{VERSION 3 0 "SGI MIPS UNIX" "3.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 }{CSTYLE " " -1 256 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 261 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 264 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 } {CSTYLE "" -1 268 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 " " 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 11 12 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 } 1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 256 1 {CSTYLE "" -1 -1 " " 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 256 48 "Exponents \+ of a differential operator at a point." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 81 "Take the point x=0. The functions we \+ consider now are formal power series at x=0:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{XPPEDIT 18 0 "y = sum(a[n]*x ^n,i = N .. infinity)" "6#/%\"yG-%$sumG6$*&&%\"aG6#%\"nG\"\"\")%\"xGF, F-/%\"iG;%\"NG%)infinityG" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 25 "where N is an element of " }{TEXT 257 1 "Z" } {TEXT -1 248 "=\{...,-3,-2,-1,0,1,2,3,....\}, and where a[n] are compl ex numbers. Right now we do not worry about convergence (formal series means: no convergence conditions on the a[i]). We will now apply a li near differential operator L on that and we get\nL(y) =" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "L(sum(a[n]*x^n,n = N .. infinity)) = sum(a[n]*L(x^n),x = N .. infinity);" "6#/-%\"LG6#-% $sumG6$*&&%\"aG6#%\"nG\"\"\")%\"xGF.F//F.;%\"NG%)infinityG-F(6$*&&F,6# F.F/-F%6#)F1F.F//F1;F4F5" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 42 "These latter two are equal because L is a " } {TEXT 258 1 "C" }{TEXT -1 59 "-linear map. Now lets focus on one parti cular term: L(x^n)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "appl y_L := proc(L, y, Dx,x)\n local i;\n add(coeff(L,Dx,i) * diff(y,[x$i ]), i=0..degree(L,Dx))\nend;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(app ly_LGR6&%\"LG%\"yG%#DxG%\"xG6#%\"iG6\"F--%$addG6$*&-%&coeffG6%9$9&8$\" \"\"-%%diffG6$9%7#-%\"$G6$9'F7F8/F7;\"\"!-%'degreeG6$F5F6F-F-F-" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "L:=x*Dx^4+6*Dx^3+x/(x-1)*Dx^ 2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"LG,(*&%\"xG\"\"\")%#DxG\"\"% \"\"\"F(*$)F*\"\"$F,\"\"'*&*&F'F,)F*\"\"#F,F,,&F'F(!\"\"F(!\"\"F(" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "apply_L( L, x^n, Dx,x);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,,*&*&%\"xG\"\"\",&*&*&)F&%\"nGF')F,\" \"#\"\"\"F/*$)F&\"\"#F/!\"\"F'*&*&F+F/F,F'F/*$)F&\"\"#F/F3!\"\"F'F/,&F &F'F9F'F3F'*&*&F+F/)F,\"\"$F/F/*$)F&\"\"$F/F3\"\"'*&*&F+F/F-F/F/*$)F& \"\"$F/F3!#=*&*&F+F/F,F/F/*$)F&\"\"$F/F3\"#7*&F&F/,**&*&F+F/)F,\"\"%F/ F/*$)F&\"\"%F/F3F'*&*&F+F/F=F/F/*$)F&\"\"%F/F3!\"'*&*&F+F/F-F/F/*$)F& \"\"%F/F3\"#6*&*&F+F/F,F/F/*$)F&\"\"%F/F3FgnF'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "normal(%/x^n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&*&%\"nG\"\"\",2*&)%\"xG\"\"#F&F%\"\"\"F,*$F)F&!\"\"*&F%F&F*F,! \"(F%\"\"(F*\"\"'!\"'F,*&)F%\"\"$F&F*F&F,*$F5F&F.F,F&*&)F*\"\"$F&,&F*F ,F.F,\"\"\"!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "convert (series(%,x=0),polynom);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,2*&*&%\"n G\"\"\",(!\"'F'F&\"\"(*$)F&\"\"$\"\"\"!\"\"F'F.*$)%\"xG\"\"$F.!\"\"F/* &*&F&F.,&F&F'F/F'F'F.F2F4F/*&F&F.F7F.F/*(F&F.F7F.F2F'F/*(F&F.F7F.)F2\" \"#F.F/*(F&F.F7F.)F2F-F.F/*(F&F.F7F.)F2\"\"%F.F/*(F&F.F7F.)F2\"\"&F.F/ " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "lowest_deg, lowest_coef f := ldegree(%,x), tcoeff(%,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>6$ %+lowest_degG%-lowest_coeffG6$!\"$,$*&%\"nG\"\"\",(!\"'F,F+\"\"(*$)F+ \"\"$\"\"\"!\"\"F,F4" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "fac tor(lowest_coeff);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#**%\"nG\"\"\",&F $F%!\"\"F%F%,&F$F%!\"#F%F%,&F$F%\"\"$F%F%" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 9 "solve(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&\"\"!\" \"\"\"\"#!\"$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 212 "We see that the lowest power of x of L(x^n)/x^n written as a power series in x equals -3, unless n in \{0,1,2, -3\} because then the lowest_coeff=0 in whic h case the lowest_deg is not -3 but will be higher than -3." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 109 "In other words : The valuation at x=0 (i.e. lowest power of x-0 with a non-zero coeff icient) of L(x^n)/x^n is:" }}{PARA 0 "" 0 "" {TEXT -1 29 " > -3 if n in \{0,1,2, -3\}" }}{PARA 0 "" 0 "" {TEXT -1 19 " = -3 otherwise ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 40 "Henc e the valuation of L(x^n) at x=0 is:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 "v0( L(x^n) ) > n-3 if n in \{0,1,2, \+ -3\}" }}{PARA 0 "" 0 "" {TEXT -1 44 "v0( L(x^n) ) = n-3 if n not in \+ \{0,1,2,-3\}." }}{PARA 0 "" 0 "" {TEXT -1 31 "Now consider again L(y) \+ where: " }{XPPEDIT 18 0 "y = sum(a[n]*x^n,i = N .. infinity)" "6#/%\"y G-%$sumG6$*&&%\"aG6#%\"nG\"\"\")%\"xGF,F-/%\"iG;%\"NG%)infinityG" }} {PARA 0 "" 0 "" {TEXT -1 51 "Assume that a[N]<>0, in other words: v0(y )=N. Then:" }}{PARA 0 "" 0 "" {TEXT -1 39 "v0( L(a[n] * x^n) ) >= n -3 for all n" }}{PARA 0 "" 0 "" {TEXT -1 119 "Note that if a[n]=0 then this number is infinity, which is also >= n-3. We see now that we hav e only two possibilities:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 56 "Case 1). v0(L)=N is in \{0,1,2, -3\}. Then v0(L(y )) > N-3" }}{PARA 0 "" 0 "" {TEXT -1 58 "Case 2). v0(L)=N is not in \{ 0,1,2,-3\}. Then v0(L(y)) = N-3" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 212 "In case 2, we see that v0(L(y)) is N-3, \+ which is less than infinity. If the valuation is not infinity, then th e function is not 0. In other words: L(y)<>0 in case 2. Therefore, y i s not a solution of L in case 2." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 98 "So: only in case 1 can y be a solution of L. The numbers \{0,1,2,-3\} are called the exponents of L." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 60 "Computing the \+ indicial equation of an operator L at a point." }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 58 "L := Dx^2-4*(3*x^4-2*x^3+8*x^2-9*x+3)/(x-1)^2/ (x^2+1)/x^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"LG,&*$)%#DxG\"\"# \"\"\"\"\"\"*&,,*$)%\"xG\"\"%F*\"\"$*$)F0F2F*!\"#*$)F0F)F*\"\")F0!\"*F 2F+F**(),&F0F+!\"\"F+\"\"#F*,&F6F+F+F+\"\"\")F0\"\"#F*!\"\"!\"%" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "points:=[-1, 0, 1, infinity] ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'pointsG7&!\"\"\"\"!\"\"\"%)inf inityG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "for p in points d o t[p]:=`if`(p=infinity,1/x,x-p) od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"tG6#!\"\",&%\"xG\"\"\"F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >&%\"tG6#\"\"!%\"xG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"tG6#\"\"\" ,&%\"xGF'!\"\"F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"tG6#%)infinit yG*&\"\"\"F)%\"xG!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "t[p] is the local parameter at x=p" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "for p in points do K[p]:=series(normal(apply_L(L, t[p]^n, Dx, x)/ t[p]^n),x=p) od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"KG6#!\"\"+/,& %\"xG\"\"\"F+F+,&%\"nGF'*$)F-\"\"#\"\"\"F+!\"##!#DF0\"\"!#!#dF0\"\"\"# !$l$\"\")\"\"##!$X#\"\"%\"\"$-%\"OG6#F+\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"KG6#\"\"!+1%\"xG,(!#7\"\"\"*$)%\"nG\"\"#\"\"\"F,F/ !\"\"!\"#\"#7!\"\"\"#;\"\"!!\")\"\"\"!#C\"\"#F+\"\"$-%\"OG6#F,\"\"%" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"KG6#\"\"\"+1,&%\"xGF'!\"\"F',(! \"'F'%\"nGF+*$)F.\"\"#\"\"\"F'!\"#!\")!\"\"\"\"&\"\"!F6\"\"\"#!#RF1\" \"#\"#N\"\"$-%\"OGF&\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"KG6# %)infinityG,,*&,(!#7\"\"\"%\"nGF,*$)F-\"\"#\"\"\"F,F1*$)%\"xG\"\"#F1! \"\"F,*&F1F1*$)F4\"\"$F1F6!#;*&F1F1*$)F4\"\"%F1F6!#S*&F1F1*$)F4\"\"&F1 F6!#O-%\"OG6#*&F1F1*$)F4\"\"'F1F6F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 73 "Now write them in terms of the local parameter t[p] using the v ariable T." }}{PARA 0 "" 0 "" {TEXT -1 18 "t[p] = x-p or 1/x" }} {PARA 0 "" 0 "" {TEXT -1 3 "So:" }}{PARA 0 "" 0 "" {TEXT -1 58 "x = T+ p or 1/T where T refers to the local parameter t[p]" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "for p in points do K[p] := convert(subs(x =`if`(p=infinity,1/T,T+p),K[p]),polynom) od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"KG6#!\"\",,*&,&%\"nGF'*$)F+\"\"#\"\"\"\"\"\"F/*$)% \"TG\"\"#F/!\"\"F0#!#DF.F0F3#!#dF.*$)F3F.F/#!$l$\"\")*$)F3\"\"$F/#!$X# \"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"KG6#\"\"!,.*&,(!#7\"\"\" *$)%\"nG\"\"#\"\"\"F,F/!\"\"F1*$)%\"TG\"\"#F1!\"\"F,*&F1F1F5F7\"#7\"#; F,F5!\")*$)F5F0F1!#C*$)F5\"\"$F1F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >&%\"KG6#\"\"\",.*&,(!\"'F'%\"nG!\"\"*$)F,\"\"#\"\"\"F'F1*$)%\"TG\"\"# F1!\"\"F'*&F1F1F4F6!\")\"\"&F'F4F9*$)F4F0F1#!#RF0*$)F4\"\"$F1\"#N" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"KG6#%)infinityG,**&,(!#7\"\"\"%\" nGF,*$)F-\"\"#\"\"\"F,F,)%\"TGF0F1F,*$)F3\"\"$F1!#;*$)F3\"\"%F1!#S*$)F 3\"\"&F1!#O" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "for p in poi nts do indicial_equation[p] := tcoeff(K[p],T) od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%2indicial_equationG6#!\"\",&%\"nGF'*$)F)\"\"#\"\"\" \"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%2indicial_equationG6#\"\" !,(!#7\"\"\"*$)%\"nG\"\"#\"\"\"F*F-!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%2indicial_equationG6#\"\"\",(!\"'F'%\"nG!\"\"*$)F*\"\"#\"\"\" F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%2indicial_equationG6#%)infini tyG,(!#7\"\"\"%\"nGF**$)F+\"\"#\"\"\"F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "for p in points do exponents[p] := \{solve(indicial_e quation[p],n)\} od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%*exponentsG6 #!\"\"<$\"\"!\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%*exponentsG6 #\"\"!<$!\"$\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%*exponentsG6# \"\"\"<$!\"#\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%*exponentsG6#% )infinityG<$\"\"$!\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "fo r p in points do integer_exponents[p] := select(type,exponents[p],inte ger) od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%2integer_exponentsG6#! \"\"<$\"\"!\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%2integer_expon entsG6#\"\"!<$!\"$\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%2integer _exponentsG6#\"\"\"<$!\"#\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%2 integer_exponentsG6#%)infinityG<$\"\"$!\"%" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 70 "for p in points do min_int_exp[p] := min(op(integer _exponents[p])) od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%,min_int_exp G6#!\"\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%,min_int_expG6#\" \"!!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%,min_int_expG6#\"\"\"!\" #" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%,min_int_expG6#%)infinityG!\"% " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "Above you can see how to comp ute:" }}{PARA 0 "" 0 "" {TEXT -1 48 "1) the indicial equation at each \+ of these points" }}{PARA 0 "" 0 "" {TEXT -1 17 "2) the exponents" }} {PARA 0 "" 0 "" {TEXT -1 35 "3) those exponents that are integer" }} {PARA 0 "" 0 "" {TEXT -1 32 "4) the smallest integer exponent" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 55 "We note t hat if L has a nonzero rational solution y in " }{TEXT 260 1 "C" } {TEXT -1 134 "(x), then the valuation of y at x=p, the number v_p(y), \+ is always an integer. This integer must be an element of the set expon ents[p]." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 468 "So if the set exponents[p] is empty for any point p, or if it doe s not contain any integers (i.e. if integer_exponents[p] is empty) the n such rational solution y can not exist. So when you compute rational solutions, and you find any point p for which integer_exponents[p]=\{ \} then there are no rational solutions. If integer_exponents[p] <> \{ \} for all p, then there may (or may not) be rational solutions, but i n any case, for any nonzero rational solution y you have:" }}{PARA 0 " " 0 "" {TEXT -1 2 " " }{TEXT 261 30 "v_p(y) in integer_exponents[p]" }}{PARA 0 "" 0 "" {TEXT -1 10 "Therefore:" }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{TEXT 262 24 "v_p(y) >= min_int_exp[p]" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 88 "Note that min_int_exp[p] \+ can also be computed by a Maple command in the DEtools package." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "with(DEtools): _Envdiffopdom ain:=[Dx,x];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%1_EnvdiffopdomainG7$ %#DxG%\"xG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 131 "for p in poi nts do\n min_int_exp[p]:=gen_exp(L,T,x=p,'restrict_to'=\{'minimal',in teger\});\n min_int_exp[p]:=min_int_exp[p][1][1]\nod;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%,min_int_expG6#!\"\"7#7$\"\"!/%\"TG,&%\"xG\"\" \"F/F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%,min_int_expG6#!\"\"\"\"! " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%,min_int_expG6#\"\"!7#7$!\"$/% \"TG%\"xG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%,min_int_expG6#\"\"!! \"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%,min_int_expG6#\"\"\"7#7$!\" #/%\"TG,&%\"xGF'!\"\"F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%,min_int _expG6#\"\"\"!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%,min_int_expG6 #%)infinityG7#7$!\"%/%\"TG*&\"\"\"F.%\"xG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%,min_int_expG6#%)infinityG!\"%" }}}{EXCHG {PARA 0 " " 0 "" {TEXT 263 41 "Singularities of a differential operator." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 44 "Consider \+ again a differential operator L in " }{TEXT 264 1 "C" }{TEXT -1 35 "(x )[Dx]. A point x=p is considered " }{TEXT 265 12 "non-singular" } {TEXT -1 3 " if" }}{PARA 0 "" 0 "" {TEXT -1 68 " *) None of the denom inators of the coefficients of L vanish at x=p" }}{PARA 0 "" 0 "" {TEXT -1 41 " *) lcoeff(L,Dx) does not vanish at x=p." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 59 "All other points, an d the point x=infinity, are considered " }{TEXT 266 17 "singular point s. " }{TEXT -1 52 "There are always only finitely many singular points ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 27 "If p is non-singular, then:" }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{TEXT 267 76 "exponents of L at p = \{0,1,2,..,N-1\} where N=degree(L,Dx) is the order of L." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "Why? Well, take t=x-p. Then take L(t^n). This will look l ike:" }}{PARA 0 "" 0 "" {TEXT -1 71 "lcoeff(L,Dx) * n*(n-1)*...*(n-(N- 1)) * t^(n-N) plus higher powers of t." }}{PARA 0 "" 0 "" {TEXT -1 32 "So the indicial equation equals:" }}{PARA 0 "" 0 "" {TEXT -1 41 " sub s(x=p, lcoeff(L,Dx)) * n * (n-1)* ..." }}{PARA 0 "" 0 "" {TEXT -1 35 " and the solutions are: n=0, n=1,..." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 95 "In our example, the singular points are \+ -1, 0, 1, infinity. All other poins p are non-singular." }}{PARA 0 "" 0 "" {TEXT -1 95 "Now lets try to compute the rational solutions of L. If y is a non-zero rational solution then:" }}{PARA 0 "" 0 "" {TEXT -1 27 " vp(y) is an element of:" }}{PARA 0 "" 0 "" {TEXT -1 47 "*) \+ \{0,..,N-1\} = \{0,1\} for a non-singular points" }}{PARA 0 "" 0 "" {TEXT -1 73 "*) The set integer_exponents[p], computed above, for all \+ singular points." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 "So: vp(y) >=" }}{PARA 0 "" 0 "" {TEXT -1 31 "0 for all no n-singular points p" }}{PARA 0 "" 0 "" {TEXT -1 10 "0 for p=-1" }} {PARA 0 "" 0 "" {TEXT -1 10 "-3 for p=0" }}{PARA 0 "" 0 "" {TEXT -1 10 "-2 for p=1" }}{PARA 0 "" 0 "" {TEXT -1 17 "-4 for p=infinity" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 74 "Therefore , if we write y= (x-1)^0*(x-0)^(-3)*(x-1)^(-2) * P for some P in " } {TEXT 268 1 "C" }{TEXT -1 10 "(x), then " }}{PARA 0 "" 0 "" {TEXT -1 63 "vp(P) >= 0 for all finite p, in other words: P is a polynomial." } }{PARA 0 "" 0 "" {TEXT -1 32 "Furthermore: if p=infinity then:" }} {PARA 0 "" 0 "" {TEXT -1 108 "vp(P) = vp(y / (x-1)^0*(x-0)^(-3)*(x-1)^ (-2)) = vp(y) - vp( (x-1)^0*(x-0)^(-3)*(x-1)^(-2) ) = vp(y)-5 >= -9." }}{PARA 0 "" 0 "" {TEXT -1 92 "So P is a polynomial with valuation >= \+ -9 at infinity, in other words: degree(P,x) <= 9. So:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "Y := P*mul( (x-p)^min_int_exp[p], p =\{op(points)\} minus \{infinity\});" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%\"YG*&%\"PG\"\"\"*&)%\"xG\"\"$F'),&F*\"\"\"!\"\"F.\"\"#F'!\"\"" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "v_inf_Y:=degree(denom(Y),x) -degree(numer(Y),x)-deg_P;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(v_inf _YG,&\"\"&\"\"\"%°_PG!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "v_inf_Y >= min_int_exp[infinity];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#1!\"%,&\"\"&\"\"\"%°_PG!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "solve(%,deg_P);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-% *RealRangeG6$,$%)infinityG!\"\"\"\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "deg_P:=solve(v_inf_Y = min_int_exp[infinity], deg_P); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%°_PG\"\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "P:=add(c[i]*x^i,i=0..deg_P);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%\"PG,6&%\"cG6#\"\"!\"\"\"*&&F'6#F*F*%\"xGF*F** &&F'6#\"\"#F*)F.F2\"\"\"F**&&F'6#\"\"$F*)F.F8F4F**&&F'6#\"\"%F*)F.F=F4 F**&&F'6#\"\"&F*)F.FBF4F**&&F'6#\"\"'F*)F.FGF4F**&&F'6#\"\"(F*)F.FLF4F **&&F'6#\"\")F*)F.FQF4F**&&F'6#\"\"*F*)F.FVF4F*" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 2 "L;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*$)%#Dx G\"\"#\"\"\"\"\"\"*&,,*$)%\"xG\"\"%F(\"\"$*$)F.F0F(!\"#*$)F.F'F(\"\")F .!\"*F0F)F(*(),&F.F)!\"\"F)\"\"#F(,&F4F)F)F)\"\"\")F.\"\"#F(!\"\"!\"% " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "Y;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,6&%\"cG6#\"\"!\"\"\"*&&F&6#F)F)%\"xGF)F)*&&F&6#\"\"# F))F-F1\"\"\"F)*&&F&6#\"\"$F))F-F7F3F)*&&F&6#\"\"%F))F-F " 0 "" {MPLTEXT 1 0 26 "normal(apply_L(L ,Y,Dx,x));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,$*&,^o*&&%\"cG6#\"\"#\" \"\"%\"xGF+!\"&*&&F(6#\"\"'F+)F,F1\"\"\"\"#:*&&F(6#\"\"(F+)F,F8F3\"#9* &&F(6#\"\")F+)F,F>F3\"#7*&&F(6#\"\"*F+)F,FDF3FD*&F/F3)F,\"\"&F3!\"$*&& F(6#FHF+)F,\"\"%F3F-*&&F(6#FNF+)F,\"\"$F3!\"'*&&F(6#FSF+)F,F*F3FT*&FBF 3F?F3FD*&F " 0 "" {MPLTEXT 1 0 28 "solve (\{coeffs(numer(%),x)\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<,/&%\"cG 6#\"\"\"\"\"!/&F&6#F)&F&6#\"\"#/&F&6#\"\"%F)/&F&6#\"\"'F)/&F&6#\"\"*F) /&F&6#\"\"(F)/&F&6#\"\")F)/&F&6#\"\"&F)/&F&6#\"\"$F)/F-F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "Y:=subs(%,Y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"YG*&,&&%\"cG6#\"\"#\"\"\"*&F'F+)%\"xGF*\"\"\"F+F/*& )F.\"\"$F/),&F.F+!\"\"F+\"\"#F/!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "diffop2de(L,y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #,&*&*&,,*$)%\"xG\"\"%\"\"\"\"\"$*$)F)F,F+!\"#*$)F)\"\"#F+\"\")F)!\"*F ,\"\"\"F5-%\"yG6#F)F5F+*(),&F)F5!\"\"F5\"\"#F+,&F0F5F5F5\"\"\")F)\"\"# F+!\"\"!\"%-%%diffG6$F6-%\"$G6$F)F2F5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "ratsols(%,y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7 #*&,*!\"\"\"\"\"%\"xGF'*$)F(\"\"#\"\"\"F&*$)F(\"\"$F,F'F,*$),&F)F'F(F& \"\"$F,!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "factor(op(% ));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&*$)%\"xG\"\"#\"\"\"\"\"\"F* F*F)*&),&F'F*!\"\"F*\"\"#F))F'\"\"$F)!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "One more example:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 144 "L:=Dx^3+6*x*(2*x^2+3)/(x^2-2)/(x^2+1)*Dx^2+3*(x^2+2)*(11*x^4- 5*x^2-2)/x^2/(x^2+1)\n/(x^2-2)^2*Dx+3*(x^2+2)*(5*x^4+5*x^2+2)/x^3/(x^2 +1)/(x^2-2)^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"LG,**$)%#DxG\"\" $\"\"\"\"\"\"*&*(%\"xGF+,&*$)F.\"\"#F*F2F)F+F+)F(F2F*F**&,&F0F+!\"#F+ \"\"\",&F0F+F+F+\"\"\"!\"\"\"\"'*&*(,&F0F+F2F+F+,(*$)F.\"\"%F*\"#6F0! \"&F6F+F+F(F+F**()F.\"\"#F*F8\"\"\")F5\"\"#F*F:F)*&*&F>F*,(F@\"\"&F0FN F2F+F+F**()F.\"\"$F*F8\"\"\")F5\"\"#F*F:F)" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 38 "factors(numer(lcoeff(L,Dx))*denom(L));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$\"\"\"7%7$,&*$)%\"xG\"\"#\"\"\"F$!\"#F$F+7$F* \"\"$7$,&F(F$F$F$F$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "[seq (i[1],i=%[2])];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%,&*$)%\"xG\"\"#\" \"\"\"\"\"!\"#F*F',&F%F*F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "singular_points:=[seq([RootOf(i,x),\"is a root of\",i],i=%),[inf inity]];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%0singular_pointsG7&7%-%' RootOfG6#,&*$)%#_ZG\"\"#\"\"\"\"\"\"!\"#F0Q-is~a~root~of6\",&*$)%\"xGF .F/F0F1F07%\"\"!F2F77%-F(6#,&F+F0F0F0F2,&F5F0F0F07#%)infinityG" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 143 "for p in singular_points do \n min_int_exp[p]:=gen_exp(L,T,x=p[1],'restrict_to'=\{'minimal',integ er\});\n min_int_exp[p]:=min_int_exp[p][1][1]\nod;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>&%,min_int_expG6#7%-%'RootOfG6#,&*$)%#_ZG\"\"#\"\"\" \"\"\"!\"#F1Q-is~a~root~of6\",&*$)%\"xGF/F0F1F2F17#7$!\"$/%\"TG,&F8F1F (!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%,min_int_expG6#7%-%'RootO fG6#,&*$)%#_ZG\"\"#\"\"\"\"\"\"!\"#F1Q-is~a~root~of6\",&*$)%\"xGF/F0F1 F2F1!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%,min_int_expG6#7%\"\"!Q -is~a~root~of6\"%\"xG7#7$!\"\"/%\"TGF+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%,min_int_expG6#7%\"\"!Q-is~a~root~of6\"%\"xG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%,min_int_expG6#7%-%'RootOfG6#,&*$)%#_ZG\" \"#\"\"\"\"\"\"F1F1Q-is~a~root~of6\",&*$)%\"xGF/F0F1F1F17#7$\"\"!/%\"T G,&F7F1F(!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%,min_int_expG6#7% -%'RootOfG6#,&*$)%#_ZG\"\"#\"\"\"\"\"\"F1F1Q-is~a~root~of6\",&*$)%\"xG F/F0F1F1F1\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%,min_int_expG6#7 #%)infinityG7#7$\"\"\"/%\"TG*&\"\"\"F/%\"xG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%,min_int_expG6#7#%)infinityG\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "P:='P'; Y:=P*mul(i[3]^min_int_exp[i], i= \{op(singular_points)\} minus \{[infinity]\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"PGF$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"YG*&%\" PG\"\"\"*&%\"xG\"\"\"),&*$)F)\"\"#F'\"\"\"!\"#F0\"\"$F'!\"\"" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "deg_P:='deg_P';degree(denom( Y),x)-degree(numer(Y),x)-deg_P;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%& deg_PGF$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&\"\"(\"\"\"%°_PG!\"\" " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "%=min_int_exp[[infinity ]];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&\"\"(\"\"\"%°_PG!\"\"F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "deg_P:=solve(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%°_PG\"\"'" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 28 "P:=add(c[i]*x^i,i=0..deg_P);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"PG,0&%\"cG6#\"\"!\"\"\"*&&F'6#F*F*%\"xGF*F**&&F' 6#\"\"#F*)F.F2\"\"\"F**&&F'6#\"\"$F*)F.F8F4F**&&F'6#\"\"%F*)F.F=F4F**& &F'6#\"\"&F*)F.FBF4F**&&F'6#\"\"'F*)F.FGF4F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "\{coeffs(numer(normal(apply_L(L,Y,Dx,x))),x)\}; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<),$&%\"cG6#\"\"&!\"$,(&F&6#\"\"#! #[&F&6#\"\"!F.&F&6#\"\"'\"$#>,$&F&6#\"\"\"\"#7,&F7!#XF%\"#g,&F%\"#:&F& 6#\"\"$FB,&F7!#:F@!#F,$F@!#7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "Y:=normal(subs(solve(%),Y));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%\"YG*&,,&%\"cG6#\"\"#!\"\"&F(6#\"\"'\"\"%*&F'\"\"\")%\"xGF*\"\"\"F1* &&F(6#F/F1)F3F/F4F1*&F,F1)F3F.F4F1F4*&F3\"\"\"),&*$F2F4F1!\"#F1\"\"$F4 !\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 103 "3-dimensional space of \+ rational solutions. Order(L)=3, therefore every solution is a rational function." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 218 "Assignment: Compute the exponents of the following operator at the points p=0 and p=1 like at the top of the worksheet. You can use \+ gen_exp to do the remaining exponents. Then compute all rational solut ions like above." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 142 "L:=Dx^ 3+2*(x^4+6*x^2+3)/x/(x-1)/(x+1)/(x^2+1)*Dx^2-3*(3*x^4+5+12*x^2)/x^2/(x -1)/(\nx+1)/(x^2+1)*Dx+3*(3*x^4+5+12*x^2)/x^3/(x-1)/(x+1)/(x^2+1);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**$)%#DxG\"\"$\"\"\"\"\"\"*&*&,(*$) %\"xG\"\"%F(F)*$)F/\"\"#F(\"\"'F'F)F))F&F3F(F(**F/\"\"\",&F/F)!\"\"F) \"\"\",&F/F)F)F)\"\"\",&F1F)F)F)\"\"\"!\"\"F3*&*&,(F-F'\"\"&F)F1\"#7F) F&F)F(**)F/\"\"#F(F8\"\"\"F;\"\"\"F=\"\"\"F?!\"$*&FBF(**)F/\"\"$F(F8\" \"\"F;\"\"\"F=\"\"\"F?F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 89 "Perha ps it is better to start with an easier one (all factors in denom(L) h ave degree 1)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "L:=Dx^3+( x-4)/x/(x-2)*Dx^2-12/x^2*Dx+12/x^2/(x-2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"LG,**$)%#DxG\"\"$\"\"\"\"\"\"*&*&,&%\"xGF+!\"%F+F+) F(\"\"#F*F**&F/\"\"\",&F/F+!\"#F+\"\"\"!\"\"F+*&F(F**$)F/\"\"#F*F8!#7* &F*F**&)F/\"\"#F*F5\"\"\"F8\"#7" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "How to compute the exponents at x=infinity?" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 12 "p:=infinity;" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%\"pG%)infinityG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "t:=1 /x; # local parameter" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"tG*&\"\" \"F&%\"xG!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "series(no rmal(apply_L(L, t^n, Dx, x)/t^n),x=p);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**&,*\"#7\"\"\"%\"nG\"#6*$)F(\"\"#\"\"\"!\"#*$)F(\"\"$F-!\"\"F- *$)%\"xG\"\"$F-!\"\"F'*&,(F(F.F*F.\"#CF'F-*$)F5\"\"%F-F7F'*&,(F(!\"%F* F@\"#[F'F-*$)F5\"\"&F-F7F'-%\"OG6#*&F-F-*$)F5\"\"'F-F7F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "convert(%,polynom);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&,*\"#7\"\"\"%\"nG\"#6*$)F(\"\"#\"\"\"!\"#*$)F( \"\"$F-!\"\"F-*$)%\"xG\"\"$F-!\"\"F'*&,(F(F.F*F.\"#CF'F-*$)F5\"\"%F-F7 F'*&,(F(!\"%F*F@\"#[F'F-*$)F5\"\"&F-F7F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "subs(x=1/T,%); # express it in terms of the local par ameter" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&,*\"#7\"\"\"%\"nG\"#6*$) F(\"\"#\"\"\"!\"#*$)F(\"\"$F-!\"\"F')%\"TGF1F-F'*&,(F(F.F*F.\"#CF'F')F 4\"\"%F-F'*&,(F(!\"%F*F<\"#[F'F')F4\"\"&F-F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "tcoeff(%,T);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# ,*\"#7\"\"\"%\"nG\"#6*$)F&\"\"#\"\"\"!\"#*$)F&\"\"$F+!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "ind_eq:=%; # indicicial equation at x=infinity" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'ind_eqG,*\"#7\"\"\"% \"nG\"#6*$)F(\"\"#\"\"\"!\"#*$)F(\"\"$F-!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "solve(%); # exponents at infinity" }}{PARA 11 " " 1 "" {XPPMATH 20 "6%!\"%\"\"$!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 107 "Now compute the exponents of L at x=0 and x=2 in the same way. Then try to compute the rational solutions.." }}}}{MARK "65 0 0" 107 }{VIEWOPTS 1 1 0 1 1 1803 }