLambd aInfty2G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I8Entry21TermLambda Infty1G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I8Entry21TermLambdaI nfty0G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I;Entry21TermLambdaOn eMinus5G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I;Entry21TermLambda OneMinus4G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I;Entry21TermLamb daOneMinus3G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I;Entry21TermLa mbdaOneMinus2G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I;Entry21Term LambdaOneMinus1G6\"*.,&I\"qGF$\"\"\"I\"tGF$!\"\"F(,4*(I\"hGF$F(I\"pGF$ F(F'\"\"#F(*(F.F(F'F/I%t110GF$F(F**(F.F(F'F/I%t120GF$F(F**(F-F(F.F(F'F (F**(F.F(F'F(F1F(F(*(F.F(F'F(F3F(F(*(F'F(F1F(F3F(F/*&I#a1GF$F(F'F(F(F9 F*F(F-F(,&F'F(F*F(!\"#,&F)F(F*F(F*F)F*" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I9Entry21TermLambdaTMinus5G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I9Entry21TermLambdaTMinus4G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I9Entry21TermLambdaTMinus3G6\",$*(I\"hGF$\"\"\"I%tt10GF$F(I%tt20GF$ F(!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I9Entry21TermLambdaTMinus2G6 \",$*&I#atGF$\"\"\"I\"hGF$F(!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I9 Entry21TermLambdaTMinus1G6\"*,,H*(I\"hGF$\"\"\"I\"pGF$F)I\"qGF$\"\"$F) **F(F)F*F)F+\"\"#I\"tGF$F)!\"\"*(F*F)F+F,I%tt10GF$F)F0*(F*F)F+F,I%tt20 GF$F)F0**F*F)F+F.F/F)F2F)F)**F*F)F+F.F/F)F4F)F)*(I#atGF$F)F+F.F/F)F.*( F8F)F+F)F/F.!\"$*&F8F)F/F,F)*(F(F)F*F)F+F.F0**F(F)F*F)F+F)F/F)F)*(F*F) F+F.F2F)F)*(F*F)F+F.F4F)F)**F*F)F+F)F/F)F2F)F0**F*F)F+F)F/F)F4F)F0*(F+ F.F2F)F4F)F.*&F8F)F+F.F0*(F8F)F+F)F/F)F)*(F+F)F2F)F4F)!\"#F)F(F)F/F0,& F/F)F0F)F0,&F+F)F/F0FF" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 ",0*(I%t120G6\"\"\"\"I%t110GF%F&,&I'lambd aGF%F&!\"\"F&!\"#F**&I#a1GF%F&F(F*F**(I%t010GF%F&I%t020GF%F&F)F+F**&I# a0GF%F&F)F*F**(I%tt20GF%F&I%tt10GF%F&,&F)F&I\"tGF%F*F+F**&I#atGF%F&F6F *F**(I\"pGF%F&I\"hGF%F&,&F)F&I\"qGF%F*F*F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 225 286 "Choice of mu removes the cubic pole at lambda=q. Term i n (lambda-t)^\{-3\} is coherent with the deformation operator \\hbar* \\partial_t. Term in (lambda-t)^\{-2\} is also coherent. Term (lambda) ^\{-1\} at infinity should be -h*\\partial_t a_0 -h*\\partial_t a_1-h* \\partial_t a_t- h^2\\partial_t p " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "factor(simplify(-residue(Entry21,lambda=infinity))); \n" }{MPLTEXT 1 0 20 "CoherenceEquation4;\n" }{MPLTEXT 1 0 19 "Coheren ceEquation5;" }}{PARA 11 "" 1 "" {XPPMATH 20 ",$*,,**&I\"hG6\"\"\"\"I \"pGF'F(F(I#a0GF'F(I#a1GF'F(I#atGF'F(F(,&I\"qGF'F(I\"tGF'!\"\"F(F&F(F/ F0,&F/F(F0F(F0F0" }}{PARA 11 "" 1 "" {XPPMATH 20 ",**&I\"hG6\"\"\"\"I \"pGF%F&F&I#a0GF%F&I#a1GF%F&I#atGF%F&" }}{PARA 11 "" 1 "" {XPPMATH 20 ",2*(I\"hG6\"\"\"\"I\"pGF%F&I\"qGF%F&F&*&I#atGF%F&I\"tGF%F&F&*&F$F&I)t infty10GF%F&!\"\"*&I%t010GF%F&I%t020GF%F&F&*&I%t110GF%F&I%t120GF%F&F&* &I%tt10GF%F&I%tt20GF%F&F&*&F-F&I)tinfty20GF%F&F.I#a1GF%F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "La0:=-factor(Entry21TermLambdaZeroM inus1):\n" }{MPLTEXT 1 0 42 "La1:=-factor(Entry21TermLambdaOneMinus1): \n" }{MPLTEXT 1 0 40 "Lat:=-factor(Entry21TermLambdaTMinus1):\n" } {MPLTEXT 1 0 57 "LpFunction:=unapply(-Entry21TermLambdaMinusq/h,a0,a1, at):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "Equation1:=simplify (Entry21TermLambdaMinusqSquare-(-p*h*Lq)):\n" }{MPLTEXT 1 0 125 "Equat ion1bis:=2*(q-1)*(q-t)*h*a0/((t-1)*t)+2*q*(q-t)*h*a1/((t-1)*t)+2*q*(q- 1)*h*at/((t-1)*t)+2*q*(q-1)*(q-t)*h*p^2/((t-1)*t)\n" }{MPLTEXT 1 0 138 "+2*((h-t010-t020)*t+(-2*h*t+t*t010+t*t020+t*t110+t*t120-2*h+t010+ t020+tt10+tt20)*q+(3*h-t010-t020-t110-t120-tt10-tt20)*q^2)*h*p/((t-1)* t)\n" }{MPLTEXT 1 0 229 "+2*h/t/(t-1)/q/(q-1)/(q-t)*(t^2*t010*t020-2*t *t010*t020*(t+1)*q+(t^2*t010*t020+t^2*t110*t120+4*t*t010*t020+t010*t02 0+tt10*tt20)*q^2+(-2*t*t010*t020-2*t*t110*t120-2*t010*t020-2*tt10*tt20 )*q^3+(t010*t020+t110*t120+tt10*tt20)*q^4);\n" }{MPLTEXT 1 0 14 "facto r(series(" }{MPLTEXT 1 0 27 "Equation1-Equation1bis,p));" }}{PARA 11 " " 1 "" {XPPMATH 20 ">I-Equation1bisG6\",.*.,&I\"qGF$\"\"\"!\"\"F)F),&F (F)I\"tGF$F*F)I\"hGF$F)I#a0GF$F),&F,F)F*F)F*F,F*\"\"#*.F(F)F+F)F-F)I#a 1GF$F)F/F*F,F*F0*.F(F)F'F)F-F)I#atGF$F)F/F*F,F*F0*0F(F)F'F)F+F)F-F)I\" pGF$F0F/F*F,F*F0*,,(*&,(F-F)I%t010GF$F*I%t020GF$F*F)F,F)F)*&,6*&F-F)F, F)!\"#*&F,F)F;F)F)*&F,F)F " 0 "" {MPLTEXT 1 0 175 "Coe ffp:=((2*((h-t010-t020)*t+(-2*h*t+t*t010+t*t020+t*t110+t*t120-2*h+t010 +t020+tt10+tt20)*q+(3*h-t010-t020-t110-t120-tt10-tt20)*q^2))*h/((t-1)* t))/(2*q*(q-1)*(q-t)*h/t/(t-1)):" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 13 "Coeffpbis:=-(" }{MPLTEXT 1 0 10 "t010+t020-" }{MPLTEXT 1 0 46 "h)/ q-(t120-h+t110)/(q-1)-(tt10+tt20-h)/(q-t);\n" }{MPLTEXT 1 0 14 "factor (Coeffp-" }{MPLTEXT 1 0 9 "Coeffpbis" }{MPLTEXT 1 0 2 ");" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I*CoeffpbisG6\",(*&,(I%t010GF$\"\"\"I%t020GF$F)I \"hGF$!\"\"F)I\"qGF$F,F,*&,(I%t110GF$F)I%t120GF$F)F+F,F),&F-F)F,F)F,F, *&,(I%tt10GF$F)I%tt20GF$F)F+F,F),&F-F)I\"tGF$F,F,F," }}{PARA 11 "" 1 " " {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "Coe ffConstant:=(2" }{MPLTEXT 1 0 260 "*h*(t^2*t010*t020-2*t*t010*t020*(t+ 1)*q+(t^2*t010*t020+t^2*t110*t120+4*t*t010*t020+t010*t020+tt10*tt20)*q ^2+(-2*t*t010*t020-2*t*t110*t120-2*t010*t020-2*tt10*tt20)*q^3+(t010*t0 20+t110*t120+tt10*tt20)*q^4)/(t*(t-1)*q*(q-1)*(q-t)))/(2*q*(q-1)*(q-t) *h/((t-1)*t));\n" }{MPLTEXT 1 0 19 "CoeffConstantbis:=(" }{MPLTEXT 1 0 52 "t010*t020/q^2+t110*t120/(q-1)^2+tt10*tt20/(q-t)^2);\n" }{MPLTEXT 1 0 9 "simplify(" }{MPLTEXT 1 0 14 "CoeffConstant-" }{MPLTEXT 1 0 18 "CoeffConstantbis);" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I.CoeffConstantG 6\"**,,*(I\"tGF$\"\"#I%t010GF$\"\"\"I%t020GF$F+F+*,F(F+F*F+F,F+,&F(F+F +F+F+I\"qGF$F+!\"#*&,,F'F+*(F(F)I%t110GF$F+I%t120GF$F+F+*(F(F+F*F+F,F+ \"\"%*&F*F+F,F+F+*&I%tt10GF$F+I%tt20GF$F+F+F+F/F)F+*&,*F6F0*(F(F+F4F+F 5F+F0F8F0F9F0F+F/\"\"$F+*&,(F8F+*&F4F+F5F+F+F9F+F+F/F7F+F+,&F/F+F(!\" \"F0F/F0,&F/F+FDF+F0" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I1CoeffConstant bisG6\",(*(I%t010GF$\"\"\"I%t020GF$F(I\"qGF$!\"#F(*(I%t120GF$F(I%t110G F$F(,&F*F(!\"\"F(F+F(*(I%tt20GF$F(I%tt10GF$F(,&F*F(I\"tGF$F0F+F(" }} {PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "simplify(CoherenceEquation4);\n" }{MPLTEXT 1 0 29 "si mplify(CoherenceEquation5);" }{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 ",**&I\"hG6\"\"\"\"I\"pGF%F&F&I#a0GF%F&I#a1GF%F&I#atGF%F&" }}{PARA 11 "" 1 "" {XPPMATH 20 ",0*&,&I\"hG6\"!\"\"I)tinfty20GF&F'\" \"\"I)tinfty10GF&F)F)*(F%F)I\"pGF&F)I\"qGF&F)F)*&I%t120GF&F)I%t110GF&F )F)*&I%tt20GF&F)I%tt10GF&F)F)*&I#atGF&F)I\"tGF&F)F)*&I%t010GF&F)I%t020 GF&F)F)I#a1GF&F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1254 "a1:=f actor((-q^4*tinfty10*tinfty20-q^4*h*tinfty10+2*q^3*tinfty10*tinfty20+2 *q^3*h*tinfty10-2*q^2*t*tinfty10*tinfty20-2*q^2*t*h*tinfty10+t*q^3*tin fty10*tinfty20+t*q^3*h*tinfty10+t*t120*t110*q^3-2*t*q^4*p*t110-2*t*q^4 *t120*p+t120*t110*q^2+2*tt20*tt10*q^2*t-tt20*tt10*q^3*t+2*q^5*p^2*t+2* q^5*p^2-q^6*p^2-q^4*p^2-t010*t020*t^2*q^2-4*q^4*p^2*t+2*q^3*p^2*t-q^2* tinfty10*tinfty20-q^2*h*tinfty10-h*q^5*p+p*t010*q^3-2*p*t010*q^4+p*t02 0*q^3-2*p*t020*q^4+h*q^4*p-q^4*p^2*t^2+2*t^2*t010*t020*q+2*h*q^4*p*t-t 010*t020*t^2-2*q^2*p*t020*t^2-2*q^2*p*t010*t^2-q^2*p^2*t^2+q*p*t020*t^ 2+q*p*t010*t^2-q^3*p*h*t^2+q*tinfty10*tinfty20*t+q*h*tinfty10*t-tt20*t t10*q*t-2*t120*t110*q^3-2*q^4*p*tt10-2*q^4*p*tt20-q^4*p*t110-q^4*t120* p+q^5*p*t110+q^5*t120*p+2*p*t110*t*q^3+p*t110*t^2*q^3+2*t120*p*t*q^3+t 120*p*t^2*q^3+2*p*tt10*t*q^3+4*p*t020*t*q^3+p*t020*t^2*q^3+p*t010*q^5+ p*t020*q^5+q^5*p*tt10+q^5*p*tt20+2*t120*t110*t*q^2-t120*t110*t^2*q^2-2 *t010*t020*t*q^2+t010*t020*t*q^3-2*q^4*p*t010*t-q^4*p*tt10*t-q^4*p*tt2 0*t-2*q^4*p*t020*t+p*tt10*q^3+t010*t020*t*q+q^2*p*h*t^2-2*q^3*p*h*t-2* q^2*p*t010*t-q^2*p*tt10*t-q^2*p*tt20*t-2*q^2*p*t020*t+p*tt20*q^3+2*p*t t20*t*q^3-q*t120*t110*t-q^2*p*t110*t^2-q^2*t120*p*t^2+4*p*t010*t*q^3+p *t010*t^2*q^3+2*q^3*p^2*t^2)/(t^2*q-t^2-q^2*t+t-q+q^2)/q);\n" } {MPLTEXT 1 0 1207 "a0:=factor(-(-q^4*tinfty10*tinfty20-q^4*h*tinfty10+ q^3*tinfty10*tinfty20+q^3*h*tinfty10-q^2*t*tinfty10*tinfty20-q^2*t*h*t infty10+t*q^3*tinfty10*tinfty20+t*q^3*h*tinfty10+t*t120*t110*q^3-2*t*q ^4*p*t110-2*t*q^4*t120*p+tt20*tt10*q^2*t-tt20*tt10*q^3*t-tt20*tt10*q^2 +2*q^5*p^2*t+2*q^5*p^2-q^6*p^2-q^4*p^2-t010*t020*t^2*q^2-4*q^4*p^2*t+2 *q^3*p^2*t-h*q^5*p+p*t010*q^3-2*p*t010*q^4+p*t020*q^3-2*p*t020*q^4+2*h *q^4*p-h*q^3*p-q^4*p^2*t^2+2*t^2*t010*t020*q+2*h*q^4*p*t-t010*t020*t^2 -2*q^2*p*t020*t^2-2*q^2*p*t010*t^2-q^2*p^2*t^2+q*p*t020*t^2+q*p*t010*t ^2-q^3*p*h*t^2-t120*t110*q^3+tt20*tt10*q^3+t010*t020*q^3-2*q^4*p*tt10- 2*q^4*p*tt20-q^4*p*t110-q^4*t120*p+q^5*p*t110+q^5*t120*p+2*p*t110*t*q^ 3+p*t110*t^2*q^3+2*t120*p*t*q^3+t120*p*t^2*q^3+2*p*tt10*t*q^3+4*p*t020 *t*q^3+p*t020*t^2*q^3+p*t010*q^5+p*t020*q^5+q^5*p*tt10+q^5*p*tt20+t120 *t110*t*q^2-t120*t110*t^2*q^2-3*t010*t020*t*q^2+t010*t020*t*q^3-2*q^4* p*t010*t-q^4*p*tt10*t-q^4*p*tt20*t-2*q^4*p*t020*t+p*tt10*q^3+2*t010*t0 20*t*q-q*p*h*t^2+2*q^2*p*h*t+2*q^2*p*h*t^2-4*q^3*p*h*t-2*q^2*p*t010*t- q^2*p*tt10*t-q^2*p*tt20*t-2*q^2*p*t020*t+p*tt20*q^3+2*p*tt20*t*q^3-q^2 *p*t110*t^2-q^2*t120*p*t^2+4*p*t010*t*q^3+p*t010*t^2*q^3+2*q^3*p^2*t^2 -t010*t020*q^2)/q/t/(t*q-t+q-q^2));\n" }{MPLTEXT 1 0 1256 "at:=factor( (q^4*tinfty10*tinfty20+q^4*h*tinfty10-q^3*tinfty10*tinfty20-q^3*h*tinf ty10+2*q^2*t*tinfty10*tinfty20+2*q^2*t*h*tinfty10-2*t*q^3*tinfty10*tin fty20-2*t*q^3*h*tinfty10-t^2*q*tinfty10*tinfty20-t^2*q*h*tinfty10+q^2* t^2*tinfty10*tinfty20+q^2*t^2*h*tinfty10+2*t*q^4*p*t110+2*t*q^4*t120*p -2*tt20*tt10*q^2*t+q*t120*t110*t^2-t^2*tt20*tt10*q^2+2*tt20*tt10*q^3*t +tt20*tt10*q^2-2*q^5*p^2*t-2*q^5*p^2+q^6*p^2+q^4*p^2+4*q^4*p^2*t-2*q^3 *p^2*t+h*q^5*p-p*t010*q^3+2*p*t010*q^4-p*t020*q^3+2*p*t020*q^4-2*h*q^4 *p+h*q^3*p+q^4*p^2*t^2-t^2*t010*t020*q-h*q^4*p*t+t010*t020*t^2+2*q^2*p *t020*t^2+2*q^2*p*t010*t^2+q^2*p^2*t^2-q*p*t020*t^2-q*p*t010*t^2+t120* t110*q^3-tt20*tt10*q^3-t010*t020*q^3+2*q^4*p*tt10+2*q^4*p*tt20+q^4*p*t 110+q^4*t120*p-q^5*p*t110-q^5*t120*p-2*p*t110*t*q^3-p*t110*t^2*q^3-2*t 120*p*t*q^3-t120*p*t^2*q^3-2*p*tt10*t*q^3-4*p*t020*t*q^3-p*t020*t^2*q^ 3-p*t010*q^5-p*t020*q^5-q^5*p*tt10-q^5*p*tt20-2*t120*t110*t*q^2+2*t010 *t020*t*q^2+2*q^4*p*t010*t+q^4*p*tt10*t+q^4*p*tt20*t+2*q^4*p*t020*t-p* tt10*q^3-2*t010*t020*t*q-q^2*p*h*t+2*q^3*p*h*t+2*q^2*p*t010*t+q^2*p*tt 10*t+q^2*p*tt20*t+2*q^2*p*t020*t+t^2*tt20*tt10*q-p*tt20*q^3-2*p*tt20*t *q^3+q^2*p*t110*t^2+q^2*t120*p*t^2-4*p*t010*t*q^3-p*t010*t^2*q^3-2*q^3 *p^2*t^2+t010*t020*q^2)/(t^2*q-t^2-q^2*t+t-q+q^2)/t/q);" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I#a1G6\"*,,du*&I\"pGF$\"\"#I\"qGF$\"\"'\"\"\"*(F(F )F*\"\"&I\"tGF$F,!\"#*(F(F)F*\"\"%F/F)F,*(I\"hGF$F,F(F,F*F.F,**F4F,F(F ,F*F2F/F,F0**F4F,F(F,F*\"\"$F/F)F,*&F(F)F*F.F0*(F(F)F*F2F/F,F2*(F(F)F* F7F/F)F0*(F(F,F*F.I%t010GF$F,!\"\"*(F(F,F*F.I%t020GF$F,F=*(F(F,F*F.I%t 110GF$F,F=*(F(F,F*F.I%t120GF$F,F=*(F(F,F*F.I%tt10GF$F,F=*(F(F,F*F.I%tt 20GF$F,F=**F(F,F*F2F/F,FI#a0G6\",$*,,du*&I\"pGF$\"\"#I\"qGF$\"\"'\"\"\"*(F)F*F+\"\"&I\"tGF$F- !\"#*(F)F*F+\"\"%F0F*F-*(I\"hGF$F-F)F-F+F/F-**F5F-F)F-F+F3F0F-F1**F5F- F)F-F+\"\"$F0F*F-*&F)F*F+F/F1*(F)F*F+F3F0F-F3*(F)F*F+F8F0F*F1*(F)F-F+F /I%t010GF$F-!\"\"*(F)F-F+F/I%t020GF$F-F>*(F)F-F+F/I%t110GF$F-F>*(F)F-F +F/I%t120GF$F-F>*(F)F-F+F/I%tt10GF$F-F>*(F)F-F+F/I%tt20GF$F-F>**F)F-F+ F3F0F-F=F-F***F)F-F+F3F0F-F@F-F***F)F-F+F3F0F-FBF-F***F)F-F+F3F0F-FDF- F***F)F-F+F3F0F-FFF-F-**F)F-F+F3F0F-FHF-F-**F)F-F+F8F0F*F=F-F>**F)F-F+ F8F0F*F@F-F>**F)F-F+F8F0F*FBF-F>**F)F-F+F8F0F*FDF-F>*(F5F-F)F-F+F3F1** F5F-F)F-F+F8F0F-F3**F5F-F)F-F+F*F0F*F1*(F5F-F+F3I)tinfty10GF$F-F-**F5F -F+F8F0F-FWF-F>*&F)F*F+F3F-*(F)F*F+F8F0F-F1*(F)F*F+F*F0F*F-*(F)F-F+F3F =F-F**(F)F-F+F3F@F-F**(F)F-F+F3FBF-F-*(F)F-F+F3FDF-F-*(F)F-F+F3FFF-F** (F)F-F+F3FHF-F***F)F-F+F8F0F-F=F-!\"%**F)F-F+F8F0F-F@F-F]o**F)F-F+F8F0 F-FBF-F1**F)F-F+F8F0F-FDF-F1**F)F-F+F8F0F-FFF-F1**F)F-F+F8F0F-FHF-F1** F)F-F+F*F0F*F=F-F***F)F-F+F*F0F*F@F-F***F)F-F+F*F0F*FBF-F-**F)F-F+F*F0 F*FDF-F-*(F+F3FWF-I)tinfty20GF$F-F-**F+F8F0F-F=F-F@F-F>**F+F8F0F-FBF-F DF-F>**F+F8F0F-FFF-FHF-F-**F+F8F0F-FWF-FhoF-F>**F+F*F0F*F=F-F@F-F-**F+ F*F0F*FBF-FDF-F-*(F5F-F)F-F+F8F-**F5F-F)F-F+F*F0F-F1**F5F-F)F-F+F-F0F* F-*(F5F-F+F8FWF-F>**F5F-F+F*F0F-FWF-F-*(F)F-F+F8F=F-F>*(F)F-F+F8F@F-F> *(F)F-F+F8FFF-F>*(F)F-F+F8FHF-F>**F)F-F+F*F0F-F=F-F***F)F-F+F*F0F-F@F- F***F)F-F+F*F0F-FFF-F-**F)F-F+F*F0F-FHF-F-**F)F-F+F-F0F*F=F-F>**F)F-F+ F-F0F*F@F-F>*(F+F8F=F-F@F-F>*(F+F8FBF-FDF-F-*(F+F8FFF-FHF-F>*(F+F8FWF- FhoF-F>**F+F*F0F-F=F-F@F-F8**F+F*F0F-FBF-FDF-F>**F+F*F0F-FFF-FHF-F>**F +F*F0F-FWF-FhoF-F-**F+F-F0F*F=F-F@F-F1*(F+F*F=F-F@F-F-*(F+F*FFF-FHF-F- **F+F-F0F-F=F-F@F-F1*(F0F*F=F-F@F-F-F-F+F>F0F>,&F+F-F0F>F>,&F+F-F>F-F> F>" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I#atG6\",$*.,du*&I\"pGF$\"\"#I\"q GF$\"\"'\"\"\"*(F)F*F+\"\"&I\"tGF$F-!\"#*(F)F*F+\"\"%F0F*F-*(I\"hGF$F- F)F-F+F/F-**F5F-F)F-F+F3F0F-!\"\"*&F)F*F+F/F1*(F)F*F+F3F0F-F3*(F)F*F+ \"\"$F0F*F1*(F)F-F+F/I%t010GF$F-F7*(F)F-F+F/I%t020GF$F-F7*(F)F-F+F/I%t 110GF$F-F7*(F)F-F+F/I%t120GF$F-F7*(F)F-F+F/I%tt10GF$F-F7*(F)F-F+F/I%tt 20GF$F-F7**F)F-F+F3F0F-F=F-F***F)F-F+F3F0F-F?F-F***F)F-F+F3F0F-FAF-F** *F)F-F+F3F0F-FCF-F***F)F-F+F3F0F-FEF-F-**F)F-F+F3F0F-FGF-F-**F)F-F+F;F 0F*F=F-F7**F)F-F+F;F0F*F?F-F7**F)F-F+F;F0F*FAF-F7**F)F-F+F;F0F*FCF-F7* (F5F-F)F-F+F3F1**F5F-F)F-F+F;F0F-F**(F5F-F+F3I)tinfty10GF$F-F-**F5F-F+ F;F0F-FUF-F1**F5F-F+F*F0F*FUF-F-*&F)F*F+F3F-*(F)F*F+F;F0F-F1*(F)F*F+F* F0F*F-*(F)F-F+F3F=F-F**(F)F-F+F3F?F-F**(F)F-F+F3FAF-F-*(F)F-F+F3FCF-F- *(F)F-F+F3FEF-F**(F)F-F+F3FGF-F***F)F-F+F;F0F-F=F-!\"%**F)F-F+F;F0F-F? F-F\\o**F)F-F+F;F0F-FAF-F1**F)F-F+F;F0F-FCF-F1**F)F-F+F;F0F-FEF-F1**F) F-F+F;F0F-FGF-F1**F)F-F+F*F0F*F=F-F***F)F-F+F*F0F*F?F-F***F)F-F+F*F0F* FAF-F-**F)F-F+F*F0F*FCF-F-*(F+F3FUF-I)tinfty20GF$F-F-**F+F;F0F-FEF-FGF -F***F+F;F0F-FUF-FgoF-F1**F+F*F0F*FEF-FGF-F7**F+F*F0F*FUF-FgoF-F-*(F5F -F)F-F+F;F-**F5F-F)F-F+F*F0F-F7*(F5F-F+F;FUF-F7**F5F-F+F*F0F-FUF-F***F 5F-F+F-F0F*FUF-F7*(F)F-F+F;F=F-F7*(F)F-F+F;F?F-F7*(F)F-F+F;FEF-F7*(F)F -F+F;FGF-F7**F)F-F+F*F0F-F=F-F***F)F-F+F*F0F-F?F-F***F)F-F+F*F0F-FEF-F -**F)F-F+F*F0F-FGF-F-**F)F-F+F-F0F*F=F-F7**F)F-F+F-F0F*F?F-F7*(F+F;F=F -F?F-F7*(F+F;FAF-FCF-F-*(F+F;FEF-FGF-F7*(F+F;FUF-FgoF-F7**F+F*F0F-F=F- F?F-F***F+F*F0F-FAF-FCF-F1**F+F*F0F-FEF-FGF-F1**F+F*F0F-FUF-FgoF-F***F +F-F0F*F=F-F?F-F7**F+F-F0F*FAF-FCF-F-**F+F-F0F*FEF-FGF-F-**F+F-F0F*FUF -FgoF-F7*(F+F*F=F-F?F-F-*(F+F*FEF-FGF-F-**F+F-F0F-F=F-F?F-F1*(F0F*F=F- F?F-F-F-F0F7,&F0F-F7F-F7,&F+F-F0F7F7F+F7,&F+F-F7F-F7F7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "simplify(C01-C01bis);\n" }{MPLTEXT 1 0 22 "simplify(C11-C11bis);\n" }{MPLTEXT 1 0 21 "simplify(Ct1-Ct1bis );" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "a0bis:=-q*(q-1)*(q-t)/t*p^2-p*(q-1) *(q-t)/t*h-q*tinfty10/t*h\n" }{MPLTEXT 1 0 77 "-p/t*( (tinfty10+tinfty 20)*q*(q-1)+t*(t010+t020)*(q-1)+(t110+t120)*(t-1)*q )\n" }{MPLTEXT 1 0 147 "-t010*t020/q+t110*t120*(t-1)/t/(q-1)-tt10*tt20*(t-1)/(q-t)-tinf ty10*tinfty20/t*q+(t010*t020-t120*t110+tt20*tt10)/t+(t010*t020+t120*t1 10-tt20*tt10)\n" }{MPLTEXT 1 0 32 "+p*q*(q-1)/t*CoherenceEquation1\n" }{MPLTEXT 1 0 2 ":\n" }{MPLTEXT 1 0 17 "factor(a0-a0bis);" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "a1bis:=q*(q-1)*(q-t)/(t-1)*p^2+p*q*(q-t)/(t-1)*h+(q-1)*tinfty10/(t -1)*h\n" }{MPLTEXT 1 0 81 "+p/(t-1)*( (tinfty10+tinfty20)*q*(q-1)+(t-1 )*(t110+t120)*q+(t010+t020)*t*(q-1)) \n" }{MPLTEXT 1 0 156 "+t010*t020 *t/(t-1)/q-t110*t120/(q-1)+tt10*tt20*t/(q-t)+tinfty10*tinfty20/(t-1)*( q-1)-(t010*t020-t110*t120-tt10*tt20)/(t-1)+(-t010*t020-t120*t110+tt20* tt10)\n" }{MPLTEXT 1 0 36 "-p*q*(q-1)/(t-1)*CoherenceEquation1\n" } {MPLTEXT 1 0 2 ":\n" }{MPLTEXT 1 0 17 "factor(a1-a1bis);" }}{PARA 11 " " 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "atbis:=-q*(q-1)*(q-t)/t/(t-1)*p^2-p*q*(q-1)/t/(t-1)*h-tinfty10*(q- t)/t/(t-1)*h\n" }{MPLTEXT 1 0 83 "-p/t/(t-1)*( (tinfty10+tinfty20)*q*( q-1)+(t-1)*(t110+t120)*q+(t010+t020)*t*(q-1) )\n" }{MPLTEXT 1 0 157 "- t010*t020/(t-1)/q+t110*t120/t/(q-1)-tt10*tt20/(q-t)-tinfty10*tinfty20/ t/(t-1)*(q-t)-(t010*t020-t120*t110+tt20*tt10)/t+(t010*t020-t120*t110-t t20*tt10)/(t-1)\n" }{MPLTEXT 1 0 39 "+p*q*(q-1)/t/(t-1)*CoherenceEquat ion1 \n" }{MPLTEXT 1 0 2 ":\n" }{MPLTEXT 1 0 17 "factor(at-atbis);" }} {PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "simplify(CoherenceEquation4);\n" }{MPLTEXT 1 0 30 "si mplify(CoherenceEquation5);\n" }{MPLTEXT 1 0 21 "simplify(Equation1); \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 44 "Lp:=factor(simplif y(LpFunction(a0,a1,at))):\n" }{MPLTEXT 1 0 42 "Lpbis:=-diff(q*(q-1)*(q -t),q)/t/(t-1)*p^2\n" }{MPLTEXT 1 0 107 "+((t110+tt10+tt20+t120)*q+(t0 10+t020+tt10+tt20)*(q-1)+(t020+t110+t120+t010)*(q-t)-(2*q-1)*h )* p/ t/(t-1)\n" }{MPLTEXT 1 0 85 "+t010*t020/(t-1)/q^2-t110*t120/t/(q-1)^2+ tt10*tt20/(q-t)^2-tinfty10*tinfty20/t/(t-1)\n" }{MPLTEXT 1 0 21 "-tinf ty10/t/(t-1)*h:\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 42 "Lpter:=-diff (q*(q-1)*(q-t),q)/t/(t-1)*p^2\n" }{MPLTEXT 1 0 77 "+((P111+Pt11)*q+(P0 11+Pt11)*(q-1)+(P011+P111)*(q-t)-(2*q-1)*h )* p/t/(t-1)\n" }{MPLTEXT 1 0 81 "+P022/(t-1)/q^2-P122/t/(q-1)^2+Pt22/(q-t)^2-(Pt22+Pt12*t+P022 +P112+P122)/t/(t-1)\n" }{MPLTEXT 1 0 20 "-tinfty10/t/(t-1)*h\n" } {MPLTEXT 1 0 28 "+CoherenceEquation3/t/(t-1)\n" }{MPLTEXT 1 0 2 ":\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 18 "factor(Lp-Lpbis);\n" }{MPLTEXT 1 0 17 "factor(Lp-Lpter);" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "Lqquater:=2*mu*(p-P1(q)/2+h/2/q+h/2/(q-1)+" }{MPLTEXT 1 0 9 "h/2/(q-t)" }{MPLTEXT 1 0 22 ")-h*nu0 -h*nuMinus1*q:" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 97 "Lpquater:=mu*(p*diff(P1(q),q)+h*p*(1/q^2 +1/(q-1)^2+1/(q-t)^2)-diff(tdP2(q),q)-C01/q^2-C11/(q-1)^2" }{MPLTEXT 1 0 12 "-Ct1/(q-t)^2" }{MPLTEXT 1 0 3 " )\n" }{MPLTEXT 1 0 15 "+h*nuMi nus1*p:\n" }{MPLTEXT 1 0 23 "simplify(Lq-Lqquater);\n" }{MPLTEXT 1 0 30 "factor(simplify(Lp-Lpquater));" }}{PARA 11 "" 1 "" {XPPMATH 20 "\" \"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 225 57 "We thus obtain the evolution of the Darboux coordinates: \n" }{TEXT 224 91 "\\hbar \\partial_t q= 2*q*(q-1)*(q-t)/t/(t-1)*p-q*( q-1)*(q-t)/t/(t-1)*P1(q)+q*(q-1)/t/(t-1)*h" }}{PARA 209 "" 0 "" {TEXT 215 56 "\\hbar \\partial_t p = -diff(q*(q-1)*(q-t),q)/t/(t-1)*p^2\n" } {TEXT 215 107 "+((t110+tt10+tt20+t120)*q+(t010+t020+tt10+tt20)*(q-1)+( t020+t110+t120+t010)*(q-t)-(2*q-1)*h )* p/t/(t-1)\n" }{TEXT 215 85 " +t010*t020/(t-1)/q^2-t110*t120/t/(q-1)^2+tt10*tt20/(q-t)^2-tinfty10*ti nfty20/t/(t-1)\n" }{TEXT 215 20 "-tinfty10/t/(t-1)*h:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "Hamiltonian:= q*(q-1)*(q-t)/t/(t-1) *p^2-q*(q-1)*(q-t)/t/(t-1)*P1(q)*p+q*(q-1)/t/(t-1)*h*p\n" }{MPLTEXT 1 0 51 "+q*(q-1)*(q-t)*P2(q)/t/(t-1)+h*tinfty10/t/(t-1)*q;\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 42 "factor(simplify(diff(Hamiltonian,p)-Lq)); \n" }{MPLTEXT 1 0 104 "factor(simplify(diff(Hamiltonian,q)+Lp -Coheren ceEquation2*(2*q-t-1)/t/(t-1)-CoherenceEquation3/t/(t-1)\n" }{MPLTEXT 1 0 4 "));\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 94 "Hamiltonianbis:= \+ mu*(p^2-P1(q)*p+h*p*(1/q+1/(q-1)+1/(q-t)) +tdP2(q) )-h*nu0*p-h*nuMinus 1*q*p \n" }{MPLTEXT 1 0 66 "-nuMinus1*(t010*t020+t110*t120+tt10*tt20-t infty10*(tinfty20+h)) ;\n" }{MPLTEXT 1 0 60 "factor(series(simplify(Lp -(-diff(Hamiltonianbis,q))),h=0));\n" }{MPLTEXT 1 0 39 "simplify(Lq-(d iff(Hamiltonianbis,p)));\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 30 "sim plify(Hamiltonianbis-Ct1);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I,Hamil tonianG6\",,*.I\"qGF$\"\"\",&F'F(!\"\"F(F(,&F'F(I\"tGF$F*F(F,F*,&F,F(F *F(F*I\"pGF$\"\"#F(*0F'F(F)F(F+F(F,F*F-F*,(*&,&I%t010GF$F(I%t020GF$F(F (F'F*F(*&,&I%t110GF$F(I%t120GF$F(F(F)F*F(*&,&I%tt10GF$F(I%tt20GF$F(F(F +F*F(F(F.F(F**.F.F(F'F(F)F(F,F*F-F*I\"hGF$F(F(*.F'F(F)F(F+F(,.*(F4F(F5 F(F'!\"#F(*&I%P012GF$F(F'F*F(*(F9F(F8F(F)FCF(*&I%P112GF$F(F)F*F(*(F=F( FI/HamiltonianbisG 6\",**(,&I\"qGF$\"\"\"I\"tGF$!\"\"F),***F'F)F*F+,&F*F)F+F)F+F(F)F)*&F' F)F.F+F+**F'F),&F*\"\"#F+F)F)F*F+F.F+F)F)F)F),.*$I\"pGF$F2F)*&,(*&,&I% t010GF$F)I%t020GF$F)F)F(F+F)*&,&I%t110GF$F)I%t120GF$F)F),&F(F)F+F)F+F) *&,&I%tt10GF$F)I%tt20GF$F)F)F'F+F)F)F5F)F+*(I\"hGF$F)F5F),(*$F(F+F)*$F @F+F)*$F'F+F)F)F)*(F:F)F;F)F(!\"#F)*(F?F)F>F)F@FLF)*(FDF)FCF)F'FLF)F)F )*(FFF),(F-F)F/FLF0F)F)F5F)F+*.FFF)F'F)F*F+F.F+F(F)F5F)F+**F'F)F*F+F.F +,**&F:F)F;F)F)*&F?F)F>F)F)*&FDF)FCF)F)*&I)tinfty10GF$F),&FFF)I)tinfty 20GF$F)F)F+F)F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 202 "simplify(series(a0bis-(-(t- 1)*Hamiltonian+P012+p*(q-1)*h)-q*(q-t)/t*CoherenceEquation3-(q-1)*(q-t )/t*CoherenceEquation2+(q-1)*(q-t)/t*CoherenceEquation3-CoherenceEquat ion3+CoherenceEquation2,P112=0) );\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 172 "simplify(series(a1bis-(t*Hamiltonian+P112-p*q*h-h*tinfty10/(t -1))+CoherenceEquation2*q^2/(t-1)-(t*q+q-t)/(t-1)*CoherenceEquation2+( q-1)/(t-1)*CoherenceEquation3,P112=0) );\n" }{MPLTEXT 1 0 1 "\n" } {MPLTEXT 1 0 170 "simplify(series(atbis-(-Hamiltonian+Pt12+h*tinfty10/ (t-1))-CoherenceEquation2*q^2/t/(t-1)+(t*q+q-t)/t/(t-1)*CoherenceEquat ion2-(q-t)/t/(t-1)*CoherenceEquation3,P112=0) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "nuMinus1;\n" }{MPLTEXT 1 0 13 "factor(nu0);\n" }{MPLTEXT 1 0 31 "s implify((0*C01+1*C11+t*Ct1));\n" }{MPLTEXT 1 0 25 "simplify((C01+C11+C t1));\n" }{MPLTEXT 1 0 63 "factor(series(simplify((C01/(q-0)+C11/(q-1) +Ct1/(q-t))),p=0)):\n" }{MPLTEXT 1 0 65 "Quantity:=p^2-P1(q)*p+p*(h*1/ (q-0)+h*1/(q-1)+h*1/(q-t))+tdP2(q):\n" }{MPLTEXT 1 0 52 "simplify((C01 /(q-0)+C11/(q-1)+Ct1/(q-t))-Quantity);\n" }{MPLTEXT 1 0 20 "CoherenceE quation3;\n" }{MPLTEXT 1 0 20 "CoherenceEquation2;\n" }{MPLTEXT 1 0 20 "CoherenceEquation1;\n" }{MPLTEXT 1 0 30 "simplify(CoherenceEquatio n4);\n" }{MPLTEXT 1 0 29 "simplify(CoherenceEquation5);" }{MPLTEXT 1 0 1 "\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "*(,&I\"qG6\"\"\"\"I\"tGF%!\" \"F&F'F(,&F'F&F(F&F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "**,&I\"qG6\"\"\" \"!\"\"F&F&,&F$F&I\"tGF%F'F&F)F',&F)F&F'F&F'" }}{PARA 11 "" 1 "" {XPPMATH 20 ",,*&,&I\"hG6\"!\"\"I)tinfty20GF&F'\"\"\"I)tinfty10GF&F)F) *(F%F)I\"pGF&F)I\"qGF&F)F)*&I%t010GF&F)I%t020GF&F)F)*&I%t120GF&F)I%t11 0GF&F)F)*&I%tt20GF&F)I%tt10GF&F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "*& I\"hG6\"\"\"\"I\"pGF$F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 ",.*&I%Pt12G6\"\"\"\"I\"tGF%F&F&*&I%t010G F%F&I%t020GF%F&F&*&I%t110GF%F&I%t120GF%F&F&*&I%tt10GF%F&I%tt20GF%F&F&* &I)tinfty10GF%F&I)tinfty20GF%F&!\"\"I%P112GF%F&" }}{PARA 11 "" 1 "" {XPPMATH 20 ",(I%Pt12G6\"\"\"\"I%P012GF$F%I%P112GF$F%" }}{PARA 11 "" 1 "" {XPPMATH 20 ",2I)tinfty10G6\"\"\"\"I%tt10GF$F%I%tt20GF$F%I%t010GF$ F%I%t020GF$F%I)tinfty20GF$F%I%t110GF$F%I%t120GF$F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 226 85 "Computation of the Lax matrices in the geometric gauge without apparent singularities" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "G1:=Matrix(2,2,0):\n" }{MPLTEXT 1 0 12 "G1[1,1] :=1:\n" }{MPLTEXT 1 0 12 "G1[1,2]:=0:\n" }{MPLTEXT 1 0 9 "G1[2,1]:=" } {MPLTEXT 1 0 8 "tinfty10" }{MPLTEXT 1 0 14 "*lambda+eta0:\n" }{MPLTEXT 1 0 12 "G1[2,2]:=1:\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 27 "dG1dlam bda:=Matrix(2,2,0):\n" }{MPLTEXT 1 0 90 "for i from 1 to 2 do for j fr om 1 to 2 do dG1dlambda[i,j]:=diff(G1[i,j],lambda): od: od:\n" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 22 "dG1dt:=Matrix(2,2,0):\n" } {MPLTEXT 1 0 122 "for i from 1 to 2 do for j from 1 to 2 do dG1dt[i,j] :=diff(G1[i,j],t)+diff(G1[i,j],q)*dqdt+diff(G1[i,j],p)*dpdt: od: od: \n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 87 "Ltilde:=simplify(Multiply(M ultiply(G1,checkL),G1^(-1))+h*Multiply(dG1dlambda,G1^(-1))):" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 73 "eta0:=facto r(solve(factor(-residue(Ltilde[2,1],lambda=infinity)),eta0)):\n" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 34 "eta0theo:=1/(tinfty10-tinfty20)* (\n" }{MPLTEXT 1 0 106 "-(2*0*P022+2*1*P122+2*t*Pt22) +(0^2*C01+1^2*C1 1+t^2*Ct1) -h*p*q^2 -tinfty10*(0*(t010+t020)+1*(t110+t120)+t" } {MPLTEXT 1 0 12 "*(tt10+tt20)" }{MPLTEXT 1 0 2 ")\n" }{MPLTEXT 1 0 49 "+tinfty10*(tinfty10-tinfty20-h)*(q-1*0-1*1-t*1)):" }{MPLTEXT 1 0 1 " \n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 136 "factor(simplify(eta0-eta0t heo+eta0theo*((tinfty10+tt10+tt20+t010+t020+tinfty20+t110+t120)/(t010+ t020+t110+t120+tt10+tt20+2*tinfty10))));" }{MPLTEXT 1 0 1 "\n" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 82 "factor(-residue(Ltilde[2,1]/lamb da,lambda=infinity)+tinfty10*CoherenceEquation1);\n" }{MPLTEXT 1 0 46 "factor(-residue(Ltilde[2,1],lambda=infinity));" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 225 51 "Ex pression of the Lax matrix in the geometric gauge" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 7 "checkL;" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 23 "simplify(checkL[1,1]);\n" }{MPLTEXT 1 0 23 "simplify(checkL[1,2]); \n" }{MPLTEXT 1 0 23 "simplify(checkL[2,2]):\n" }{MPLTEXT 1 0 152 "che ckL22bis:=((t010+t020)- q*(q-1)*(q-t)*p/t)/lambda + (t110+t120+q*(q-1) *(q-t)/(t-1)*p )/(lambda-1)+(tt10+tt20+(-q*(q-1)*(q-t)/(t-1)/t)*p)/(la mbda-t):\n" }{MPLTEXT 1 0 33 "factor(checkL[2,2]-checkL22bis);\n" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 86 "checkL21bis:=(t010*t020*t/q-(t01 0+t020)*(q-1)*(q-t)*p+q*(q-1)^2*(q-t)^2/t*p^2)/lambda\n" }{MPLTEXT 1 0 86 "-(t120*t110*(t-1)/(q-1)+(t110+t120)*q*(q-t)*p+(q-t)^2*(q-1)*q^2/ (t-1)*p^2)/(lambda-1)\n" }{MPLTEXT 1 0 90 "+(tt10*tt20*t*(t-1)/(q-t)-( tt10+tt20)*q*(q-1)*p+(q-1)^2*(q-t)*q^2/(t-1)/t*p^2)/(lambda-t)\n" } {MPLTEXT 1 0 31 "-tinfty10*tinfty20 -h*tinfty10\n" }{MPLTEXT 1 0 2 ": \n" }{MPLTEXT 1 0 34 "factor((checkL[2,1]-checkL21bis));" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 54 "factor(-residue(checkL [2,1]/lambda,lambda=infinity));\n" }{MPLTEXT 1 0 47 "factor(-residue(c heckL[2,1],lambda=infinity)):\n" }{MPLTEXT 1 0 60 "factor(residue(1/2* Trace(checkL^2)*(lambda-1)^0,lambda=1)):\n" }{MPLTEXT 1 0 66 "simplify (residue(1/2*Trace(checkL^2)*(lambda-1)^0,lambda=1)- C11);" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 59 "checkL11bis:=-Q2(lam bda)/(lambda-0)/(lambda-1)/(lambda-t);\n" }{MPLTEXT 1 0 58 "checkL12bi s:=(lambda-q)/(lambda-0)/(lambda-1)/(lambda-t);\n" }{MPLTEXT 1 0 69 "c heckL22bis:=P1(lambda)+Q2(lambda)/(lambda-0)/(lambda-1)/(lambda-t);\n" }{MPLTEXT 1 0 134 "checkL21bis:=h*diff(Q2(lambda)/(lambda-q),lambda)+ L[2,1]*(lambda-0)*(lambda-1)*(lambda-t)/(lambda-q)-P1(lambda)*Q2(lambd a)/(lambda-q)\n" }{MPLTEXT 1 0 59 "-Q2(lambda)^2/(lambda-q)/(lambda-0) /(lambda-1)/(lambda-t):\n" }{MPLTEXT 1 0 35 "simplify(checkL[1,1]-chec kL11bis);\n" }{MPLTEXT 1 0 35 "simplify(checkL[1,2]-checkL12bis);\n" } {MPLTEXT 1 0 35 "simplify(checkL[2,2]-checkL22bis);\n" }{MPLTEXT 1 0 34 "simplify(checkL[2,1]-checkL21bis);" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I(mfencedG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6'-I%mrowGF$ 6#-I'mtableGF$66-I$mtrGF$6'-I$mtdGF$6(-I&mfracGF$6)-F,6)-I#miGF$6&Q\"p F'/%'italicGQ%trueF'/%+foregroundGQ([0,0,0]F'/%,mathvariantGQ'italicF' -I#moGF$6-Q1⁢F'/FGQ'normalF'/%&fenceGQ&falseF'/%*separa torGFQ/%)stretchyGFQ/%*symmetricGFQ/%(largeopGFQ/%.movablelimitsGFQ/%' accentGFQ/%'lspaceGQ&0.0emF'/%'rspaceGFjn-F=6&Q\"qF'F@FCFFFI-F#6%-F,6% F]o-FJ6.Q(−F'FCFMFOFRFTFVFXFZFfn/FinQ,0.2222222emF'/F\\oFho-I#mn GF$6%Q\"1F'FCFMFCFMFI-F#6%-F,6%F]oFdo-F=6&Q\"tF'F@FCFFFCFM-F,6'-F=6&Q) λF'/FAFQFCFMFI-F#6%-F,6%FgpFdoFjoFCFMFI-F#6%-F,6%FgpFdoFbpFCFM/ %.linethicknessGF]p/%+denomalignGQ'centerF'/%)numalignGFgq/%)bevelledG FQFC/%)rowalignGQ!F'/%,columnalignGF^r/%+groupalignGF^r/%(rowspanGF]p/ %+columnspanGF]p-F56(-F86)-F,6#-F,6&-FJ6.Q*&uminus0;F'FCFMFOFRFTFVFXFZ FfnFgoFioFgp-FJ6.Q\"+F'FCFMFOFRFTFVFXFZFfnFgoFioF]o-F,6'FgpFIF[qFI-F#6 %-F,6&F_sFgpFbsFbpFCFMFcqFeqFhqFjqFCF\\rF_rFarFcrFerF\\rF_rFar-F26'-F5 6(-F86)-F,6#-F,6--F,6%-F#6%-F,6*F_s-F,6%-F=6&Q\"hF'F@FCFFFIFI,checkL11bisG6\"*0I\"pGF$\"\"\"I\"qGF$ F',&F(F'!\"\"F'F',&F(F'I\"tGF$F*F'I'lambdaGF$F*,&F-F'F*F'F*,&F-F'F,F*F *" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I,checkL12bisG6\"**,&I'lambdaGF$\" \"\"I\"qGF$!\"\"F(F'F*,&F'F(F*F(F*,&F'F(I\"tGF$F*F*" }}{PARA 11 "" 1 " " {XPPMATH 20 ">I,checkL22bisG6\",**&,&I%t010GF$\"\"\"I%t020GF$F)F)I'l ambdaGF$!\"\"F)*&,&I%t110GF$F)I%t120GF$F)F),&F+F)F,F)F,F)*&,&I%tt10GF$ F)I%tt20GF$F)F),&F+F)I\"tGF$F,F,F)*0I\"pGF$F)I\"qGF$F),&F:F)F,F)F),&F: F)F7F,F)F+F,F1F,F6F,F," }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 " \"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "C:=0:\n" }{MPLTEXT 1 0 31 "factor(simplify(checkA[ 1,1])):\n" }{MPLTEXT 1 0 51 "checkA11bis:=C-q*(q-1)*(q-t)/t/(t-1)*p/(l ambda-t);\n" }{MPLTEXT 1 0 35 "simplify(checkA[1,1]-checkA11bis);\n" } {MPLTEXT 1 0 36 "checkA[1,2]:=simplify(checkA[1,2]);\n" }{MPLTEXT 1 0 42 "factor(simplify(checkA[2,2]+checkA[1,1])):" }{MPLTEXT 1 0 1 "\n" } {MPLTEXT 1 0 117 "checkA22bis:=C +(-(tt10+tt20)+q*(q-1)*(q-t)/t/(t-1)* p)/(lambda-t)-(q-t)/t/(t-1)*h-(tinfty10+tinfty20)*(q-t)/t/(t-1);\n" } {MPLTEXT 1 0 76 "factor(simplify(checkA[2,2]-checkA22bis-(q-t)/t/(t-1) *CoherenceEquation1));\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 32 "check A21:=simplify(checkA[2,1]):" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 103 "c heckA21bis:=(-tt10*tt20*t*(t-1)/(q-t)+(q-1)*q*(tt10+tt20)*p-(q-1)^2*(q -t)*q^2/(t-1)/t*p^2)/(lambda-t)\n" }{MPLTEXT 1 0 40 "-tinfty10*tinfty2 0*(q-t)/t/(t-1)*lambda\n" }{MPLTEXT 1 0 29 "+(q-1)*q*(q-t)^2/t/(t-1)*p ^2\n" }{MPLTEXT 1 0 97 "+(q-t)/t/(t-1)*((tinfty10+tinfty20)*q*(q-1)+(( t110+t120)*(t-1)+t*(t010+t020))*q-(t010+t020)*t)*p\n" }{MPLTEXT 1 0 143 "-t/(t-1)*t010*t020/q+(t-1)/t*t110*t120/(q-1)-tinfty10*tinfty20/t/ (t-1)*(q-t)^2-tinfty10*tinfty20/t*(q-t)+tt10*tt20-t110*t120/t+t010*t02 0/(t-1)\n" }{MPLTEXT 1 0 70 "-(q-t)*tinfty10*lambda/t/(t-1)*h+(q-t)*(q -1)/t/(t-1)*(p*q-tinfty10)*h\n" }{MPLTEXT 1 0 1 ";" }{MPLTEXT 1 0 1 " \n" }{MPLTEXT 1 0 72 "factor(checkA21-checkA21bis+(q-1)*(q-t)/t/(t-1)* q*p*CoherenceEquation1);" }{MPLTEXT 1 0 1 "\n" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I,checkA11bisG6\",$*0I\"pGF$\"\"\"I\"qGF$F(,&F)F(!\"\"F( F(,&F)F(I\"tGF$F+F(,&F-F(F+F(F+F-F+,&I'lambdaGF$F(F-F+F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I'check AG6\"6$\"\"\"\"\"#**,&I\"qGF%F'I\"tGF%!\"\"F',&F,F'F-F'F-F,F-,&I'lambd aGF%F'F,F-F-" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I,checkA22bisG6\",(*&,( I%tt10GF$!\"\"I%tt20GF$F)*.I\"qGF$\"\"\"I\"pGF$F-,&F,F-F)F-F-,&F,F-I\" tGF$F)F-F1F),&F1F-F)F-F)F-F-,&I'lambdaGF$F-F1F)F)F-**F0F-F1F)F2F)I\"hG F$F-F)**,&I)tinfty10GF$F-I)tinfty20GF$F-F-F0F-F1F)F2F)F)" }}{PARA 11 " " 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I,checkA21 bisG6\",<*&,(*,I%tt10GF$\"\"\"I%tt20GF$F*I\"tGF$F*,&F,F*!\"\"F*F*,&I\" qGF$F*F,F.F.F.**,&F)F*F+F*F*F0F*,&F0F*F.F*F*I\"pGF$F*F**.F3\"\"#F/F*F0 F6F-F.F,F.F4F6F.F*,&I'lambdaGF$F*F,F.F.F**.I)tinfty10GF$F*I)tinfty20GF $F*F/F*F,F.F-F.F8F*F.*.F3F*F0F*F/F6F,F.F-F.F4F6F**,F/F*F,F.F-F.,(*(,&F :F*F;F*F*F0F*F3F*F**&,&*&,&I%t110GF$F*I%t120GF$F*F*F-F*F**&,&I%t010GF$ F*I%t020GF$F*F*F,F*F*F*F0F*F*FGF.F*F4F*F**,FIF*FJF*F,F*F-F.F0F.F.*,FEF *FFF*F-F*F,F.F3F.F**,F:F*F;F*F,F.F-F.F/F6F.**F:F*F;F*F,F.F/F*F.*&F+F*F )F*F**(FEF*FFF*F,F.F.*(FIF*FJF*F-F.F**.F/F*F:F*F8F*F,F.F-F.I\"hGF$F*F. *.F/F*F3F*F,F.F-F.,&*&F4F*F0F*F*F:F.F*FSF*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "facto r(-residue(checkA[2,2]/lambda^2,lambda=infinity));\n" }{MPLTEXT 1 0 53 "factor(-residue(checkA21/lambda^2,lambda=infinity));\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 58 "factor(-residue((checkA[1,1]-C)/lambda,la mbda=infinity));\n" }{MPLTEXT 1 0 54 "factor(-residue(checkA[1,2]/lamb da,lambda=infinity));\n" }{MPLTEXT 1 0 58 "factor(-residue((checkA[2,2 ]-C)/lambda,lambda=infinity));\n" }{MPLTEXT 1 0 74 "ConstantCoeffcheck A21:=factor(-residue(checkA21/lambda,lambda=infinity)):\n" }{MPLTEXT 1 0 21 "ConstantCoeffcheckA21" }{MPLTEXT 1 0 33 "bis:=q*(q-1)*(q-t)^2/ t/(t-1)*p^2\n" }{MPLTEXT 1 0 97 "+p*(q-t)/t/(t-1)*((tinfty10+tinfty20) *q^2+(t010+t020)*(t*(q-1)+q)+(t110+t120)*t*q+(tt10+tt20)*q)\n" } {MPLTEXT 1 0 45 "-t010*t020*t/(t-1)/q+t110*t120*(t-1)/t/(q-1)\n" } {MPLTEXT 1 0 39 "-tinfty10*tinfty20*(q-t)*(q-1)/t/(t-1)\n" }{MPLTEXT 1 0 39 "+t010*t020/(t-1)-t110*t120/t+tt10*tt20\n" }{MPLTEXT 1 0 40 "-( q-1)*(q-t)*(-p*q+tinfty10)/(t-1)/t*h \n" }{MPLTEXT 1 0 39 "-CoherenceE quation1*p*q^2*(q-t)/t/(t-1)" }{MPLTEXT 1 0 2 ";\n" }{MPLTEXT 1 0 7 "f actor(" }{MPLTEXT 1 0 21 "ConstantCoeffcheckA21" }{MPLTEXT 1 0 1 "-" } {MPLTEXT 1 0 21 "ConstantCoeffcheckA21" }{MPLTEXT 1 0 5 "bis);" }} {PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ", $*,I)tinfty10G6\"\"\"\",&I\"qGF%F&I\"tGF%!\"\"F&,&I\"hGF%F&I)tinfty20G F%F&F&,&F)F&F*F&F*F)F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ", $**,0I\"hG6\"\"\"\"I%t110GF&!\"\"I%t120GF&F)I%tt10GF&F)I%tt20GF&F)I%t0 10GF&F)I%t020GF&F)F',&I\"qGF&F'I\"tGF&F)F',&F1F'F)F'F)F1F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I9ConstantCoeffcheckA21bisG6\",6*.,&I\"qGF$\" \"\"!\"\"F)F)F(F),&F(F)I\"tGF$F*\"\"#F,F*,&F,F)F*F)F*I\"pGF$F-F)*,F/F) F+F)F,F*F.F*,**&,&I)tinfty10GF$F)I)tinfty20GF$F)F)F(F-F)*&,&I%t010GF$F )I%t020GF$F)F),&*&F,F)F'F)F)F(F)F)F)*(,&I%t110GF$F)I%t120GF$F)F)F,F)F( F)F)*&,&I%tt10GF$F)I%tt20GF$F)F)F(F)F)F)F)*,F8F)F9F)F,F)F.F*F(F*F**,F> F)F?F)F.F)F,F*F'F*F)*.F4F)F5F)F+F)F'F)F,F*F.F*F**(F8F)F9F)F.F*F)*(F>F) F?F)F,F*F**&FCF)FBF)F)*.F'F)F+F),&*&F/F)F(F)F*F4F)F)F.F*F,F*I\"hGF$F)F **.,2F4F)FBF)FCF)F8F)F9F)F5F)F>F)F?F)F)F/F)F(F-F+F)F,F*F.F*F*" }} {PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 226 31 "Getting the Painlev\351 6 equation" }}{EXCHG }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "L2q:=diff(Lq,q)*Lq+diff(Lq,p)*Lp+ diff(Lq,t )*h:\n" }{MPLTEXT 1 0 8 "Painleve" }{MPLTEXT 1 0 56 "Equation:=simplif y( L2q- 1/2*(1/q+1/(q-1)+1/(q-t))*Lq^2\n" }{MPLTEXT 1 0 117 "+h*Lq*(1/ t+1/(t-1)+1/(q-t))-q*(q-1)*(q-t)/t^2/(t-1)^2*(alpha+beta*t/q^2+gammaa* (t-1)/(q-1)^2+delta*t*(t-1)/(q-t)^2)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "beta:=-1/2*(t010-t020)^2;\n" }{MPLTEXT 1 0 27 "gammaa :=1/2*(t110-t120)^2;\n" }{MPLTEXT 1 0 31 "delta:=-(tt10-tt20)^2/2+h^2/ 2;\n" }{MPLTEXT 1 0 36 "alpha:=1/2*(tinfty10-tinfty20-h)^2:\n" } {MPLTEXT 1 0 7 "factor(" }{MPLTEXT 1 0 8 "Painleve" }{MPLTEXT 1 0 106 "Equation+q*(q-1)*(q-t)/2/t^2/(t-1)^2*(tinfty20-t010-tt20-t120-tt10-t0 20-t110+tinfty10)*CoherenceEquation1\n" }{MPLTEXT 1 0 48 "+q*(q-1)*(q- t)*h/t^2/(t-1)^2*CoherenceEquation1\n" }{MPLTEXT 1 0 2 ");" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I%betaG6\",$*$,&I%t010GF$\"\"\"I%t020GF$!\"\"\" \"##F+F," }}{PARA 11 "" 1 "" {XPPMATH 20 ">I'gammaaG6\",$*$,&I%t110GF$ \"\"\"I%t120GF$!\"\"\"\"##F)F," }}{PARA 11 "" 1 "" {XPPMATH 20 ">I&del taG6\",&*$,&I%tt10GF$\"\"\"I%tt20GF$!\"\"\"\"##F+F,*$I\"hGF$F,#F)F," } }{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "RHSP6:=-q*(q-1)*(q-t)/t^2/(t-1)^2*(alpha+beta*t/q^2+g ammaa*(t-1)/(q-1)^2+delta*t*(t-1)/(q-t)^2);\n" }{MPLTEXT 1 0 81 "RHSP6 bis:=-2*q*(q-1)*(q-t)/t^2/(t-1)^2*diff(q*(q-1)*(q-t)*(P1(q)^2/4-P2(q)) ,q) \n" }{MPLTEXT 1 0 110 "+q*(q-1)*(q-t)*(tinfty10-tinfty20)/t^2/(t- 1)^2*h-q*(q-1)*(q-t)/2/t^2/(t-1)^2*h^2-q*(q-1)/(q-t)/2/t/(t-1)*h^2:\n" }{MPLTEXT 1 0 57 "factor(RHSP6-RHSP6bis+4*(Pt12+P012+P112)/t^2/(t-1)^ 2*q^4\n" }{MPLTEXT 1 0 157 "+(-2*(1+t)/t/(t-1)^2*CoherenceEquation2+2/ t/(t-1)^2*CoherenceEquation3+1/2*(tinfty20-t010-t020-t110-tt10-tt20-t1 20+tinfty10)/t/(t-1)^2*CoherenceEquation1)*q\n" }{MPLTEXT 1 0 181 "-(- 2*(1+4*t+t^2)/t^2/(t-1)^2*CoherenceEquation2+2*(1+t)/t^2/(t-1)^2*Coher enceEquation3+(t+1)/2*(tinfty20-t010-t020-t110-tt10-tt20-t120+tinfty10 )/t^2/(t-1)^2*CoherenceEquation1)*q^2\n" }{MPLTEXT 1 0 164 "-(6*(1+t)/ t^2/(t-1)^2*CoherenceEquation2-2/t^2/(t-1)^2*CoherenceEquation3-1/2*(t infty20-t010-t020-t110-tt10-tt20-t120+tinfty10)/t^2/(t-1)^2*CoherenceE quation1)*q^3\n" }{MPLTEXT 1 0 3 ");\n" }{MPLTEXT 1 0 85 "Termq:=facto r(residue((RHSP6-RHSP6bis+4*(Pt12+P012+P112)/t^2/(t-1)^2*q^4)/q^2,q=0) ):\n" }{MPLTEXT 1 0 86 "Termq2:=factor(residue((RHSP6-RHSP6bis+4*(Pt12 +P012+P112)/t^2/(t-1)^2*q^4)/q^3,q=0)):\n" }{MPLTEXT 1 0 86 "Termq3:=f actor(residue((RHSP6-RHSP6bis+4*(Pt12+P012+P112)/t^2/(t-1)^2*q^4)/q^4, q=0)):\n" }{MPLTEXT 1 0 166 "factor(Termq-2*(1+t)/t/(t-1)^2*CoherenceE quation2+2/t/(t-1)^2*CoherenceEquation3+1/2*(tinfty20-t010-t020-t110-t t10-tt20-t120+tinfty10)/t/(t-1)^2*CoherenceEquation1);\n" }{MPLTEXT 1 0 173 "factor(Termq3-6*(1+t)/t^2/(t-1)^2*CoherenceEquation2+2/t^2/(t-1 )^2*CoherenceEquation3+1/2*(tinfty20-t010-t020-t110-tt10-tt20-t120+tin fty10)/t^2/(t-1)^2*CoherenceEquation1);\n" }{MPLTEXT 1 0 188 "factor(T ermq2+2*(1+4*t+t^2)/t^2/(t-1)^2*CoherenceEquation2-2*(1+t)/t^2/(t-1)^2 *CoherenceEquation3-(t+1)/2*(tinfty20-t010-t020-t110-tt10-tt20-t120+ti nfty10)/t^2/(t-1)^2*CoherenceEquation1);" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I&RHSP6G6\",$*.I\"qGF$\"\"\",&F'F(!\"\"F(F(,&F'F(I\"tGF$F*F(F,!\" #,&F,F(F*F(F-,**$,(I)tinfty10GF$F(I)tinfty20GF$F*I\"hGF$F*\"\"##F(F5*( ,&I%t010GF$F(I%t020GF$F*F5F,F(F'F-#F*F5*(,&I%t110GF$F(I%t120GF$F*F5F.F (F)F-F6**,&*$,&I%tt10GF$F(I%tt20GF$F*F5F;*$F4F5F6F(F,F(F.F(F+F-F(F(F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 210 "" 0 "" {TEXT 215 80 "Conclusi on: q satisfies the Painlev\351 6 equation whose r.h.s. may be rewritt en as" }{TEXT 215 1 " " }}{PARA 211 "" 0 "" {TEXT 215 89 "-q*(q-1)*(q- t)/t^2/(t-1)^2*(alpha+beta*t/q^2+gammaa*(t-1)/(q-1)^2+delta*t*(t-1)/(q -t)^2) " }}{PARA 212 "" 0 "" {TEXT 215 71 "=-2*q*(q-1)*(q-t)/t^2/(t-1) ^2*diff(q*(q-1)*(q-t)*(P1(q)^2/4-P2(q)),q) " }{TEXT 215 109 "+q*(q-1) *(q-t)*(tinfty10-tinfty20)/t^2/(t-1)^2*h-q*(q-1)*(q-t)/2/t^2/(t-1)^2*h ^2-q*(q-1)/(q-t)/2/t/(t-1)*h^2:" }}}{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 }