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{PARA 224 "> " 0 "" {MPLTEXT 1 224 21 "f := (x+1)^2*(x-5)^3;" }}{PARA 223 "" 1 "" {XPPMATH 20 "6#>I\"fG6\"*&,&I\"xGF%\"\"\"F)F)\"\"#,&F(F)!\"&F)\"\"$" } {TEXT 223 1 " " }{TEXT 223 1 " " }}}{EXCHG {PARA 222 "" 0 "" {TEXT 214 99 "We apply a change of variables and a gauge transformation to t he modified Bessel operator with nu=2" }}}{EXCHG {PARA 224 "> " 0 "" {MPLTEXT 1 224 36 "M := changeOfVars(subs(nu=2,LB), f):" }}}{EXCHG {PARA 224 "> " 0 "" {MPLTEXT 1 224 20 "L := gauge(M, 1, 1):" }}} {EXCHG {PARA 222 "" 0 "" }{PARA 222 "" 0 "" {TEXT 214 63 "From the exp -irregular singularities we get possibilities for f" }}}{EXCHG {PARA 224 "> " 0 "" {MPLTEXT 1 224 32 "Sirr := irregularSing(L, t, \{\}):" } }}{EXCHG {PARA 224 "> " 0 "" {MPLTEXT 1 224 26 "f:=besselsubst(Sirr,t, \{\});" }}{PARA 223 "" 1 "" {XPPMATH 20 "6#>I\"fG6\"7#,,*$I\"xGF%\"\"& !\"\"*$F)\"\"%\"#8*$F)\"\"$!#Y*$F)\"\"#!#5F)\"$v\"" }{TEXT 223 1 " " } {TEXT 223 1 " " }}}{EXCHG {PARA 222 "> " 0 "" {MPLTEXT 1 225 8 "f:=f[1 ]:" }}}{EXCHG {PARA 222 "" 0 "" {TEXT 214 63 "The constant can be comp uted from the exp-regular singularities" }}}{EXCHG {PARA 224 "> " 0 "" {MPLTEXT 1 224 29 "seq( subs(x=i,f) ,i=\{-1,5\} );" }}{PARA 223 "" 1 "" {XPPMATH 20 "6$!$D\"F#" }{TEXT 223 1 " " }{TEXT 223 1 " " }}} {EXCHG {PARA 224 "> " 0 "" {MPLTEXT 1 224 5 "f:=f-" }{MPLTEXT 1 224 12 "subs(x=5,f);" }}{PARA 223 "" 1 "" {XPPMATH 20 "6#>I\"fG6\",.*$I\"x GF%\"\"&!\"\"*$F(\"\"%\"#8*$F(\"\"$!#Y*$F(\"\"#!#5F(\"$v\"\"$D\"\"\"\" " }{TEXT 223 1 " " }{TEXT 223 1 " " }}}{EXCHG {PARA 222 "" 0 "" } {PARA 222 "" 0 "" {TEXT 214 34 "If we solve this with nu=0 we get:" }} }{EXCHG {PARA 224 "> " 0 "" {MPLTEXT 1 224 36 "M := changeOfVars(subs( nu=0,LB), f):" }}}{EXCHG {PARA 224 "> " 0 "" {MPLTEXT 1 224 11 "equiv( M,L);" }}{PARA 223 "" 1 "" {XPPMATH 20 "6$,$**,(*$I\"xG6\"\"\"#\"#bF'! $g\"\"#t\"\"\"F-,&F'F-!\"&F-!\"\",&F'\"\"&!\"(F-F0,&F'F-F-F-F0F0,&*&,: !'$R4\"F-F'!'(4G#F&\"&t=#*$F'\"\"$\"'DcC*$F'\"\"%\"&]U\"*$F'F2!'!H+\"* $F'\"\"'\"&eb\"*$F'\"\"(\"&id\"*$F'\"\")!%2x*$F'\"\"*\"%([\"*$F'\"#5!$ P\"*$F'\"#6F2F-I\"DG6$I*protectedGFVI(_syslibGF(F-F-**F1F-,(F&F-F'!#9F MF-F-F4F " 0 "" {MPLTEXT 1 224 18 "gen_exp(L,t,x=-1);" }}{PARA 225 "" 1 "" {XPPMATH 20 "6#7#7%! \"&\"\"$/I\"tG6\",&I\"xGF)\"\"\"F,F," }{TEXT 226 1 " " }{TEXT 226 1 " \+ " }{TEXT 226 1 " " }}}{EXCHG {PARA 224 "> " 0 "" {MPLTEXT 1 224 17 "ge n_exp(L,t,x=5);" }}{PARA 225 "" 1 "" {XPPMATH 20 "6#7#7%!\"(\"\"&/I\"t G6\",&I\"xGF)\"\"\"!\"&F," }{TEXT 226 1 " " }{TEXT 226 1 " " }{TEXT 226 1 " " }}}{EXCHG {PARA 222 "" 0 "" {TEXT 214 30 "another value for \+ nu is likely" }}}{EXCHG {PARA 224 "> " 0 "" {MPLTEXT 1 224 34 "nu := ( 12+8)/2/degree(numer(f),x);" }}{PARA 225 "" 1 "" {XPPMATH 20 "6#>I#nuG 6\"\"\"#" }{TEXT 226 1 " " }{TEXT 226 1 " " }{TEXT 226 1 " " }}} {EXCHG {PARA 222 "" 0 "" {TEXT 214 36 "With this value we get the solu tions" }}}{EXCHG {PARA 224 "> " 0 "" {MPLTEXT 1 224 36 "M := changeOfV ars(subs(nu=2,LB), f):" }}}{EXCHG {PARA 224 "> " 0 "" 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