{VERSION 6 0 "Windows 10" "6.0" } {USTYLETAB {PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 0 0 1 }1 1 0 0 8 4 2 0 2 0 2 2 -1 1 }{PSTYLE "War ning" -1 7 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Left Justified Maple Outp ut" -1 12 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Fixed Width" -1 17 1 {CSTYLE " " -1 -1 "Courier" 1 10 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }3 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Help" -1 10 1 {CSTYLE "" -1 -1 "Courier" 1 9 0 0 255 1 2 2 2 2 2 1 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Head ing 4" -1 20 1 {CSTYLE "" -1 -1 "Times" 1 10 0 0 0 1 1 1 2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Line Printed Output" -1 6 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "T imes" 1 14 0 0 0 1 2 1 2 2 2 2 1 0 0 1 }1 1 0 0 8 2 2 0 2 0 2 2 -1 1 } {PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }3 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Heading 3" -1 5 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 1 1 2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Diagnostic" -1 9 1 {CSTYLE "" -1 -1 "Courier" 1 10 64 128 64 1 2 2 2 2 2 1 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Ordered List 1" -1 200 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 3 3 2 0 2 0 2 2 -1 1 }{PSTYLE "Text Output" -1 2 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Ordered List 2" -1 201 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 3 3 2 36 2 0 2 2 -1 1 }{PSTYLE "Ordered List 3" -1 202 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 3 3 2 72 2 0 2 2 -1 1 }{PSTYLE "Ordered List 4" -1 203 1 {CSTYLE "" -1 -1 "Times " 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 3 3 2 108 2 0 2 2 -1 1 } {PSTYLE "Ordered List 5" -1 204 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 3 3 2 144 2 0 2 2 -1 1 }{PSTYLE "Annota tion Title" -1 205 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 0 0 1 }3 1 0 0 12 12 2 0 2 0 2 2 -1 1 }{PSTYLE "Maple Output12" -1 206 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "HyperlinkError" -1 207 1 {CSTYLE "" -1 -1 "Courier New" 1 12 255 0 255 1 2 2 1 2 2 1 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "HyperlinkWarning" -1 208 1 {CSTYLE "" -1 -1 "Courier New" 1 12 0 0 255 1 2 2 1 2 2 1 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Bullet Item" -1 15 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 3 3 2 0 2 0 2 2 -1 1 }{PSTYLE "M aple Plot" -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }3 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "List Item" -1 14 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 3 3 2 0 2 0 2 2 -1 1 }{PSTYLE "Normal257" -1 209 1 {CSTYLE "" -1 -1 "Time s" 1 18 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 "Normal256" -1 210 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 "Dash Item" -1 16 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 3 3 2 0 2 0 2 2 -1 1 }{PSTYLE "Error" -1 8 1 {CSTYLE "" -1 -1 "Cour ier" 1 10 255 0 255 1 2 2 2 2 2 1 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Title" -1 18 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 1 2 2 2 1 0 0 1 }3 1 0 0 12 12 2 0 2 0 2 2 -1 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Author" -1 19 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }3 1 0 0 8 8 2 0 2 0 2 2 -1 1 } {CSTYLE "Help Variable" -1 25 "Courier" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Help Underlined Bold" -1 41 "Times" 1 12 0 0 0 1 1 1 2 2 2 2 0 0 0 1 }{CSTYLE "Copyright" -1 34 "Times" 1 10 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Maple Comment" -1 21 "Courier" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "Popup" -1 31 "Times" 1 12 0 128 128 1 1 2 1 2 2 2 0 0 0 1 }{CSTYLE "Atomic Variable" -1 200 "Times" 1 12 175 0 175 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "Dictionary Hyperlink" -1 45 "Times" 1 12 147 0 15 1 2 2 1 2 2 2 0 0 0 1 }{CSTYLE "Plot Text" -1 28 "Times" 1 8 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "2D Input" -1 19 "Times" 1 12 255 0 0 1 2 2 2 2 1 2 0 0 0 1 }{CSTYLE "Code" -1 201 "Courier New" 1 12 255 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math Italic" -1 3 "Times " 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math Small" -1 7 "Time s" 1 1 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math Bold Small" -1 10 "Times" 1 1 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "Help Bold" -1 39 "T imes" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "Help Menus" -1 36 "Ti mes" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "Help Heading" -1 26 "T imes" 1 14 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "2D Output" -1 20 "Tim es" 1 12 0 0 255 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "2D Inert Output" -1 202 "Times" 1 12 144 144 144 1 2 2 2 2 1 2 0 0 0 1 }{CSTYLE "Help Norm al" -1 30 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Maple Inp ut" -1 0 "Courier" 1 12 255 0 0 1 2 1 2 2 1 2 0 0 0 1 }{CSTYLE "Page N umber" -1 33 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Header and Footer" -1 203 "Times" 1 10 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math Italic Small204" -1 204 "Times" 1 1 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "Help Fixed" -1 23 "Courier" 1 10 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math Italic Small206" -1 205 "Times" 1 1 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "Output Labels" -1 29 "Times" 1 8 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math Italic Small205" -1 206 "Times" 1 1 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "Help Notes" -1 37 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "Help Underlined" -1 44 "Times" 1 12 0 0 0 1 2 2 1 2 2 2 0 0 0 1 }{CSTYLE "2D Math Italic Small" -1 207 "Times" 1 1 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math Symbol 2" -1 16 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math Itali c Small207" -1 208 "Times" 1 1 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "2 D Math" -1 2 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Help N onterminal" -1 24 "Courier" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "Annotation Text" -1 209 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 } {CSTYLE "Help Maple Name" -1 35 "Times" 1 12 104 64 92 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "Text" -1 210 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Plot Title" -1 27 "Times" 1 10 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "Help Underlined Italic" -1 43 "Times" 1 12 0 0 0 1 1 2 1 2 2 2 0 0 0 1 }{CSTYLE "Caption Reference" -1 211 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "Default" -1 38 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Help Italic Bold" -1 40 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "Maple Input Placeholder" -1 212 "Courier \+ New" 1 12 200 0 200 1 2 1 2 2 1 2 0 0 0 1 }{CSTYLE "LaTeX" -1 32 "Time s" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Help Emphasized" -1 22 " Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "Equation Label" -1 213 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "2D Comment" -1 18 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Help Italic" -1 42 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "Prompt" -1 1 "Co urier" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math Bold" -1 5 " Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "Ti mes" 1 12 0 128 128 1 2 2 1 2 2 2 0 0 0 1 }{CSTYLE "Caption Text" -1 214 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 215 "Times " 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 216 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 217 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 218 "Times" 1 14 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 219 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 } {CSTYLE "" -1 220 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 221 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 222 "T imes" 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 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 225 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 226 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 } } {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT 210 168 "In this Maple file, we \+ conmpute the Lax pair in the oper gauge in the Painlev\351 1 case. We \+ also check that the formulas proposed in the article are correct in th is case." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "restart:\n" } {MPLTEXT 1 0 21 "with(LinearAlgebra):\n" }{MPLTEXT 1 0 12 "tinfty10:=0 ;" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 27 "Pinfty11:=-(2*tinfty14)/2;\n " }{MPLTEXT 1 0 26 "Pinfty01:=-(2*tinfty12)/2;" }{MPLTEXT 1 0 1 "\n" } {MPLTEXT 1 0 31 "Pinfty32 := -(1/4)*tinfty15^2;\n" }{MPLTEXT 1 0 55 "P infty22 := -(1/2)*tinfty15*tinfty13+(1/4)*tinfty14^2;\n" }{MPLTEXT 1 0 79 "Pinfty12 := -(1/2)*tinfty15*tinfty11+(1/2)*tinfty14*tinfty12-(1/ 4)*tinfty13^2;\n" }{MPLTEXT 1 0 29 "P1:=x-> Pinfty01+Pinfty11*x;\n" } {MPLTEXT 1 0 55 "P2:=x-> Pinfty02+Pinfty12*x+Pinfty22*x^2+Pinfty32*x^3 ;\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 11 "Unknownn:=-" }{MPLTEXT 1 0 9 "Unknown:\n" }{MPLTEXT 1 0 1 "L" }{MPLTEXT 1 0 12 "Unknownn:=-L" } {MPLTEXT 1 0 8 "Unknown:" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 11 "Unkno wnn2:=" }{MPLTEXT 1 0 10 "Unknown2:\n" }{MPLTEXT 1 0 1 "L" }{MPLTEXT 1 0 22 "Unknownn2:=LUnknown2:\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "L" }{MPLTEXT 1 0 22 "tinfty25:=-Ltinfty15;\n" }{MPLTEXT 1 0 1 "L" } {MPLTEXT 1 0 22 "tinfty23:=-Ltinfty13;\n" }{MPLTEXT 1 0 1 "L" } {MPLTEXT 1 0 22 "tinfty21:=-Ltinfty11;\n" }{MPLTEXT 1 0 1 "L" } {MPLTEXT 1 0 22 "tinfty20:=-Ltinfty10;\n" }{MPLTEXT 1 0 1 "L" } {MPLTEXT 1 0 21 "tinfty24:=Ltinfty14;\n" }{MPLTEXT 1 0 1 "L" }{MPLTEXT 1 0 21 "tinfty22:=Ltinfty12;\n" }{MPLTEXT 1 0 1 "L" }{MPLTEXT 1 0 12 "tinfty10:=0:" }}{PARA 0 "" 0 "" {TEXT 227 0 "" }{MPLTEXT 1 0 1 "\n" } }{PARA 11 "" 1 "" {XPPMATH 20 ">I)tinfty10G6\"\"\"!" }}{PARA 11 "" 1 " " {XPPMATH 20 ">I)Pinfty11G6\",$I)tinfty14GF$!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I)Pinfty01G6\",$I)tinfty12GF$!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I)Pinfty32G6\",$*$I)tinfty15GF$\"\"##!\"\"\"\"%" }} {PARA 11 "" 1 "" {XPPMATH 20 ">I)Pinfty22G6\",&*&I)tinfty15GF$\"\"\"I) tinfty13GF$F(#!\"\"\"\"#*$I)tinfty14GF$F,#F(\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I)Pinfty12G6\",(*&I)tinfty15GF$\"\"\"I)tinfty11GF$F(#!\" \"\"\"#*&I)tinfty14GF$F(I)tinfty12GF$F(#F(F,*$I)tinfty13GF$F,#F+\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I#P1G6\"f*6#I\"xGF$F$6$I)operatorGF$ I&arrowGF$F$,&I)Pinfty01GF$\"\"\"*&I)Pinfty11GF$F-9$F-F-F$F$F$" }} {PARA 11 "" 1 "" {XPPMATH 20 ">I#P2G6\"f*6#I\"xGF$F$6$I)operatorGF$I&a rrowGF$F$,*I)Pinfty02GF$\"\"\"*&I)Pinfty12GF$F-9$F-F-*&I)Pinfty22GF$F- F0\"\"#F-*&I)Pinfty32GF$F-F0\"\"$F-F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I*Ltinfty25G6\",$I*Ltinfty15GF$!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I*Ltinfty23G6\",$I*Ltinfty13GF$!\"\"" }}{PARA 11 "" 1 " " {XPPMATH 20 ">I*Ltinfty21G6\",$I*Ltinfty11GF$!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I*Ltinfty20G6\",$I*Ltinfty10GF$!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I*Ltinfty24G6\"I*Ltinfty14GF$" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I*Ltinfty22G6\"I*Ltinfty12GF$" }}}{EXCHG {PARA 210 "" 0 "" {TEXT 228 17 "Study at infinity" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "espilon:=1:\n" }{MPLTEXT 1 0 210 "logPsi1Infty:=-1/5* tinfty15/h*lambda^(5/2)-1/4*tinfty14/h*lambda^2-1/3*tinfty13/h*lambda^ (3/2)-1/2*tinfty12/h*lambda^1-tinfty11/h*lambda^(1/2)+1/4*epsilon*ln(l ambda)+A10+ Unknown/lambda^(1/2)+ Unknown2/lambda;\n" }{MPLTEXT 1 0 145 "logPsi2Infty:=1/5*tinfty15/h*lambda^(5/2)-1/4*tinfty14/h*lambda^2 +1/3*tinfty13/h*lambda^(3/2)-1/2*tinfty12/h*lambda^1+tinfty11/h*lambda ^(1/2)+1/4" }{MPLTEXT 1 0 8 "*epsilon" }{MPLTEXT 1 0 56 "*ln(lambda)+A 20- Unknown/lambda^(1/2)+ Unknown2/lambda;\n" }{MPLTEXT 1 0 1 "\n" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 202 "GrosLlogpsi1Infty:=-1/5*Ltinfty 15/h*lambda^(5/2)-1/4*Ltinfty14/h*lambda^2-1/3*Ltinfty13/h*lambda^(3/2 )-1/2*Ltinfty12/h*lambda^1-Ltinfty11/h*lambda^(1/2)+LA10+ LUnknown/lam bda^(1/2)+ LUnknown2/lambda ;\n" }{MPLTEXT 1 0 199 "GrosLlogpsi2Infty :=1/5*Ltinfty15/h*lambda^(5/2)-1/4*Ltinfty14/h*lambda^2+1/3*Ltinfty13/ h*lambda^(3/2)-1/2*Ltinfty12/h*lambda^1+Ltinfty11/h*lambda^(1/2)+LA20- LUnknown/lambda^(1/2)+ LUnknown2/lambda;\n" }{MPLTEXT 1 0 1 "\n" } {MPLTEXT 1 0 155 "GrosLpsi1Infty := exp(-1/5*tinfty15/h*lambda^(5/2)-1 /4*tinfty14/h*lambda^2-1/3*tinfty13/h*lambda^(3/2)-1/2*tinfty12/h*lamb da^1-tinfty11/h*lambda^(1/2)+1/4*" }{MPLTEXT 1 0 8 "epsilon*" } {MPLTEXT 1 0 238 "ln(lambda)+A10+ Unknown/lambda^(1/2)+ Unknown2/lambd a)*(-1/5*Ltinfty15/h*lambda^(5/2)-1/4*Ltinfty14/h*lambda^2-1/3*Ltinfty 13/h*lambda^(3/2)-1/2*Ltinfty12/h*lambda^1-Ltinfty11/h*lambda^(1/2)+LA 10+ LUnknown/lambda^(1/2)+ LUnknown2/lambda);\n" }{MPLTEXT 1 0 153 "Gr osLpsi2Infty := exp(1/5*tinfty15/h*lambda^(5/2)-1/4*tinfty14/h*lambda^ 2+1/3*tinfty13/h*lambda^(3/2)-1/2*tinfty12/h*lambda^1+tinfty11/h*lambd a^(1/2)+1/4" }{MPLTEXT 1 0 8 "*epsilon" }{MPLTEXT 1 0 237 "*ln(lambda) +A20- Unknown/lambda^(1/2)+ Unknown2/lambda)*(1/5*Ltinfty15/h*lambda^( 5/2)-1/4*Ltinfty14/h*lambda^2+1/3*Ltinfty13/h*lambda^(3/2)-1/2*Ltinfty 12/h*lambda^1+Ltinfty11/h*lambda^(1/2)+LA20- LUnknown/lambda^(1/2)+ LU nknown2/lambda);" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 30 "psi1Infty:=ex p(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(psi2Infty,lambda):\n" } {MPLTEXT 1 0 47 "d2psi1dlambda2Infty:=diff(psi1Infty,lambda$2):\n" } {MPLTEXT 1 0 47 "d2psi2dlambda2Infty:=diff(psi2Infty,lambda$2):\n" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 89 "WronskianLa mbdaInfty:=h*factor(psi1Infty*dpsi2dlambdaInfty-psi2Infty*dpsi1dlambda Infty):\n" }{MPLTEXT 1 0 132 "WronskianLambdabisInfty:=h*simplify(fact or( (diff(logPsi2Infty,lambda)-diff(logPsi1Infty,lambda))*exp(logPsi1I nfty+logPsi2Infty))):\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 116 "Wrons kianTildeLambdaInfty:=h^3*factor(dpsi2dlambdaInfty*d2psi1dlambda2Infty -dpsi1dlambdaInfty*d2psi2dlambda2Infty):\n" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I-logPsi1InftyG6\",4*(I)tinfty15GF$\"\"\"I\"hGF$!\"\"I'l ambdaGF$#\"\"&\"\"##F*F-*(I)tinfty14GF$F(F)F*F+F.#F*\"\"%*(I)tinfty13G F$F(F)F*F+#\"\"$F.#F*F7*(I)tinfty12GF$F(F)F*F+F(#F*F.*(I)tinfty11GF$F( F)F*F+#F(F.F**&I(epsilonGF$F(-I#lnG6$%*protectedGI(_syslibGF$6#F+F(#F( F3I$A10GF$F(*&I(UnknownGF$F(F+F;F(*&I)Unknown2GF$F(F+F*F(" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I-logPsi2InftyG6\",4*(I)tinfty15GF$\"\"\"I\"hGF$ !\"\"I'lambdaGF$#\"\"&\"\"##F(F-*(I)tinfty14GF$F(F)F*F+F.#F*\"\"%*(I)t infty13GF$F(F)F*F+#\"\"$F.#F(F7*(I)tinfty12GF$F(F)F*F+F(#F*F.*(I)tinft y11GF$F(F)F*F+#F(F.F(*&I(epsilonGF$F(-I#lnG6$%*protectedGI(_syslibGF$6 #F+F(#F(F3I$A20GF$F(*&I(UnknownGF$F(F+F;F**&I)Unknown2GF$F(F+F*F(" }} {PARA 11 "" 1 "" {XPPMATH 20 ">I2GrosLlogpsi1InftyG6\",2*(I*Ltinfty15G F$\"\"\"I\"hGF$!\"\"I'lambdaGF$#\"\"&\"\"##F*F-*(I*Ltinfty14GF$F(F)F*F +F.#F*\"\"%*(I*Ltinfty13GF$F(F)F*F+#\"\"$F.#F*F7*(I*Ltinfty12GF$F(F)F* F+F(#F*F.*(I*Ltinfty11GF$F(F)F*F+#F(F.F*I%LA10GF$F(*&I)LUnknownGF$F(F+ F;F(*&I*LUnknown2GF$F(F+F*F(" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I2GrosL logpsi2InftyG6\",2*(I*Ltinfty15GF$\"\"\"I\"hGF$!\"\"I'lambdaGF$#\"\"& \"\"##F(F-*(I*Ltinfty14GF$F(F)F*F+F.#F*\"\"%*(I*Ltinfty13GF$F(F)F*F+# \"\"$F.#F(F7*(I*Ltinfty12GF$F(F)F*F+F(#F*F.*(I*Ltinfty11GF$F(F)F*F+#F( F.F(I%LA20GF$F(*&I)LUnknownGF$F(F+F;F**&I*LUnknown2GF$F(F+F*F(" }} {PARA 11 "" 1 "" {XPPMATH 20 ">I/GrosLpsi1InftyG6\"*&-I$expG6$%*protec tedGI(_syslibGF$6#,4*(I)tinfty15GF$\"\"\"I\"hGF$!\"\"I'lambdaGF$#\"\"& \"\"##F1F4*(I)tinfty14GF$F/F0F1F2F5#F1\"\"%*(I)tinfty13GF$F/F0F1F2#\" \"$F5#F1F>*(I)tinfty12GF$F/F0F1F2F/#F1F5*(I)tinfty11GF$F/F0F1F2#F/F5F1 *&I(epsilonGF$F/-I#lnGF(6#F2F/#F/F:I$A10GF$F/*&I(UnknownGF$F/F2FBF/*&I )Unknown2GF$F/F2F1F/F/,2*(I*Ltinfty15GF$F/F0F1F2F3F6*(I*Ltinfty14GF$F/ F0F1F2F5F9*(I*Ltinfty13GF$F/F0F1F2F=F?*(I*Ltinfty12GF$F/F0F1F2F/FB*(I* Ltinfty11GF$F/F0F1F2FEF1I%LA10GF$F/*&I)LUnknownGF$F/F2FBF/*&I*LUnknown 2GF$F/F2F1F/F/" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I/GrosLpsi2InftyG6\"* &-I$expG6$%*protectedGI(_syslibGF$6#,4*(I)tinfty15GF$\"\"\"I\"hGF$!\" \"I'lambdaGF$#\"\"&\"\"##F/F4*(I)tinfty14GF$F/F0F1F2F5#F1\"\"%*(I)tinf ty13GF$F/F0F1F2#\"\"$F5#F/F>*(I)tinfty12GF$F/F0F1F2F/#F1F5*(I)tinfty11 GF$F/F0F1F2#F/F5F/*&I(epsilonGF$F/-I#lnGF(6#F2F/#F/F:I$A20GF$F/*&I(Unk nownGF$F/F2FBF1*&I)Unknown2GF$F/F2F1F/F/,2*(I*Ltinfty15GF$F/F0F1F2F3F6 *(I*Ltinfty14GF$F/F0F1F2F5F9*(I*Ltinfty13GF$F/F0F1F2F=F?*(I*Ltinfty12G F$F/F0F1F2F/FB*(I*Ltinfty11GF$F/F0F1F2FEF/I%LA20GF$F/*&I)LUnknownGF$F/ F2FBF1*&I*LUnknown2GF$F/F2F1F/F/" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I*p si1InftyG6\"-I$expG6$%*protectedGI(_syslibGF$6#,4*(I)tinfty15GF$\"\"\" I\"hGF$!\"\"I'lambdaGF$#\"\"&\"\"##F0F3*(I)tinfty14GF$F.F/F0F1F4#F0\" \"%*(I)tinfty13GF$F.F/F0F1#\"\"$F4#F0F=*(I)tinfty12GF$F.F/F0F1F.#F0F4* (I)tinfty11GF$F.F/F0F1#F.F4F0*&I(epsilonGF$F.-I#lnGF'6#F1F.#F.F9I$A10G F$F.*&I(UnknownGF$F.F1FAF.*&I)Unknown2GF$F.F1F0F." }}{PARA 11 "" 1 "" {XPPMATH 20 ">I*psi2InftyG6\"-I$expG6$%*protectedGI(_syslibGF$6#,4*(I) tinfty15GF$\"\"\"I\"hGF$!\"\"I'lambdaGF$#\"\"&\"\"##F.F3*(I)tinfty14GF $F.F/F0F1F4#F0\"\"%*(I)tinfty13GF$F.F/F0F1#\"\"$F4#F.F=*(I)tinfty12GF$ F.F/F0F1F.#F0F4*(I)tinfty11GF$F.F/F0F1#F.F4F.*&I(epsilonGF$F.-I#lnGF'6 #F1F.#F.F9I$A20GF$F.*&I(UnknownGF$F.F1FAF0*&I)Unknown2GF$F.F1F0F." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "L21Infty:=simplify(Wronskian TildeLambdaInfty/WronskianLambdabisInfty):\n" }{MPLTEXT 1 0 75 "L21Inf tyOrdrelambda5:=factor(-residue(L21Infty/lambda^6,lambda=infinity));\n " }{MPLTEXT 1 0 75 "L21InftyOrdrelambda4:=factor(-residue(L21Infty/lam bda^5,lambda=infinity));\n" }{MPLTEXT 1 0 75 "L21InftyOrdrelambda3:=fa ctor(-residue(L21Infty/lambda^4,lambda=infinity));\n" }{MPLTEXT 1 0 75 "L21InftyOrdrelambda2:=factor(-residue(L21Infty/lambda^3,lambda=inf inity));\n" }{MPLTEXT 1 0 75 "L21InftyOrdrelambda1:=factor(-residue(L2 1Infty/lambda^2,lambda=infinity));\n" }{MPLTEXT 1 0 74 "L21InftyOrdrel ambda0:=factor(-residue(L21Infty/lambda^1,lambda=infinity));" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I5L21InftyOrdrelambda5G6\"\"\"!" }}{PARA 11 " " 1 "" {XPPMATH 20 ">I5L21InftyOrdrelambda4G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I5L21InftyOrdrelambda3G6\",$*$I)tinfty15GF$\"\"##\"\" \"\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I5L21InftyOrdrelambda2G6\",& *&I)tinfty15GF$\"\"\"I)tinfty13GF$F(#F(\"\"#*$I)tinfty14GF$F+#!\"\"\" \"%" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I5L21InftyOrdrelambda1G6\",(*&I) tinfty15GF$\"\"\"I)tinfty11GF$F(#F(\"\"#*&I)tinfty14GF$F(I)tinfty12GF$ F(#!\"\"F+*$I)tinfty13GF$F+#F(\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "> I5L21InftyOrdrelambda0G6\",,*(I(UnknownGF$\"\"\"I\"hGF$F(I)tinfty15GF$ F(#F(\"\"#*(I(epsilonGF$F(F)F(I)tinfty14GF$F(#F(\"\"%*&F)F(F/F(F0*&I)t infty11GF$F(I)tinfty13GF$F(F+*$I)tinfty12GF$F,#!\"\"F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "factor(simplify(" }{MPLTEXT 1 0 20 "L21InftyOrdrelambda4" }{MPLTEXT 1 0 10 "*lambda^4+" }{MPLTEXT 1 0 20 "L21InftyOrdrelambda3" }{MPLTEXT 1 0 10 "*lambda^3+" }{MPLTEXT 1 0 20 "L21InftyOrdrelambda2" }{MPLTEXT 1 0 10 "*lambda^2+" }{MPLTEXT 1 0 20 "L21InftyOrdrelambda1" }{MPLTEXT 1 0 34 "*lambda- (-P2(lambda)+Pinfty0 2)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "" 0 " " {TEXT 210 39 "We conclude that L_\{2,1\}is of the form " }{TEXT 210 54 "-P2(lambda)+O(1) at infinity. Let us now study L_\{2,2\}" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "L22Infty:=factor(h*simplify( diff(WronskianLambdabisInfty,lambda)/WronskianLambdabisInfty)):\n" } {MPLTEXT 1 0 75 "L22InftyOrdrelambda5:=factor(-residue(L22Infty/lambda ^6,lambda=infinity));\n" }{MPLTEXT 1 0 75 "L22InftyOrdrelambda4:=facto r(-residue(L22Infty/lambda^5,lambda=infinity));\n" }{MPLTEXT 1 0 75 "L 22InftyOrdrelambda3:=factor(-residue(L22Infty/lambda^4,lambda=infinity ));\n" }{MPLTEXT 1 0 75 "L22InftyOrdrelambda2:=factor(-residue(L22Inft y/lambda^3,lambda=infinity));\n" }{MPLTEXT 1 0 75 "L22InftyOrdrelambda 1:=factor(-residue(L22Infty/lambda^2,lambda=infinity));\n" }{MPLTEXT 1 0 75 "L22InftyOrdrelambda0:=factor(-residue(L22Infty/lambda^1,lambda =infinity));\n" }{MPLTEXT 1 0 80 "L22InftyOrdrelambdaMoins1:=factor(-r esidue(L22Infty/lambda^0,lambda=infinity));\n" }{MPLTEXT 1 0 82 "L22In ftyOrdrelambdaMoins2:=factor(-residue(L22Infty/lambda^(-1),lambda=infi nity));" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I5L22InftyOrdrelambda5G6\"\" \"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I5L22InftyOrdrelambda4G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I5L22InftyOrdrelambda3G6\"\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 ">I5L22InftyOrdrelambda2G6\"\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 ">I5L22InftyOrdrelambda1G6\",$I)tinfty14G F$!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I5L22InftyOrdrelambda0G6\",$ I)tinfty12GF$!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I:L22InftyOrdrela mbdaMoins1G6\",$*&,&I(epsilonGF$\"\"\"\"\"$F)F)I\"hGF$F)#F)\"\"#" }} {PARA 11 "" 1 "" {XPPMATH 20 ">I:L22InftyOrdrelambdaMoins2G6\",$*(I\"h GF$\"\"\",&*&I)Unknown2GF$F(I)tinfty15GF$F(\"\"#I)tinfty13GF$F(F(F,!\" \"F/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 2 140 "We conclude that L_\{2,2 \} behaves at infinity like -tinfty14*lambda -tinfty12+2*h/lambda +O(1 /lambda^2) =P1(lambda) +2*h/lambda+O(1/lambda^2)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 215 58 "We end with the explicit formulas for L_\{2,2\} an d L_\{2,1\}:" }{TEXT 216 39 "\nL_\{2,2\}= P_1(lambda) +2*h/(lambda-q )" }{TEXT 2 6 " and " }{TEXT 2 1 " " }{TEXT 217 7 "L_\{2,1\}" }{TEXT 218 2 "= " }{TEXT 219 23 "-P_2(lambda)+Pinfty02+C" }{TEXT 220 17 " -p *h/(lambda-q)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "L21Form:=- P2(lambda)+Pinfty02- p*h/(lambda-q);\n" }{MPLTEXT 1 0 38 "L22Form:=P1( lambda) +2*h/(lambda-q);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I(L21For mG6\",**&,(*&I)tinfty15GF$\"\"\"I)tinfty11GF$F*#!\"\"\"\"#*&I)tinfty14 GF$F*I)tinfty12GF$F*#F*F.*$I)tinfty13GF$F.#F-\"\"%F*I'lambdaGF$F*F-*&, &*&F)F*F4F*F,*$F0F.#F*F6F*F7F.F-*&F)F.F7\"\"$F<*(I\"pGF$F*I\"hGF$F*,&F 7F*I\"qGF$F-F-F-" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I(L22FormG6\",(*&I) tinfty14GF$\"\"\"I'lambdaGF$F(!\"\"I)tinfty12GF$F**&I\"hGF$F(,&F)F(I\" qGF$F*F*\"\"#" }}}{EXCHG {PARA 209 "" 0 "" {TEXT 221 56 "Conputation f or the auxiliary matrix A in the oper gauge" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 2 183 "The deformation operator is \\mathcal\{L\}=\\hbar (alph a15\\partial_\{t_\{\\infty^\{(1)\},5\} +alpha14\\partial_\{t_\{\\infty ^\{(1)\},4\}+e*\\partial_\{t_\{\\infty^\{(1)\},3\} +f*\\partial_\{t_\{ \\infty^\{(1)\},2\}" }{TEXT 2 32 "+g*\\partial_\{t_\{\\infty^\{(1)\},1 \})" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "WronskianGrosLInfty: =factor(psi1Infty*GrosLpsi2Infty-psi2Infty*GrosLpsi1Infty):\n" } {MPLTEXT 1 0 70 "A12Infty:=factor(simplify(WronskianGrosLInfty/Wronski anLambdaInfty)):\n" }{MPLTEXT 1 0 48 "Y1Infty:=h*factor(dpsi1dlambdaIn fty/psi1Infty):\n" }{MPLTEXT 1 0 48 "Y2Infty:=h*factor(dpsi2dlambdaInf ty/psi2Infty):\n" }{MPLTEXT 1 0 43 "Z1Infty:=factor(GrosLpsi1Infty/psi 1Infty):\n" }{MPLTEXT 1 0 43 "Z2Infty:=factor(GrosLpsi2Infty/psi2Infty ):\n" }{MPLTEXT 1 0 68 "A12bisInfty:=factor(simplify((Z2Infty-Z1Infty) /(Y2Infty-Y1Infty))):\n" }{MPLTEXT 1 0 82 "A11Infty:=factor(simplify( \+ (Y2Infty*Z1Infty-Y1Infty*Z2Infty)/(Y2Infty-Y1Infty) )):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "factor(simplify(A12bisInfty-A12Inft y));" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "Ltinfty15:=h*alpha15:\n" }{MPLTEXT 1 0 22 "Ltinfty 14:=h*alpha14:\n" }{MPLTEXT 1 0 22 "Ltinfty13:=h*alpha13:\n" }{MPLTEXT 1 0 22 "Ltinfty12:=h*alpha12:\n" }{MPLTEXT 1 0 21 "Ltinfty11:=h*alpha 11:" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 14 "Ltinfty10:=0:\n" }{MPLTEXT 1 0 14 "Ltinfty20:=0:\n" }{MPLTEXT 1 0 12 "LA20:=LA10:\n" }{MPLTEXT 1 0 1 "\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "A12InftyLambda 3:=factor(-residue(A12Infty/lambda^4,lambda=infinity));\n" }{MPLTEXT 1 0 70 "A12InftyLambda2:=factor(-residue(A12Infty/lambda^3,lambda=infi nity));\n" }{MPLTEXT 1 0 70 "A12InftyLambda1:=factor(-residue(A12Infty /lambda^2,lambda=infinity));\n" }{MPLTEXT 1 0 70 "A12InftyLambda0:=fac tor(-residue(A12Infty/lambda^1,lambda=infinity));\n" }{MPLTEXT 1 0 75 "A12InftyLambdaMoins1:=factor(-residue(A12Infty/lambda^0,lambda=infini ty));\n" }{MPLTEXT 1 0 78 "A12InftyLambdaMoins2:=factor(-residue(A12In fty/lambda^(-1),lambda=infinity));\n" }}{PARA 11 "" 1 "" {XPPMATH 20 " >I0A12InftyLambda3G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I0A12Inf tyLambda2G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I0A12InftyLambda1 G6\",$*&I(alpha15GF$\"\"\"I)tinfty15GF$!\"\"#\"\"#\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I0A12InftyLambda0G6\",$*&,&*&I(alpha13GF$\"\"\"I)t infty15GF$F*\"\"&*&I(alpha15GF$F*I)tinfty13GF$F*!\"$F*F+!\"##\"\"#\"#: " }}{PARA 11 "" 1 "" {XPPMATH 20 ">I5A12InftyLambdaMoins1G6\",$*&,**&I (alpha11GF$\"\"\"I)tinfty15GF$\"\"#\"#:*(I(alpha13GF$F*I)tinfty13GF$F* F+F*!\"&*(I(alpha15GF$F*I)tinfty11GF$F*F+F*!\"$*&F3F*F0F,\"\"$F*F+F5#F ,F-" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I5A12InftyLambdaMoins2G6\",$*&,0 **I(UnknownGF$\"\"\"I(alpha15GF$F*I\"hGF$F*I)tinfty15GF$\"\"#\"\"$*(I( alpha11GF$F*I)tinfty13GF$F*F-F.\"#:*(I(alpha13GF$F*I)tinfty11GF$F*F-F. \"\"&*(F5F*F2F.F-F*!\"&**F+F*F6F*F2F*F-F*!\"'*&F+F*F2F/F/*&I)LUnknownG F$F*F-F/F3F*F-!\"%#!\"#F3" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 2 12 "We ge t that " }{TEXT 222 54 "A_\{1,2\}=2*alpha15/5/tinfty15*lambda+nu +mu/( lambda-q)\n" }{TEXT 223 8 "with nu=" }{XPPEDIT 2 0 "Typesetting:-mrow( Typesetting:-mo(\"&uminus0;\", mathvariant = \"normal\", fence = \"fal se\", separator = \"false\", stretchy = \"false\", symmetric = \"false \", largeop = \"false\", movablelimits = \"false\", accent = \"false\" , lspace = \"0.2222222em\", rspace = \"0.2222222em\"), Typesetting:-mf rac(Typesetting:-mrow(Typesetting:-mn(\"2\", mathvariant = \"normal\") , Typesetting:-mo(\"⁢\", mathvariant = \"normal\", fenc e = \"false\", separator = \"false\", stretchy = \"false\", symmetric \+ = \"false\", largeop = \"false\", movablelimits = \"false\", accent = \+ \"false\", lspace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mfenc ed(Typesetting:-mrow(Typesetting:-mi(\"\"), Typesetting:-mrow(Typesett ing:-mn(\"3\", mathvariant = \"normal\"), Typesetting:-mo(\"&Invisible Times;\", mathvariant = \"normal\", fence = \"false\", separator = \"f alse\", stretchy = \"false\", symmetric = \"false\", largeop = \"false \", movablelimits = \"false\", accent = \"false\", lspace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mi(\"α\", italic = \"false\", mathvariant = \"normal\"), Typesetting:-mo(\"⁢\", math variant = \"normal\", fence = \"false\", separator = \"false\", stretc hy = \"false\", symmetric = \"false\", largeop = \"false\", movablelim its = \"false\", accent = \"false\", lspace = \"0.0em\", rspace = \"0. 0em\"), Typesetting:-mi(\"tinfty13\", italic = \"true\", mathvariant = \"italic\")), Typesetting:-mo(\"−\", mathvariant = \"normal\", \+ fence = \"false\", separator = \"false\", stretchy = \"false\", symmet ric = \"false\", largeop = \"false\", movablelimits = \"false\", accen t = \"false\", lspace = \"0.2222222em\", rspace = \"0.2222222em\"), Ty pesetting:-mrow(Typesetting:-mn(\"5\", mathvariant = \"normal\"), Type setting:-mo(\"⁢\", mathvariant = \"normal\", fence = \" false\", separator = \"false\", stretchy = \"false\", symmetric = \"fa lse\", largeop = \"false\", movablelimits = \"false\", accent = \"fals e\", lspace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mi(\"alpha1 3\", italic = \"true\", mathvariant = \"italic\"), Typesetting:-mo(\"& InvisibleTimes;\", mathvariant = \"normal\", fence = \"false\", separa tor = \"false\", stretchy = \"false\", symmetric = \"false\", largeop \+ = \"false\", movablelimits = \"false\", accent = \"false\", lspace = \+ \"0.0em\", rspace = \"0.0em\"), Typesetting:-mi(\"tinfty15\", italic = \"true\", mathvariant = \"italic\")), Typesetting:-mi(\"\")), mathvar iant = \"normal\")), Typesetting:-mrow(Typesetting:-mn(\"15\", mathvar iant = \"normal\"), Typesetting:-mo(\"⁢\", mathvariant \+ = \"normal\", fence = \"false\", separator = \"false\", stretchy = \"f alse\", symmetric = \"false\", largeop = \"false\", movablelimits = \" false\", accent = \"false\", lspace = \"0.0em\", rspace = \"0.0em\"), \+ Typesetting:-msup(Typesetting:-mi(\"tinfty15\", italic = \"true\", mat hvariant = \"italic\"), Typesetting:-mn(\"2\", mathvariant = \"normal \"), superscriptshift = \"0\")), linethickness = \"1\", denomalign = \+ \"center\", numalign = \"center\", bevelled = \"false\"));" "-I%mrowG6 #/I+modulenameG6\"I,TypesettingGI(_syslibGF'6$-I#moGF$6-Q*&uminus0;F'/ %,mathvariantGQ'normalF'/%&fenceGQ&falseF'/%*separatorGF4/%)stretchyGF 4/%*symmetricGF4/%(largeopGF4/%.movablelimitsGF4/%'accentGF4/%'lspaceG Q,0.2222222emF'/%'rspaceGFC-I&mfracGF$6(-F#6%-I#mnGF$6$Q\"2F'F/-F,6-Q1 ⁢F'F/F2F5F7F9F;F=F?/FBQ&0.0emF'/FEFS-I(mfencedGF$6$-F#6 '-I#miGF$6#Q!F'-F#6'-FL6$Q\"3F'F/FO-Fen6%Q(αF'/%'italicGF4F/FO-F en6%Q)tinfty13F'/FaoQ%trueF'/F0Q'italicF'-F,6-Q(−F'F/F2F5F7F9F;F =F?FAFD-F#6'-FL6$Q\"5F'F/FO-Fen6%Q(alpha13F'FeoFgoFO-Fen6%Q)tinfty15F' FeoFgoFZF/-F#6%-FL6$Q#15F'F/FO-I%msupGF$6%FdpFK/%1superscriptshiftGQ\" 0F'/%.linethicknessGQ\"1F'/%+denomalignGQ'centerF'/%)numalignGFgq/%)be velledGF4" }{TEXT 224 5 " and\n" }{TEXT 225 3 "mu=" }{XPPEDIT 2 0 "Typ esetting:-mrow(Typesetting:-mi(\"\"), Typesetting:-mo(\"&uminus0;\", m athvariant = \"normal\", fence = \"false\", separator = \"false\", str etchy = \"false\", symmetric = \"false\", largeop = \"false\", movable limits = \"false\", accent = \"false\", lspace = \"0.2222222em\", rspa ce = \"0.2222222em\"), Typesetting:-mfrac(Typesetting:-mrow(Typesettin g:-mn(\"2\", mathvariant = \"normal\"), Typesetting:-mo(\"&InvisibleTi mes;\", mathvariant = \"normal\", fence = \"false\", separator = \"fal se\", stretchy = \"false\", symmetric = \"false\", largeop = \"false\" , movablelimits = \"false\", accent = \"false\", lspace = \"0.0em\", r space = \"0.0em\"), Typesetting:-mfenced(Typesetting:-mrow(Typesetting :-mi(\"\"), Typesetting:-mrow(Typesetting:-mn(\"3\", mathvariant = \"n ormal\"), Typesetting:-mo(\"⁢\", mathvariant = \"normal \", fence = \"false\", separator = \"false\", stretchy = \"false\", sy mmetric = \"false\", largeop = \"false\", movablelimits = \"false\", a ccent = \"false\", lspace = \"0.0em\", rspace = \"0.0em\"), Typesettin g:-mi(\"α\", italic = \"false\", mathvariant = \"normal\"), Type setting:-mo(\"⁢\", mathvariant = \"normal\", fence = \" false\", separator = \"false\", stretchy = \"false\", symmetric = \"fa lse\", largeop = \"false\", movablelimits = \"false\", accent = \"fals e\", lspace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mi(\"tinfty 11\", italic = \"true\", mathvariant = \"italic\"), Typesetting:-mo(\" ⁢\", mathvariant = \"normal\", fence = \"false\", separ ator = \"false\", stretchy = \"false\", symmetric = \"false\", largeop = \"false\", movablelimits = \"false\", accent = \"false\", lspace = \+ \"0.0em\", rspace = \"0.0em\"), Typesetting:-mi(\"tinfty15\", italic = \"true\", mathvariant = \"italic\")), Typesetting:-mo(\"−\", ma thvariant = \"normal\", fence = \"false\", separator = \"false\", stre tchy = \"false\", symmetric = \"false\", largeop = \"false\", movablel imits = \"false\", accent = \"false\", lspace = \"0.2222222em\", rspac e = \"0.2222222em\"), Typesetting:-mrow(Typesetting:-mn(\"3\", mathvar iant = \"normal\"), Typesetting:-mo(\"⁢\", mathvariant \+ = \"normal\", fence = \"false\", separator = \"false\", stretchy = \"f alse\", symmetric = \"false\", largeop = \"false\", movablelimits = \" false\", accent = \"false\", lspace = \"0.0em\", rspace = \"0.0em\"), \+ Typesetting:-mi(\"α\", italic = \"false\", mathvariant = \"norma l\"), Typesetting:-mo(\"⁢\", mathvariant = \"normal\", \+ fence = \"false\", separator = \"false\", stretchy = \"false\", symmet ric = \"false\", largeop = \"false\", movablelimits = \"false\", accen t = \"false\", lspace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-m sup(Typesetting:-mi(\"tinfty13\", italic = \"true\", mathvariant = \"i talic\"), Typesetting:-mn(\"2\", mathvariant = \"normal\"), superscrip tshift = \"0\")), Typesetting:-mo(\"+\", mathvariant = \"normal\" , fence = \"false\", separator = \"false\", stretchy = \"false\", symm etric = \"false\", largeop = \"false\", movablelimits = \"false\", acc ent = \"false\", lspace = \"0.2222222em\", rspace = \"0.2222222em\"), \+ Typesetting:-mrow(Typesetting:-mn(\"5\", mathvariant = \"normal\"), Ty pesetting:-mo(\"⁢\", mathvariant = \"normal\", fence = \+ \"false\", separator = \"false\", stretchy = \"false\", symmetric = \" false\", largeop = \"false\", movablelimits = \"false\", accent = \"fa lse\", lspace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mi(\"alph a13\", italic = \"true\", mathvariant = \"italic\"), Typesetting:-mo( \"⁢\", mathvariant = \"normal\", fence = \"false\", sep arator = \"false\", stretchy = \"false\", symmetric = \"false\", large op = \"false\", movablelimits = \"false\", accent = \"false\", lspace \+ = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mi(\"tinfty13\", italic = \"true\", mathvariant = \"italic\"), Typesetting:-mo(\"&InvisibleTi mes;\", mathvariant = \"normal\", fence = \"false\", separator = \"fal se\", stretchy = \"false\", symmetric = \"false\", largeop = \"false\" , movablelimits = \"false\", accent = \"false\", lspace = \"0.0em\", r space = \"0.0em\"), Typesetting:-mi(\"tinfty15\", italic = \"true\", m athvariant = \"italic\")), Typesetting:-mo(\"−\", mathvariant = \+ \"normal\", fence = \"false\", separator = \"false\", stretchy = \"fal se\", symmetric = \"false\", largeop = \"false\", movablelimits = \"fa lse\", accent = \"false\", lspace = \"0.2222222em\", rspace = \"0.2222 222em\"), Typesetting:-mrow(Typesetting:-mn(\"15\", mathvariant = \"no rmal\"), Typesetting:-mo(\"⁢\", mathvariant = \"normal \", fence = \"false\", separator = \"false\", stretchy = \"false\", sy mmetric = \"false\", largeop = \"false\", movablelimits = \"false\", a ccent = \"false\", lspace = \"0.0em\", rspace = \"0.0em\"), Typesettin g:-mi(\"alpha11\", italic = \"true\", mathvariant = \"italic\"), Types etting:-mo(\"⁢\", mathvariant = \"normal\", fence = \"f alse\", separator = \"false\", stretchy = \"false\", symmetric = \"fal se\", largeop = \"false\", movablelimits = \"false\", accent = \"false \", lspace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-msup(Typeset ting:-mi(\"tinfty15\", italic = \"true\", mathvariant = \"italic\"), T ypesetting:-mn(\"2\", mathvariant = \"normal\"), superscriptshift = \" 0\")), Typesetting:-mi(\"\")), mathvariant = \"normal\")), Typesetting :-mrow(Typesetting:-mn(\"15\", mathvariant = \"normal\"), Typesetting: -mo(\"⁢\", mathvariant = \"normal\", fence = \"false\", separator = \"false\", stretchy = \"false\", symmetric = \"false\", l argeop = \"false\", movablelimits = \"false\", accent = \"false\", lsp ace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-msup(Typesetting:-m i(\"tinfty15\", italic = \"true\", mathvariant = \"italic\"), Typesett ing:-mn(\"3\", mathvariant = \"normal\"), superscriptshift = \"0\")), \+ linethickness = \"1\", denomalign = \"center\", numalign = \"center\", bevelled = \"false\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI( _syslibGF'6%-I#miGF$6#Q!F'-I#moGF$6-Q*&uminus0;F'/%,mathvariantGQ'norm alF'/%&fenceGQ&falseF'/%*separatorGF8/%)stretchyGF8/%*symmetricGF8/%(l argeopGF8/%.movablelimitsGF8/%'accentGF8/%'lspaceGQ,0.2222222emF'/%'rs paceGFG-I&mfracGF$6(-F#6%-I#mnGF$6$Q\"2F'F3-F06-Q1⁢F'F3 F6F9F;F=F?FAFC/FFQ&0.0emF'/FIFW-I(mfencedGF$6$-F#6+F+-F#6)-FP6$Q\"3F'F 3FS-F,6%Q(αF'/%'italicGF8F3FS-F,6%Q)tinfty11F'/FaoQ%trueF'/F4Q'i talicF'FS-F,6%Q)tinfty15F'FeoFgo-F06-Q(−F'F3F6F9F;F=F?FAFCFEFH-F #6'FjnFSF]oFS-I%msupGF$6%-F,6%Q)tinfty13F'FeoFgoFO/%1superscriptshiftG Q\"0F'-F06-Q'+F'F3F6F9F;F=F?FAFCFEFH-F#6)-FP6$Q\"5F'F3FS-F,6%Q(al pha13F'FeoFgoFSFdpFSFioF\\p-F#6'-FP6$Q#15F'F3FS-F,6%Q(alpha11F'FeoFgoF S-Fbp6%FioFOFgpF+F3-F#6%FgqFS-Fbp6%FioFjnFgp/%.linethicknessGQ\"1F'/%+ denomalignGQ'centerF'/%)numalignGFhr/%)bevelledGF8" }{TEXT 226 2 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "A12Form:=" }{MPLTEXT 1 0 27 "2*alpha15/5/tinfty15*lambda" }{MPLTEXT 1 0 20 "+nu+ mu/(lambda-q); \n" }{MPLTEXT 1 0 71 "simplify(-residue(A12Form/lambda^2,lambda=infini ty)-A12InftyLambda1); \n" }{MPLTEXT 1 0 154 "solve(\{factor(-residue(A 12Form/lambda,lambda=infinity))=A12InftyLambda0,factor(-residue(A12For m,lambda=infinity))=factor(A12InftyLambdaMoins1)\},\{mu,nu\});\n" } {MPLTEXT 1 0 1 "\n" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I(A12FormG6\",(*( I(alpha15GF$\"\"\"I)tinfty15GF$!\"\"I'lambdaGF$F(#\"\"#\"\"&I#nuGF$F(* &I#muGF$F(,&F+F(I\"qGF$F*F*F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "<$/I#muG6\",$*&,**&I(alpha11GF%\"\"\"I )tinfty15GF%\"\"#\"#:*(I(alpha13GF%F+I)tinfty13GF%F+F,F+!\"&*(I(alpha1 5GF%F+I)tinfty11GF%F+F,F+!\"$*&F4F+F1F-\"\"$F+F,F6#F-F./I#nuGF%,$*&,&* &F0F+F,F+\"\"&*&F4F+F1F+F6F+F,!\"#F9" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 129 "mu := -(2*(3*alpha15*tinfty11*tinfty15-3*alpha15*tin fty13^2+5*alpha13*tinfty13*tinfty15-15*alpha11*tinfty15^2))/(15*tinfty 15^3);\n" }{MPLTEXT 1 0 127 "mubis := -2*(1/5*alpha15*tinfty11/tinfty1 5-1/5*alpha15*tinfty13^2/tinfty15^2 +1/3*alpha13*tinfty13/tinfty15-alp ha11)/tinfty15;\n" }{MPLTEXT 1 0 28 "factor(simplify(mu-mubis));\n" } {MPLTEXT 1 0 67 "nu := -(2*(3*alpha15*tinfty13-5*alpha13*tinfty15))/(1 5*tinfty15^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I#muG6\",$*&,**&I(alp ha11GF$\"\"\"I)tinfty15GF$\"\"#!#:*(I(alpha13GF$F*I)tinfty13GF$F*F+F* \"\"&*(I(alpha15GF$F*I)tinfty11GF$F*F+F*\"\"$*&F3F*F0F,!\"$F*F+F7#!\"# \"#:" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I&mubisG6\",$*&,**(I(alpha15GF$ \"\"\"I)tinfty15GF$!\"\"I)tinfty11GF$F*#F*\"\"&*(F)F*I)tinfty13GF$\"\" #F+!\"##F,F/*(I(alpha13GF$F*F1F*F+F,#F*\"\"$I(alpha11GF$F,F*F+F,F3" }} {PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "> I#nuG6\",$*&,&*&I(alpha13GF$\"\"\"I)tinfty15GF$F*!\"&*&I(alpha15GF$F*I )tinfty13GF$F*\"\"$F*F+!\"##F1\"#:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 210 99 "These are theoretical results to compute the nu_i's, i.e. the \+ behavior of A_\{1,2\} at lambda->\\infty" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "Minfty:=Matrix(3,3,0):\n" }{MPLTEXT 1 0 23 "Minfty[1, 1]:=tinfty15:\n" }{MPLTEXT 1 0 23 "Minfty[2,2]:=tinfty15:\n" }{MPLTEXT 1 0 23 "Minfty[3,3]:=tinfty15:\n" }{MPLTEXT 1 0 23 "Minfty[2,1]:=tinf ty13:\n" }{MPLTEXT 1 0 23 "Minfty[3,2]:=tinfty13:\n" }{MPLTEXT 1 0 23 "Minfty[3,1]:=tinfty11:\n" }{MPLTEXT 1 0 8 "Minfty;\n" }{MPLTEXT 1 0 25 "NuVector:=Matrix(3,1,0):\n" }{MPLTEXT 1 0 25 "NuVector[1,1]:=nuMoi ns1:\n" }{MPLTEXT 1 0 20 "NuVector[2,1]:=nu0:\n" }{MPLTEXT 1 0 20 "NuV ector[3,1]:=nu1:\n" }{MPLTEXT 1 0 10 "NuVector;\n" }{MPLTEXT 1 0 26 "R HSVector:=Matrix(3,1,0):\n" }{MPLTEXT 1 0 31 "RHSVector[1,1]:=2*alpha1 515/5:\n" }{MPLTEXT 1 0 31 "RHSVector[2,1]:=2*alpha1513/3:\n" } {MPLTEXT 1 0 31 "RHSVector[3,1]:=2*alpha1511/1:\n" }{MPLTEXT 1 0 11 "R HSVector;\n" }{MPLTEXT 1 0 46 "NuVectorSol:=Multiply(Minfty^(-1),RHSVe ctor);\n" }{MPLTEXT 1 0 20 "alpha1515:=alpha15:\n" }{MPLTEXT 1 0 20 "a lpha1514:=alpha14:\n" }{MPLTEXT 1 0 20 "alpha1513:=alpha13:\n" } {MPLTEXT 1 0 20 "alpha1512:=alpha12:\n" }{MPLTEXT 1 0 19 "alpha1511:=a lpha11:" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 9 "mu1:=mu:\n" }{MPLTEXT 1 0 14 "NuVector[1,1]=" }{MPLTEXT 1 0 18 "NuVectorSol[1,1];\n" } {MPLTEXT 1 0 32 "NuVector[2,1]=NuVectorSol[2,1];\n" }{MPLTEXT 1 0 32 " NuVector[3,1]=NuVectorSol[3,1];\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 12 "nuMoins1 := " }{MPLTEXT 1 0 17 "NuVectorSol[1,1];" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 7 "nu0 := " }{MPLTEXT 1 0 17 "NuVectorSol[2,1];" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 5 "nu1:=" }{MPLTEXT 1 0 17 "NuVecto rSol[3,1];" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I(mfencedG6#/I+modulenameG6\"I,TypesettingGI(_syslibG F'6'-I%mrowGF$6#-I'mtableGF$67-I$mtrGF$6(-I$mtdGF$6(-I#miGF$6&Q)tinfty 15F'/%'italicGQ%trueF'/%+foregroundGQ([0,0,0]F'/%,mathvariantGQ'italic F'/%)rowalignGQ!F'/%,columnalignGFF/%+groupalignGFF/%(rowspanGQ\"1F'/% +columnspanGFM-F56(-I#mnGF$6%Q\"0F'F>/FBQ'normalF'FDFGFIFKFNFPFDFGFI-F 26(-F56(-F86&Q)tinfty13F'F;F>FAFDFGFIFKFNF4FPFDFGFI-F26(-F56(-F86&Q)ti nfty11F'F;F>FAFDFGFIFKFNFZF4FDFGFI/%&alignGQ%axisF'/FEQ)baselineF'/FHQ 'centerF'/FJQ'|frleft|hrF'/%/alignmentscopeGF=/%,columnwidthGQ%autoF'/ %&widthGF]p/%+rowspacingGQ&1.0exF'/%.columnspacingGQ&0.8emF'/%)rowline sGQ%noneF'/%,columnlinesGFhp/%&frameGFhp/%-framespacingGQ,0.4em~0.5exF '/%*equalrowsGQ&falseF'/%-equalcolumnsGFbq/%-displaystyleGFbq/%%sideGQ &rightF'/%0minlabelspacingGFepF>FV/%%openGQ\"[F'/%&closeGQ\"]F'" }} {PARA 11 "" 1 "" {XPPMATH 20 "-I(mfencedG6#/I+modulenameG6\"I,Typesett ingGI(_syslibGF'6'-I%mrowGF$6#-I'mtableGF$67-I$mtrGF$6&-I$mtdGF$6(-I#m iGF$6&Q)nuMoins1F'/%'italicGQ%trueF'/%+foregroundGQ([0,0,0]F'/%,mathva riantGQ'italicF'/%)rowalignGQ!F'/%,columnalignGFF/%+groupalignGFF/%(ro wspanGQ\"1F'/%+columnspanGFMFDFGFI-F26&-F56(-F86&Q&ν0F'F;F>FAFDFGFI FKFNFDFGFI-F26&-F56(-F86&Q&ν1F'F;F>FAFDFGFIFKFNFDFGFI/%&alignGQ%axi sF'/FEQ)baselineF'/FHQ'centerF'/FJQ'|frleft|hrF'/%/alignmentscopeGF=/% ,columnwidthGQ%autoF'/%&widthGFeo/%+rowspacingGQ&1.0exF'/%.columnspaci ngGQ&0.8emF'/%)rowlinesGQ%noneF'/%,columnlinesGF`p/%&frameGF`p/%-frame spacingGQ,0.4em~0.5exF'/%*equalrowsGQ&falseF'/%-equalcolumnsGFjp/%-dis playstyleGFjp/%%sideGQ&rightF'/%0minlabelspacingGF]pF>/FBQ'normalF'/%% openGQ\"[F'/%&closeGQ\"]F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I(mfenced G6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6'-I%mrowGF$6#-I'mtableG F$67-I$mtrGF$6&-I$mtdGF$6(-F,6%-I&mfracGF$6)-I#mnGF$6%Q\"2F'/%+foregro undGQ([0,0,0]F'/%,mathvariantGQ'normalF'-F=6%Q\"5F'F@FC/%.linethicknes sGQ\"1F'/%+denomalignGQ'centerF'/%)numalignGFN/%)bevelledGQ&falseF'F@- I#moGF$6-Q1⁢F'FC/%&fenceGFS/%*separatorGFS/%)stretchyGF S/%*symmetricGFS/%(largeopGFS/%.movablelimitsGFS/%'accentGFS/%'lspaceG Q&0.0emF'/%'rspaceGFbo-F,6#-I#miGF$6&Q,α1515F'/%'italicGQ%trueF' F@/FDQ'italicF'/%)rowalignGQ!F'/%,columnalignGFbp/%+groupalignGFbp/%(r owspanGFK/%+columnspanGFKF`pFcpFep-F26&-F56(-F,6%-F:6)F<-F=6%Q\"3F'F@F CFIFLFOFQF@FT-F,6#-Fho6&Q,α1513F'F[pF@F^pF`pFcpFepFgpFipF`pFcpFe p-F26&-F56(-F,6%FI)nuMoins1G6\",$*&I(alpha1 5GF$\"\"\"I)tinfty15GF$!\"\"#\"\"#\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I$nu0G6\",&*(I)tinfty15GF$!\"#I)tinfty13GF$\"\"\"I(alpha15GF$F*#F (\"\"&*&F'!\"\"I(alpha13GF$F*#\"\"#\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I$nu1G6\",(*(,&*&I)tinfty11GF$\"\"\"I)tinfty15GF$F*F**$I)tinfty13 GF$\"\"#!\"\"F*F+!\"$I(alpha15GF$F*#!\"#\"\"&*(F+F3F-F*I(alpha13GF$F*# F3\"\"$*&F+F/I(alpha11GF$F*F." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 210 58 "We now check that the formula for the nu_\{i\}'s is correct." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "simplify(-residue(" } {MPLTEXT 1 0 44 "A12Form/lambda^2,lambda=infinity)-nuMoins1);" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 56 "simplify(-residue(A12Form/lambda ,lambda=infinity)-nu0);\n" }{MPLTEXT 1 0 57 "simplify(-residue(A12Form *lambda^0,lambda=infinity)-nu1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\" !" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "NuMuVector:=Ma trix(3,1,0):\n" }{MPLTEXT 1 0 27 "NuMuVector[1,1]:=nuMoins1:\n" } {MPLTEXT 1 0 22 "NuMuVector[2,1]:=nu0:\n" }{MPLTEXT 1 0 21 "NuMuVector [3,1]:=mu1:" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 18 "R:=Matrix(3,3,0): \n" }{MPLTEXT 1 0 11 "R[1,1]:=1:\n" }{MPLTEXT 1 0 11 "R[2,2]:=1:\n" } {MPLTEXT 1 0 11 "R[3,3]:=1:\n" }{MPLTEXT 1 0 54 "NuMuVectorTheo:=Multi ply(Multiply(R^(-1),Minfty^(-1))," }{MPLTEXT 1 0 12 "RHSVector);\n" } {MPLTEXT 1 0 9 "simplify(" }{MPLTEXT 1 0 11 "NuMuVector-" }{MPLTEXT 1 0 16 "NuMuVectorTheo);" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I(mfencedG6#/ I+modulenameG6\"I,TypesettingGI(_syslibGF'6'-I%mrowGF$6#-I'mtableGF$67 -I$mtrGF$6&-I$mtdGF$6(-F,6%-I&mfracGF$6)-I#mnGF$6%Q\"2F'/%+foregroundG Q([0,0,0]F'/%,mathvariantGQ'normalF'-F=6%Q\"5F'F@FC/%.linethicknessGQ \"1F'/%+denomalignGQ'centerF'/%)numalignGFN/%)bevelledGQ&falseF'F@-I#m oGF$6-Q1⁢F'FC/%&fenceGFS/%*separatorGFS/%)stretchyGFS/% *symmetricGFS/%(largeopGFS/%.movablelimitsGFS/%'accentGFS/%'lspaceGQ&0 .0emF'/%'rspaceGFbo-F:6)-F,6#-I#miGF$6&Q*α15F'/%'italicGQ%trueF' F@/FDQ'italicF'-F,6#-Fjo6&Q)tinfty15F'F]pF@F`pFIFLFOFQF@/%)rowalignGQ! F'/%,columnalignGFip/%+groupalignGFip/%(rowspanGFK/%+columnspanGFKFgpF jpF\\q-F26&-F56(-F,6&-FU6.Q*&uminus0;F'F@FCFXFZFfnFhnFjnF\\oF^o/FaoQ,0 .2222222emF'/FdoF\\r-F,6%F9FT-F:6)-F,6%-Fjo6&Q)tinfty13F'F]pF@F`pFTFio -F,6#-I%msupGF$6%FdpF/%%openGQ\"[F'/%&closeGQ\"]F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 210 65 "We have checked that Proposition dealing with A_\{1,2\} is correct ." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "A11InftyLambda4:=facto r(-residue(A11Infty/lambda^5,lambda=infinity));\n" }{MPLTEXT 1 0 70 "A 11InftyLambda3:=factor(-residue(A11Infty/lambda^4,lambda=infinity));\n " }{MPLTEXT 1 0 70 "A11InftyLambda2:=factor(-residue(A11Infty/lambda^3 ,lambda=infinity));\n" }{MPLTEXT 1 0 70 "A11InftyLambda1:=factor(-resi due(A11Infty/lambda^2,lambda=infinity));\n" }{MPLTEXT 1 0 70 "A11Infty Lambda0:=factor(-residue(A11Infty/lambda^1,lambda=infinity));\n" } {MPLTEXT 1 0 75 "A11InftyLambdaMoins1:=factor(-residue(A11Infty/lambda ^0,lambda=infinity)):\n" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I0A11InftyLa mbda4G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I0A11InftyLambda3G6\" \"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I0A11InftyLambda2G6\",$*&,&*&I (alpha14GF$\"\"\"I)tinfty15GF$F*\"\"&*&I(alpha15GF$F*I)tinfty14GF$F*! \"%F*F+!\"\"#F1\"#?" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I0A11InftyLambda 1G6\",$*&,**&I(alpha12GF$\"\"\"I)tinfty15GF$\"\"#\"#:*(I(alpha13GF$F*I )tinfty14GF$F*F+F*!#5*(I(alpha15GF$F*I)tinfty12GF$F*F+F*!\"'*(F3F*I)ti nfty13GF$F*F0F*\"\"'F*F+!\"##!\"\"\"#I" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I0A11InftyLambda0G6\",$*&,2**I(alpha15GF$\"\"\"I(epsilonGF$F*I\"hGF $F*I)tinfty15GF$\"\"#!\"$*&I%LA10GF$F*F-\"\"$\"#I*(I(alpha11GF$F*I)tin fty14GF$F*F-F.F3*(I(alpha13GF$F*I)tinfty12GF$F*F-F.\"#5**F8F*I)tinfty1 3GF$F*F6F*F-F*!#5**F)F*I)tinfty11GF$F*F6F*F-F*!\"'**F)F*F9F*F " 0 "" {MPLTEXT 1 0 9 "A11Form: =" }{MPLTEXT 1 0 53 "(4*alpha15*tinfty14-5*alpha14*tinfty15)/(20*tinft y15)" }{MPLTEXT 1 0 40 "*lambda^2+c1*lambda+c0+ rho/(lambda-q);\n" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 71 "simplify(-residue(A11Form/lambda ^4,lambda=infinity)-A11InftyLambda3); \n" }{MPLTEXT 1 0 148 "solve(\{f actor(-residue(A11Form/lambda^3,lambda=infinity))=A11InftyLambda2,fact or(-residue(A11Form/lambda^2,lambda=infinity))=A11InftyLambda1\},\{c1 \});" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I(A11FormG6\",**(,&*&I(alpha14G F$\"\"\"I)tinfty15GF$F*!\"&*&I(alpha15GF$F*I)tinfty14GF$F*\"\"%F*F+!\" \"I'lambdaGF$\"\"##F*\"#?*&I#c1GF$F*F2F*F*I#c0GF$F**&I$rhoGF$F*,&F2F*I \"qGF$F1F1F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 " " {XPPMATH 20 "<#/I#c1G6\",$*&,**&I(alpha12G6\"\"\"\"I)tinfty15G6\"\" \"#\"#:*(I(alpha13G6\"\"\"\"I)tinfty14G6\"\"\"\"I)tinfty15G6\"\"\"\"!# 5*(I(alpha15G6\"\"\"\"I)tinfty12G6\"\"\"\"I)tinfty15G6\"\"\"\"!\"'*(I( alpha15G6\"\"\"\"I)tinfty13G6\"\"\"\"I)tinfty14G6\"\"\"\"\"\"'\"\"\"I) tinfty15G6\"!\"##!\"\"\"#I" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 131 "c1 := (6*alpha15*tinfty12*tinfty15-6*alpha15*tinfty13*tinfty14+10 *alpha13*tinfty14*tinfty15-15*alpha12*tinfty15^2)/(30*tinfty15^2);" }} {PARA 11 "" 1 "" {XPPMATH 20 ">I#c1G6\",$*&,**&I(alpha12GF$\"\"\"I)tin fty15GF$\"\"#!#:*(I(alpha13GF$F*I)tinfty14GF$F*F+F*\"#5*(I(alpha15GF$F *I)tinfty12GF$F*F+F*\"\"'*(F3F*I)tinfty13GF$F*F0F*!\"'F*F+!\"##F*\"#I" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "c2:=" }{MPLTEXT 1 0 54 "( 4*alpha15*tinfty14-5*alpha14*tinfty15)/(20*tinfty15);" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I#c2G6\",$*&,&*&I(alpha14GF$\"\"\"I)tinfty15GF$F*!\" &*&I(alpha15GF$F*I)tinfty14GF$F*\"\"%F*F+!\"\"#F*\"#?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "c2theo:=1/tinfty15*(alpha1515/5*tin fty14-alpha1514/4*tinfty15);\n" }{MPLTEXT 1 0 20 "c1theo:=1/tinfty15*( " }{MPLTEXT 1 0 18 "-tinfty13*c2theo " }{MPLTEXT 1 0 22 "+ alpha1515/ 5*tinfty12" }{MPLTEXT 1 0 21 "-alpha1514/4*tinfty13" }{MPLTEXT 1 0 45 "+alpha1513/3*tinfty14-alpha1512/2*tinfty15);\n" }{MPLTEXT 1 0 21 "sim plify(c2-c2theo);\n" }{MPLTEXT 1 0 20 "simplify(c1-c1theo);" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I'c2theoG6\"*&I)tinfty15GF$!\"\",&*&I(alpha15 GF$\"\"\"I)tinfty14GF$F+#F+\"\"&*&I(alpha14GF$F+F&F+#F'\"\"%F+" }} {PARA 11 "" 1 "" {XPPMATH 20 ">I'c1theoG6\"*&I)tinfty15GF$!\"\",,*(I)t infty13GF$\"\"\"F&F',&*&I(alpha15GF$F+I)tinfty14GF$F+#F+\"\"&*&I(alpha 14GF$F+F&F+#F'\"\"%F+F'*&F.F+I)tinfty12GF$F+F0*&F3F+F*F+F4*&I(alpha13G F$F+F/F+#F+\"\"$*&I(alpha12GF$F+F&F+#F'\"\"#F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 210 61 "We have checked that the formula for A_ \{1,1\} is also correct." }}}{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 }