{VERSION 2 3 "DEC ALPHA UNIX" "2.3" } {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 "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 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 "2 D Input" 2 19 "" 0 1 255 0 0 1 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 "" -1 257 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 274 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 275 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 276 "" 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 "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }1 0 0 0 6 6 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 4 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Out put" 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 "T itle" 0 18 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 1 0 0 0 0 0 0 }3 0 0 -1 12 12 0 0 0 0 0 0 19 0 }{PSTYLE "Author" 0 19 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 8 8 0 0 0 0 0 0 -1 0 } {PSTYLE "" 0 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 } 3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 257 1 {CSTYLE "" -1 -1 " " 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 0 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 } 3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 49 "An Integral of a Product \+ of four Bessel Functions" }}{PARA 19 "" 0 "" {TEXT -1 0 "" }}{PARA 19 "" 0 "" {TEXT -1 15 "Fr\351d\351ric Chyzak" }}{PARA 257 "" 0 "" {TEXT -1 28 "(Version of January 8, 1998)" }}{PARA 258 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 86 "In (Glasser, M. L. and Montal di E. (1994): Some Integrals Involving Bessel Functions, " }{TEXT 259 20 "J. Math. Anal. Appl." }{TEXT -1 2 ", " }{TEXT 276 3 "183" }{TEXT -1 179 ":577-590), Glasser and Montaldi compute a closed form for an i ntegral of a product of two Bessel functions, and suggest that their t reatment should extend to the following example" }}}{EXCHG {PARA 256 " " 0 "" {XPPEDIT 18 0 "int(x*J[1](a*x)*I[1](a*x)*Y[0](x)*K[0](x),x=0..i nfinity)=-ln(1-a^4)/2/Pi/a^2" "/-%$intG6$*,%\"xG\"\"\"-&%\"JG6#\"\"\"6 #*&%\"aGF(F'F(F(-&%\"IG6#\"\"\"6#*&F0F(F'F(F(-&%\"YG6#\"\"!6#F'F(-&%\" KG6#F<6#F'F(/F';F<%)infinityG,$**-%#lnG6#,&\"\"\"F(*$F0\"\"%!\"\"F(\" \"#FO%#PiGFO*$F0\"\"#FOFO" }{TEXT -1 1 "," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 255 "which is of interest because it contains each of the fou r types of Bessel functions. This integral is one of numerous integra ls containing four (or more) Bessel functions. See for instance (Prud nikov, A. P., Brychkov, Yu. A. and Marichev, O. I. (1986): " }{TEXT 260 49 "Integrals and Series. Volume 2: Special functions" }{TEXT -1 35 ", Gordon and Breach; Sec. 2.16.47)." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 91 "In this session, we deal with the integral above and deri ve a closed form for it using our " }{HYPERLNK 17 "Mgfun" 2 "Mgfun" " " }{TEXT -1 1 " " }{TEXT -1 44 "package in an intimate interaction wit h the " }{HYPERLNK 17 "gfun" 2 "gfun" "" }{TEXT -1 1 " " }{TEXT -1 8 " package." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(Mgfun);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#7(%,diag_of_sysG%+int_of_sysG%+pol_to _sysG%+sum_of_sysG%(sys*sysG%(sys+sysG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "with(gfun);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7V%(La placeG%.algebraicsubsG%.algeqtodiffeqG%.algeqtoseriesG%.algfuntoalgeqG %&borelG%.cauchyproductG%.diffeq*diffeqG%.diffeq+diffeqG%2diffeqtohomd iffeqG%,diffeqtorecG%)guesseqnG%(guessgfG%0hadamardproductG%0holexprto diffeqG%)invborelG%,listtoalgeqG%-listtodiffeqG%0listtohypergeomG%+lis ttolistG%.listtoratpolyG%*listtorecG%-listtoseriesG%5listtoseries/Lapl aceG%1listtoseries/egfG%4listtoseries/lgdegfG%4listtoseries/lgdogfG%1l isttoseries/ogfG%4listtoseries/revegfG%4listtoseries/revogfG%,maxdegco effG%*maxdegeqnG%,maxordereqnG%,mindegcoeffG%*mindegeqnG%,minordereqnG %*optionsgfG%,poltodiffeqG%)poltorecG%/ratpolytocoeffG%(rec*recG%(rec+ recG%,rectodiffeqG%,rectohomrecG%*rectoprocG%.seriestoalgeqG%/seriesto diffeqG%2seriestohypergeomG%-seriestolistG%0seriestoratpolyG%,seriesto recG%/seriestoseriesG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "More spe cifically, the " }{TEXT 264 4 "gfun" }{TEXT -1 74 " package will be us ed to prepare a system of PDE's for the application of " }{TEXT 263 5 "Mgfun" }{TEXT -1 55 " functions, and to solve the ODE that is output \+ by the " }{TEXT 265 5 "Mgfun" }{TEXT -1 9 " package." }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 56 "Search for a System of PDE's Satisfied by the Integrand " }{XPPEDIT 18 0 "x*J[1](a*x)*I[1](a*x)*Y[0](x)*K[0](x) " "*,%\"xG\"\"\"-&%\"JG6#\"\"\"6#*&%\"aGF$F#F$F$-&%\"IG6#\"\"\"6#*&F,F $F#F$F$-&%\"YG6#\"\"!6#F#F$-&%\"KG6#F86#F#F$" }}{EXCHG {PARA 0 "" 0 " " {TEXT -1 11 "We use the " }{HYPERLNK 17 "gfun" 2 "gfun" "" }{TEXT -1 1 " " }{TEXT -1 98 "package to compute a system of PDE's satisfied \+ by each factor of the integrand. Next, we use the " }{HYPERLNK 17 "Mg fun" 2 "Mgfun" "" }{TEXT -1 2 " p" }{TEXT -1 62 "ackage to derive a sy stem of PDE's satisfied by their product." }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 29 "System of PDE's Satisfied by " }{XPPEDIT 18 0 "x" "I\"xG6 \"" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "The identity function " } {XPPEDIT 18 0 "x" "I\"xG6\"" }{TEXT -1 54 " trivially satisfies the fo llowing differential system" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "sys1:=\{x*diff(h(x,a),x)-1,diff(h(x,a),a)\}:" ">%%sys1G<$,&*&%\"xG \"\"\"-%%diffG6$-%\"hG6$F'%\"aGF'F(F(\"\"\"!\"\"-F*6$-F-6$F'F/F/" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "where each entry " }{XPPEDIT 18 0 "expr" "I%exprG6\"" }{TEXT -1 33 " in the set denotes the equation " } {XPPEDIT 18 0 "expr=0" "/%%exprG\"\"!" }{TEXT -1 1 "." }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 29 "System of PDE's Satisfied by " }{XPPEDIT 18 0 "J[1](a*x)" "-&%\"JG6#\"\"\"6#*&%\"aG\"\"\"%\"xGF*" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "We compute a system of PDE's for" }}} {EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "f:=BesselJ(1,a*x):" ">%\"fG-%( BesselJG6$\"\"\"*&%\"aG\"\"\"%\"xGF*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "by first computing an ODE with respect to the variable " } {XPPEDIT 18 0 "x" "I\"xG6\"" }{TEXT -1 7 " using " }{HYPERLNK 17 "gfun [holexprtodiffeq]" 2 "gfun[holexprtodiffeq]" "" }{TEXT -1 1 "," } {TEXT -1 36 " and next considering symmetries of " }{XPPEDIT 18 0 "f" "I\"fG6\"" }{TEXT -1 29 " to derive a complete system." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "holexprtodiffeq(f,y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<%,(*&,&!\"\"\"\"\"*&%\"aG\"\"#%\"xGF+F(F(-% \"yG6#F,F(F(*&F,F(-%%diffG6$F-F,F(F(*&F,F+-F26$F1F,F(F(/-F.6#\"\"!F:/- -%\"DG6#F.F9,$F*#F(F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "de q:=op(remove(type,\",equation)):" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "subs(y(x)=h(x,a),deq);" "-%%subsG6$/-%\"yG6#%\"xG-%\"hG 6$F)%\"aG%$deqG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&,&!\"\"\"\"\"*& %\"aG\"\"#%\"xGF*F'F'-%\"hG6$F+F)F'F'*&F+F'-%%diffG6$F,F+F'F'*&F+F*-F1 6$F0F+F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "subs(\{x=a,a=x, y(x)=h(x,a)\},deq);" "-%%subsG6$<%/%\"xG%\"aG/F(F'/-%\"yG6#F'-%\"hG6$F 'F(%$deqG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&,&!\"\"\"\"\"*&%\"aG \"\"#%\"xGF*F'F'-%\"hG6$F+F)F'F'*&F)F'-%%diffG6$F,F)F'F'*&F)F*-F16$F0F )F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "x*diff(h(x,a),x)-a*d iff(h(x,a),a):" ",&*&%\"xG\"\"\"-%%diffG6$-%\"hG6$F$%\"aGF$F%F%*&F,F%- F'6$-F*6$F$F,F,F%!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "s ys2:=\{\"\"\",\"\",\"\}:" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 29 "Sys tem of PDE's Satisfied by " }{XPPEDIT 18 0 "I[1](a*x)" "-&%\"IG6#\"\" \"6#*&%\"aG\"\"\"%\"xGF*" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "We com pute a system of PDE's for" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "f:=BesselI(1,a*x):" ">%\"fG-%(BesselIG6$\"\"\"*&%\"aG\"\"\"%\"xGF*" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "in the same way as in the previo us section." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "holexprtodif feq(f,y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<%,(*&,&!\"\"\"\"\"*&% \"aG\"\"#%\"xGF+F'F(-%\"yG6#F,F(F(*&F,F(-%%diffG6$F-F,F(F(*&F,F+-F26$F 1F,F(F(/-F.6#\"\"!F:/--%\"DG6#F.F9,$F*#F(F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "deq:=op(remove(type,\",equation)):" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "subs(y(x)=h(x,a),deq);" "-%%subsG6$/- %\"yG6#%\"xG-%\"hG6$F)%\"aG%$deqG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#, (*&,&!\"\"\"\"\"*&%\"aG\"\"#%\"xGF*F&F'-%\"hG6$F+F)F'F'*&F+F'-%%diffG6 $F,F+F'F'*&F+F*-F16$F0F+F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "subs(\{x=a,a=x,y(x)=h(x,a)\},deq);" "-%%subsG6$<%/%\"xG%\"aG/F(F'/- %\"yG6#F'-%\"hG6$F'F(%$deqG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&,&! \"\"\"\"\"*&%\"aG\"\"#%\"xGF*F&F'-%\"hG6$F+F)F'F'*&F)F'-%%diffG6$F,F)F 'F'*&F)F*-F16$F0F)F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "x*d iff(h(x,a),x)-a*diff(h(x,a),a):" ",&*&%\"xG\"\"\"-%%diffG6$-%\"hG6$F$% \"aGF$F%F%*&F,F%-F'6$-F*6$F$F,F,F%!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "sys3:=\{\"\"\",\"\",\"\}:" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 29 "System of PDE's Satisfied by " }{XPPEDIT 18 0 "Y[0](x) " "-&%\"YG6#\"\"!6#%\"xG" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "We com pute a system of PDE's for" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "f:=BesselY(0,x):" ">%\"fG-%(BesselYG6$\"\"!%\"xG" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 55 "by first computing an ODE with respect to the vari able " }{XPPEDIT 18 0 "x" "I\"xG6\"" }{TEXT -1 10 " by using " } {HYPERLNK 17 "gfun[holexprtodiffeq]" 2 "gfun[holexprtodiffeq]" "" } {TEXT -1 25 ", and next encoding that " }{XPPEDIT 18 0 "f" "I\"fG6\"" }{TEXT -1 20 " does not depend on " }{XPPEDIT 18 0 "a" "I\"aG6\"" } {TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "holexprtod iffeq(f,y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&%\"xG\"\"\"-%%di ffG6$-F(6$-%\"yG6#F%F%F%F&F&F*F&*&F%F&F,F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "sys4:=\{subs(y(x)=h(x,a),\"),diff(h(x,a),a)\};" "> %%sys4G<$-%%subsG6$/-%\"yG6#%\"xG-%\"hG6$F,%\"aG%\"\"G-%%diffG6$-F.6$F ,F0F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%sys4G<$-%%diffG6$-%\"hG6$% \"xG%\"aGF-,(*&F,\"\"\"-F'6$-F'6$F)F,F,F0F0F3F0*&F,F0F)F0F0" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 29 "System of PDE's Satisfied by " } {XPPEDIT 18 0 "K[0](x)" "-&%\"KG6#\"\"!6#%\"xG" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "We compute a system of PDE's for" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "f:=BesselK(0,x):" ">%\"fG-%(BesselKG6$\"\"!%\"x G" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "in the same way as in the pr evious section." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "holexprt odiffeq(f,y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&%\"xG\"\"\"-%% diffG6$-F(6$-%\"yG6#F%F%F%F&F&F*F&*&F%F&F,F&!\"\"" }}}{EXCHG {PARA 0 " > " 0 "" {XPPEDIT 19 1 "sys5:=\{subs(y(x)=h(x,a),\"),diff(h(x,a),a)\}; " ">%%sys5G<$-%%subsG6$/-%\"yG6#%\"xG-%\"hG6$F,%\"aG%\"\"G-%%diffG6$-F .6$F,F0F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%sys5G<$-%%diffG6$-%\"h G6$%\"xG%\"aGF-,(*&F,\"\"\"-F'6$-F'6$F)F,F,F0F0F3F0*&F,F0F)F0!\"\"" }} }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 40 "System of PDE's Satisfied by th e Product" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 59 "Computing a system fo r the product is now a simple call to " }{HYPERLNK 17 "Mgfun[`sys*sys` ]" 2 "Mgfun[`sys*sys`]" "" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 41 "sys:=`sys*sys`(sys1,sys2,sys3,sys4,sys5);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$sysG<',D*&,(*&%\"aG\"\"%%\"xGF+!$_%\"%\"4 #\"\"\"*$F,F+\"$S(F/-%\"hG6$F,F*F/F/*(F*\"\"#F,F6-%%diffG6$-F86$-F86$- F86$F2F*F*F,F,F/\"$5#*&,(*$F,\"\"$!$3\"*&F*F+F,\"\"(\"$+(*$F,FG\"#?F/- F86$-F86$-F86$F2F,F,F,F/F/*&F,\"\"&-F86$-F86$FKF,F,F/!#)**(F*F/F,F+-F8 6$-F86$-F86$-F86$F>F,F,F,F,F/\"$S\"*&,(*$F,\"\"'F[o*$F,F6\"$0)*&F,F_oF *F+\"$?%F/FMF/F/*&F,F_o-F86$FSF,F/\"#N*&,(*&F*F/F,F/!%3>*&F*FRF,FR\"%+ G*&F*F/F,FR\"$+%F/FinF/F/*(F*F/F,FDFenF/!$q(*&F,F+FUF/\"$v\"*&,(*&F*F/ F,F+!%G:*&F*FRF,F+!%sKF*\"%77F/F>F/F/*&,(*$F,FR!$7%*&F*F+F,FRFHF,!%\"4 #F/FOF/F/*&F,FG-F86$FeoF,F/FR*&,(*$F*FD!#[*&F*FGF,F+!$+%*&F*FDF,F+F_pF /-F86$FF/!#E*&F,F/FOF/FD*&F*FDF[rF/!\")*(F*F/F,F6FgnF/!#7*&F *F6FF/F/*&,(F,!#@F]qFhtF_qF`uF/FOF/F/FetFftFgt\"#kFitFJF[u!#W*(F,FDF *F6-F86$F7F,F/!#5,B*&,(FDF/F)\"#7F0FJF/F2F/F/F5\"#;FcuFDFQFis*&,(F^oFi sF`o!\"#FboFJF/FMF/F/FdoF\\t*&,(F^pFftFjo!$!>F\\p!#CF/FinF/F/F`pF`uFbp F+*&,(Ffp!#;FhpFhvF*\"$Q#F/F>F/F/*&,(F,FhsF]qFbwF_qFhvF/FOF/F/FetF+Fgt \"#w*&,(Fir\"#)*F[sFiuFgrFftF/FFhs*&F*F6F[rF/Fis*&F*F/F " 0 "" {MPLTEXT 1 0 61 "ode:=op(int_of_sys(sys,x=-infinity..infinity,takayama_algo));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$odeG,,*&%\"aG\"\"$-%\"hG6#F'\"\"\" \"#K*&,&*$F'\"\"(F,*$F'F(!\"\"F,-%%diffG6$-F56$-F56$-F56$F)F'F'F'F'F,F ,*&,&!\"$F,*$F'\"\"%\"$.\"F,F;F,F,*&,&*$F'\"\"#!\"%*$F'\"\"'\"#;F,F7F, F,*&,&*$F'\"\"&\"#tF'F(F,F9F,F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 69 "However, the justification that the algorithm selected by the opti on " }{XPPEDIT 18 0 "takayama_algo" "I.takayama_algoG6\"" }{TEXT -1 0 "" }{TEXT -1 97 " applies to the integral under consideration is rathe r technical and is beyond this presentation." }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 27 "Resolution of the Final ODE" }}{EXCHG {PARA 0 "" 0 " " {TEXT -1 11 "We use the " }{HYPERLNK 17 "gfun" 2 "gfun" "" }{TEXT -1 1 " " }{TEXT -1 34 "package to solve for the solution " }{XPPEDIT 18 0 "h(a)" "-%\"hG6#%\"aG" }{TEXT -1 199 " which corresponds to the i ntegral to be computed. Due to its integral representation, this func tion is analytic at 0, hence admits a Taylor expansion at 0. We proce ed to compute a closed form for " }{XPPEDIT 18 0 "h(a)" "-%\"hG6#%\"aG " }{TEXT -1 132 " by summation of this expansion. To this end, we det ermine a recurrence equation on the coefficients of the Taylor expansi on using " }{HYPERLNK 17 "gfun[diffeqtorec]" 2 "gfun[diffeqtorec]" "" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "ore:=diffeq torec(ode,h(a),u(n));" ">%$oreG-%,diffeqtorecG6%%$odeG-%\"hG6#%\"aG-% \"uG6#%\"nG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$oreG<%,&*&,&%\"nG\" \"\"\"\"#F*F*-%\"uG6#F)F*F**&,&F)!\"\"!\"'F*F*-F-6#,&F)F*\"\"%F*F*F*/- F-6#F*\"\"!/-F-6#\"\"$F:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 274 5 "Maple " }{TEXT -1 32 " readily solves this recurrence:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "rsol:=rsolve(ore,u(n));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%rsolG,**(,&-%\"uG6#\"\"##\"\"\"F+-F)6#\"\"!#F-\"\"%F --%&GAMMAG6#,&%\"nGF-F+F-F--F46#,&F7F-\"\"$F-!\"\"F+**,&F.F1F(#F F-),$FAF " 0 "" {MPLTEXT 1 0 44 "collect(map(normal,rsol,expanded),u,factor); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&**,&\"\"\"F&)-%'RootOfG6#,&*$%#_ ZG\"\"#F&F&F&%\"nGF&F&,&F&F&)!\"\"F/F&F&,&F/F&F.F&F2-%\"uG6#\"\"!F&#F& F.**,&F2F&F'F&F&F0F&F3F2-F56#F.F&F2" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "We perform the corresponding change of variable, " }{XPPEDIT 18 0 "n=2*p" "/%\"nG*&\"\"#\"\"\"%\"pGF&" }{TEXT -1 1 "," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "subs(\{n=2*p,(-1)^n=1,RootOf(_Z^2+1 )^n=(-1)^p\},\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*(,&\"\"\"F&)! \"\"%\"pGF&F&,&F)\"\"#F+F&F(-%\"uG6#\"\"!F&F&*(,&F(F&F'F&F&F*F(-F-6#F+ F&!\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 8 "so that " }{TEXT 275 5 " Maple" }{TEXT -1 27 " can sum the Taylor series:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "sum(\"*a^(2*p),p=0..infinity);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#-%$sumG6$*&,&*(,&\"\"\"F*)!\"\"%\"pGF*F*,&F-\"\" #F/F*F,-%\"uG6#\"\"!F*F**(,&F,F*F+F*F*F.F,-F16#F/F*!\"#F*)%\"aG,$F-F/F */F-;F3%)infinityG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "h:=co llect(value(expand(\")),u);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hG, &*&,&*&%\"aG!\"#-%#lnG6#,&*$F)\"\"#!\"\"\"\"\"F2F2#F1F0*&F)F*-F,6#,&F/ F2F2F2F2#F2F0F2-%\"uG6#\"\"!F2F2*&,&F(F1F4F1F2-F:6#F0F2F2" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "It only remains to evaluate " }{XPPEDIT 18 0 "u[0]" "&%\"uG6#\"\"!" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "u[2]" "&%\"uG6#\"\"#" }{TEXT -1 20 ". We first compute " }{XPPEDIT 18 0 "u[ 0]" "&%\"uG6#\"\"!" }{TEXT -1 46 " and find it is 0 by inversion of li mits. Let" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "f:=x*BesselJ(1, a*x)*BesselI(1,a*x)*BesselY(0,x)*BesselK(0,x):" ">%\"fG*,%\"xG\"\"\"-% (BesselJG6$\"\"\"*&%\"aGF&F%F&F&-%(BesselIG6$\"\"\"*&F,F&F%F&F&-%(Bess elYG6$\"\"!F%F&-%(BesselKG6$F5F%F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "be the integrand. We have:" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "int(limit(f,a=0),x=0..infinity);" "-%$intG6$-%&limitG6$ %\"fG/%\"aG\"\"!/%\"xG;F+%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 124 "In the same way, each coefficient of the Taylor series for the integral is obtained by inve rsion of limits. In particular, " }{XPPEDIT 18 0 "kappa=u[2]" "/%&kap paG&%\"uG6#\"\"#" }{TEXT -1 6 ", but " }{TEXT 257 5 "Maple" }{TEXT -1 31 " is not capable of integrating:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "kappa=int(coeff(series(normal(diff(f,a,a)),a=0),a,0)/ 2,x=0..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%&kappaG-%$intG6 $,$*(%\"xG\"\"$-%(BesselYG6$\"\"!F*\"\"\"-%(BesselKGF.F0#F0\"\"%/F*;F/ %)infinityG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "(This integral for " }{XPPEDIT 18 0 "kappa" "I&kappaG6\"" }{TEXT -1 33 " cannot be compu ted by a call to " }{TEXT 258 3 "int" }{TEXT -1 82 " using the Release 4, but the next release will probably be able to integrate it.)" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "We obtain the following form for \+ " }{XPPEDIT 18 0 "h(a)" "-%\"hG6#%\"aG" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "-combine(normal(-subs(\{u(0)=0,u(2) =kappa\},h)),ln,symbolic);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(-%#l nG6#*&,&*$%\"aG\"\"#!\"\"\"\"\"F.F.,&F*F.F.F.F.F.%&kappaGF.F+!\"#F-" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "It only remains to be proved tha t " }{XPPEDIT 18 0 "kappa=1/2/Pi" "/%&kappaG*(\"\"\"\"\"\"\"\"#!\"\"%# PiGF(" }{TEXT -1 300 ". We do not do it, since computing this last in tegral which is a constant lies outside the scope of the theory of hol onomy. With this example, we have reduced the problem of evaluating a parametrized integral to the evaluation of a non-parametrized integra l. In case there were no closed form for " }{XPPEDIT 18 0 "kappa" "I& kappaG6\"" }{TEXT -1 136 ", we could at least perform a simple numeric al evaluation and return a result in terms of this numerical value and the series above for " }{XPPEDIT 18 0 "kappa=1" "/%&kappaG\"\"\"" } {TEXT -1 1 "." }}}}}{MARK "0 1 0" 0 }{VIEWOPTS 1 1 0 3 2 1804 }