{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 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 "" 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 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT 256 47 "Integration, the case of \+ exponential functions." }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "restart; theta:=exp(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&thetaG-%$expG6#%\"xG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "F:=(x^2+1/x)*theta^2+(theta^2-x)/(theta^2+1)^2+1/theta^2+2*ln(th eta^2+1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FG,**&,&*$)%\"xG\"\"# \"\"\"\"\"\"*&F,F,F*!\"\"F-F-)-%$expG6#F*F+F,F-*&,&*$F0F,F-F*!\"\"F,*$ ),&F6F-F-F-\"\"#F,F/F-*&F,F,*$)F1\"\"#F,F/F--%#lnG6#F:F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "f:=normal(diff(F,x));" }}{PARA 12 " " 1 "" {XPPMATH 20 "6#>%\"fG*&,L*&)-%$expG6#%\"xG\"\"%\"\"\"F,\"\"\"\" \"#*&)F,F0F.)F)F0F.!\"(*&)F,\"\"$F.)F)\"\"'F.F9*$F(F.!\"\"*$F8F.!\"$*$ )F)\"#5F.F;*$)F)\"\")F.F=*$F2F.!\"#*&F6F.F(F.F9*&F?F.F6F.F0*&FBF.F6F.F 9*&F?F.)F,F-F.F0*&FBF.FJF.F9*&F8F.FJF.F9*&F(F.FJF.F0*&F?F.F,F.F0*&FBF. F,F.F9*&F8F.F,F.F9*&F2F.F8F.F-*&F2F.F(F.F;*&F2F.FBF.F-F.*()F,\"\"#F.), &*$F3F.F/F/F/\"\"$F.)F)\"\"#F.!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "usev:=proc(f) global theta,Theta; subs(theta=Theta,f) end;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%usevGR6#%\"fG6\"F(F(-%%sub sG6$/%&thetaG%&ThetaG9$F(6$F-F.F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "usef:=proc(f) global theta,Theta; subs(Theta=theta,f) end;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%usefGR6#%\"fG6\"F(F(-%%sub sG6$/%&ThetaG%&thetaG9$F(6$F.F-F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "sqrfree(denom(usev(f)),Theta);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$*$)%\"xG\"\"#\"\"\"7$7$,&*$)%&ThetaGF'F(\"\"\"F/F/\" \"$7$F.F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "d:=Theta^2+1; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"dG,&*$)%&ThetaG\"\"#\"\"\"\"\" \"F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "A1:=add(c[i]*Thet a^i,i=0..degree(d,Theta)-1)/d^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% #A1G*&,&&%\"cG6#\"\"!\"\"\"*&&F(6#F+F+%&ThetaGF+F+\"\"\"*$),&*$)F/\"\" #F0F+F+F+\"\"#F0!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "A1 :=usef(A1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#A1G*&,&&%\"cG6#\"\"! \"\"\"*&&F(6#F+F+-%$expG6#%\"xGF+F+\"\"\"*$),&*$)F/\"\"#F3F+F+F+\"\"#F 3!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "normal(f-diff(A1, x));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#*&,R*&)-%$expG6#%\"xG\"\"%\"\" \"F*\"\"\"\"\"#*&)F*F.F,)F'F.F,!\"(*&)F*\"\"$F,)F'\"\"'F,F7*$F&F,!\"\" *$F6F,!\"$*$)F'\"#5F,F9*$)F'\"\")F,F;*$F0F,!\"#*&F4F,F&F,F7*&F=F,F4F,F .*&F@F,F4F,F7*&F=F,)F*F+F,F.*&F@F,FHF,F7*&F6F,FHF,F7*&F&F,FHF,F.*&F=F, F*F,F.*&F@F,F*F,F7*&F6F,F*F,F7*&F0F,F6F,F+*&F0F,F&F,F9*&F0F,F@F,F+*(&% \"cG6#F-F-)F'F5F,F0F,F9*(FSF,)F'\"\"&F,F0F,F5*(F0F,F&F,&FT6#\"\"!F-F+F ,*()F*\"\"#F,),&*$F1F,F-F-F-\"\"$F,)F'\"\"#F,!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "numer(usev(%));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,R*&)%&ThetaG\"\"%\"\"\"%\"xG\"\"\"\"\"#*&)F)F+F()F&F+F (!\"(*&)F)\"\"$F()F&\"\"'F(F4*$F%F(!\"\"*$F3F(!\"$*$)F&\"#5F(F6*$)F&\" \")F(F8*$F-F(!\"#*&F1F(F%F(F4*&F:F(F1F(F+*&F=F(F1F(F4*&F:F()F)F'F(F+*& F=F(FEF(F4*&F3F(FEF(F4*&F%F(FEF(F+*&F:F(F)F(F+*&F=F(F)F(F4*&F3F(F)F(F4 *&F-F(F3F(F'*&F-F(F%F(F6*&F-F(F=F(F'*(&%\"cG6#F*F*)F&F2F(F-F(F6*(FPF() F&\"\"&F(F-F(F2*(F-F(F%F(&FQ6#\"\"!F*F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "coeffs(rem(%,d,Theta),Theta);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$,(*$)%\"xG\"\"#\"\"\"\"\"%*$)F&\"\"$F(F)*&F%F(&%\"cG6# \"\"!\"\"\"F),$*&&F/6#F2F2F%F(F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "solve( \{%\}, \{seq(c[i],i=0..degree(d,Theta)-1)\} ); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$/&%\"cG6#\"\"\"\"\"!/&F&6#F),&! \"\"F(%\"xGF." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "A1:=usef(s ubs(%,A1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#A1G*&,&!\"\"\"\"\"% \"xGF'\"\"\"*$),&*$)-%$expG6#F)\"\"#F*F(F(F(\"\"#F*!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "newf:=normal(f-diff(A1,x));" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#>%%newfG*&,@*&)-%$expG6#%\"xG\"\")\"\" \")F,\"\"%F.\"\"#*&F(F.)F,\"\"$F.F1*$F(F.!\"\"*&F(F.F,\"\"\"F1*&)F,F1F .)F)\"\"'F.F0*$F;F.!\"#*&F;F.F/F.F0*&F3F.F;F.F0*&F;F.F,F.F0*&)F)F0F.F/ F.F1*&F3F.FCF.F1*$FCF.F6*&FCF.F,F.F1*&F:F.)F)F1F.!\"%*$F:F.F>F.*()F)\" \"#F.),&*$FHF.F8F8F8\"\"#F.)F,\"\"#F.!\"\"" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 15 "denom(usev(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# *()%&ThetaG\"\"#\"\"\"),&*$F$F'\"\"\"F+F+F&F')%\"xGF&F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "sqrfree(%,Theta);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$*$)%\"xG\"\"#\"\"\"7$7$%&ThetaGF'7$,&*$)F+F'F(\"\" \"F0F0F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 128 "Now it looks like d= Theta*(Theta^2+1). But that won't work, because we can't have a factor Theta in d, so d should be Theta^2+1." }}{PARA 0 "" 0 "" {TEXT -1 71 "Let's try to take d=Theta*(Theta^2+1) to see if it indeed doesn't wor k." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "d:=Theta*(Theta^2+1); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"dG*&%&ThetaG\"\"\",&*$)F&\"\"# \"\"\"F'F'F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "A2:=add(c [i]*Theta^i,i=0..degree(d,Theta)-1)/d^1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#A2G*&,(&%\"cG6#\"\"!\"\"\"*&&F(6#F+F+%&ThetaGF+F+*&&F(6#\"\"# F+)F/F3\"\"\"F+F5*&F/\"\"\",&*$F4F5F+F+F+\"\"\"!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "A2:=usef(A2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#A2G*&,(&%\"cG6#\"\"!\"\"\"*&&F(6#F+F+-%$expG6#%\"xGF +F+*&&F(6#\"\"#F+)F/F6\"\"\"F+F8*&F/\"\"\",&*$F7F8F+F+F+\"\"\"!\"\"" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "numer( normal(usev(newf-di ff(A2,x))) );" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,J*()%\"xG\"\"#\"\"\" )%&ThetaG\"\"$F(&%\"cG6#F'\"\"\"!\"\"*(F*F/F%F(&F-6#\"\"!F/F/*(F%F()F* \"\"&F(F,F(F/*(F)F(F%F(F2F(F+*$F%F(!\"#*(F%F()F*\"\"%F(&F-6#F/F/F'*&F< F(F&F/F'*&F%F()F*F'F(!\"%*&)F&F+F()F*\"\"'F(F=*&FEF(F " 0 "" {MPLTEXT 1 0 29 "coeffs(rem(%,d,Theta),Theta);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%,$*$)%\"xG\"\"#\"\"\"!\"#,&*&F%F(&%\"cG6#\"\"!\"\"\"F)* &F%F(&F-6#F'F0F',$*&&F-6#F0F0F%F(F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "solve(\{%\},\{c[0],c[1],c[2]\});" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 64 "No solution. So indeed, taking d=Theta*(Theta^2+1) doesn't work." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "Now try d=Theta^2+1 instead." }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 13 "d:=Theta^2+1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% \"dG,&*$)%&ThetaG\"\"#\"\"\"\"\"\"F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "A2:=add(c[i]*Theta^i,i=0..degree(d,Theta)-1)/d^1;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#A2G*&,&&%\"cG6#\"\"!\"\"\"*&&F(6#F+ F+%&ThetaGF+F+\"\"\",&*$)F/\"\"#F0F+F+F+!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "A2:=usef(A2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%#A2G*&,&&%\"cG6#\"\"!\"\"\"*&&F(6#F+F+-%$expG6#%\"xGF+F+\"\"\",&*$) F/\"\"#F3F+F+F+!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "num er( normal(usev(newf-diff(A2,x))) );" }}{PARA 12 "" 1 "" {XPPMATH 20 " 6#,F*&)%&ThetaG\"\")\"\"\")%\"xG\"\"%F(\"\"#*&F%F()F*\"\"$F(F,*$F%F(! \"\"*&F%F(F*\"\"\"F,*&)F*F,F()F&\"\"'F(F+*$F6F(!\"#*&F6F(F)F(F+*&F.F(F 6F(F+*&F6F(F*F(F+*&)F&F+F(F)F(F,*&F.F(F>F(F,*$F>F(F1*&F>F(F*F(F,*&F5F( )F&F,F(!\"%*$F5F(F9*(&%\"cG6#F3F3)F&\"\"&F(F5F(F3*(FGF()F&F/F(F5F(F1*( F5F(F>F(&FH6#\"\"!F3F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "c oeffs(rem(%,d,Theta),Theta);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$,&*$)% \"xG\"\"#\"\"\"!\"#*&F%F(&%\"cG6#\"\"!\"\"\"F',$*&&F,6#F/F/F%F(F'" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "solve(\{%\},\{c[0],c[1]\}); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$/&%\"cG6#\"\"!\"\"\"/&F&6#F)F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "A2:=subs(%,A2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#A2G*&\"\"\"F&,&*$)-%$expG6#%\"xG\"\"#F&\" \"\"F/F/!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "normal(new f-diff(A2,x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,8*&)-%$expG6#%\"x G\"\"'\"\"\")F*\"\"%F,\"\"#*&)F*\"\"$F,F&F,F/*$F&F,!\"\"*&F&F,F*\"\"\" F/*&)F*F/F,)F'F.F,F.*$F9F,F4*&F9F,F-F,F/*&F1F,F9F,F/*&F9F,F*F,F/*&F8F, )F'F/F,!\"#*$F8F,F@F,*()F*\"\"#F,,&*$F?F,F6F6F6\"\"\")F'\"\"#F,!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 108 "Note: make sure that the power of d in the denominator has decreased before you do the following ass ignment:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "newf:=%;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%newfG*&,8*&)-%$expG6#%\"xG\"\"'\"\" \")F,\"\"%F.\"\"#*&)F,\"\"$F.F(F.F1*$F(F.!\"\"*&F(F.F,\"\"\"F1*&)F,F1F .)F)F0F.F0*$F;F.F6*&F;F.F/F.F1*&F3F.F;F.F1*&F;F.F,F.F1*&F:F.)F)F1F.!\" #*$F:F.FBF.*()F,\"\"#F.,&*$FAF.F8F8F8\"\"\")F)\"\"#F.!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "usev(denom(newf));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*()%\"xG\"\"#\"\"\",&*$)%&ThetaGF&F'\"\"\"F,F,F,F* F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "sqrfree(%,Theta);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#7$*$)%\"xG\"\"#\"\"\"7$7$,&*$)%&ThetaG F'F(\"\"\"F/F/F/7$F.F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "%[ 2];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$7$,&*$)%&ThetaG\"\"#\"\"\"\" \"\"F+F+F+7$F(F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 97 "Again, we don 't care about the factor Theta. So we have d=Theta^2+1 but now we have pole order 1." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "normal(us ev(newf));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,8*&)%&ThetaG\"\"'\"\" \")%\"xG\"\"%F)\"\"#*&)F+\"\"$F)F&F)F-*$F&F)!\"\"*&F&F)F+\"\"\"F-*&)F+ F-F))F'F,F)F,*$F7F)F2*&F7F)F*F)F-*&F/F)F7F)F-*&F7F)F+F)F-*&F6F))F'F-F) !\"#*$F6F)F>F)*()F+\"\"#F),&*$F=F)F4F4F4\"\"\")F'\"\"#F)!\"\"" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "a,b := numer(%), denom(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>6$%\"aG%\"bG6$,8*&)%&ThetaG\"\"'\" \"\")%\"xG\"\"%F-\"\"#*&)F/\"\"$F-F*F-F1*$F*F-!\"\"*&F*F-F/\"\"\"F1*&) F/F1F-)F+F0F-F0*$F;F-F6*&F;F-F.F-F1*&F3F-F;F-F1*&F;F-F/F-F1*&F:F-)F+F1 F-!\"#*$F:F-FB*(F:F-,&*$FAF-F8F8F8F8FAF-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "b_prime := usev(diff(usef(b),x));" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%(b_primeG,(*(%\"xG\"\"\",&*$)%&ThetaG\"\"#\"\"\"F(F (F(F(F+F.F-*&)F'F-F.)F,\"\"%F.F-*(F0F.F)F.F+F.F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "resultant(a-z*b_prime,b,Theta);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#,$*&)%\"xG\"#;\"\"\"),&*$)F&\"\"#F(!\"%*&%\"zG\" \"\"F,F(F-F-F(\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "solv e(%,z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"#F#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "residues:=\{%\};" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)residuesG<#\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "add(r*log(gcd(a-r*b_prime,b)),r=residues);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$-%#lnG6#,&*$)%&ThetaG\"\"#\"\"\"\"\"\"F-F- F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "A3:=usef(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#A3G,$-%#lnG6#,&*$)-%$expG6#%\"xG\" \"#\"\"\"\"\"\"F2F2F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "no rmal(newf-diff(A3,x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,,*&)-%$ex pG6#%\"xG\"\"%\"\"\")F*F+F,\"\"#*&)F*\"\"$F,F&F,F.*$F&F,!\"\"*&F&F,F* \"\"\"F.*$)F*F.F,!\"#F,*&)F*\"\"#F,)F'\"\"#F,!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 197 "Again, before you do the assignment, first look \+ if everything is correct (i.e. look if the poles of order 1 are gone, \+ except of course the pole at Theta=0 because we simply can't get rid o f those)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "newf:=%;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%newfG*&,,*&)-%$expG6#%\"xG\"\"%\"\" \")F,F-F.\"\"#*&)F,\"\"$F.F(F.F0*$F(F.!\"\"*&F(F.F,\"\"\"F0*$)F,F0F.! \"#F.*&)F,\"\"#F.)F)\"\"#F.!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "newf:=normal(usev(newf));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%newfG*&,,*&)%&ThetaG\"\"%\"\"\")%\"xGF*F+\"\"#*&)F- \"\"$F+F(F+F.*$F(F+!\"\"*&F(F+F-\"\"\"F.*$)F-F.F+!\"#F+*&)F-\"\"#F+)F) \"\"#F+!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "newf:=colle ct(newf,Theta,normal);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%newfG,&*& *&,**$)%\"xG\"\"%\"\"\"\"\"#*$)F+\"\"$F-F.!\"\"\"\"\"F+F.F3)%&ThetaGF. F-F-*$)F+\"\"#F-!\"\"F3*&F-F-*$)F5\"\"#F-F9!\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "Now newf is a " }{TEXT 257 18 "Laurent polynomial" } {TEXT -1 83 " in Theta. A Laurent polynomial is a polynomial where you allow negative exponents." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 102 "We have to integrate term by term. There are onl y two terms, one with degree 2 and one with degree -2." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "term2:=coeff(newf,Theta,2) * Theta^ 2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&term2G*&*&,**$)%\"xG\"\"%\"\" \"\"\"#*$)F*\"\"$F,F-!\"\"\"\"\"F*F-F2)%&ThetaGF-F,F,*$)F*\"\"#F,!\"\" " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "term2:=usef(term2);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%&term2G*&*&,**$)%\"xG\"\"%\"\"\"\"\" #*$)F*\"\"$F,F-!\"\"\"\"\"F*F-F2)-%$expG6#F*F-F,F,*$)F*\"\"#F,!\"\"" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "Now we have to find a function c (x) such that:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "int( term2, x) = c(x) * theta^2" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "In other words:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 "diff( c(x) * th eta^2 , x) = term2." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "int_ term2:=c(x) * theta^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*int_term2 G*&-%\"cG6#%\"xG\"\"\")-%$expGF(\"\"#\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "diff(int_term2,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&-%%diffG6$-%\"cG6#%\"xGF+\"\"\")-%$expGF*\"\"#\"\"\"F,*&F(F, F-F1F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "should_be_zero := %-term2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%/should_be_zeroG,(*&-%% diffG6$-%\"cG6#%\"xGF-\"\"\")-%$expGF,\"\"#\"\"\"F.*&F*F.F/F3F2*&*&,** $)F-\"\"%F3F2*$)F-\"\"$F3F2!\"\"F.F-F2F.F/F3F3*$)F-\"\"#F3!\"\"F>" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 94 "Now notice that all terms in shoul d_be_zero contain a factor theta^2, which we can divide out." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "should_be_zero:=normal(shoul d_be_zero / theta^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%/should_be_ zeroG*&,.*&-%%diffG6$-%\"cG6#%\"xGF.\"\"\")F.\"\"#\"\"\"F/*&F+F/F0F2F1 *$)F.\"\"%F2!\"#*$)F.\"\"$F2F7F/F/F.F7F2*$)F.\"\"#F2!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "dsolve(should_be_zero, c(x));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"cG6#%\"xG,(*$)F'\"\"#\"\"\"\"\"\" *&F,F,F'!\"\"F-*&-%$expG6#,$F'!\"#F-%$_C1GF-F-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 79 "dsolve does more than what we need. We started with \+ the field K(theta) where K=" }{TEXT 258 1 "C" }{TEXT -1 57 "(x), so we 're only interested in solutions in K, we want " }{TEXT 259 18 "ration al solutions" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(DEtools);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7io%)DEnormalG %'DEplotG%)DEplot3dG%/DEplot_polygonG%(DFactorG%+DchangevarG%%GCRDG%%L CLMG%0PDEchangecoordsG%-RiemannPsolsG%(abelsolG%(adjointG%+autonomousG %-bernoullisolG%)buildsolG%)buildsymG%'canoniG%)chinisolG%,clairautsol G%/constcoeffsolsG%+convertAlgG%+convertsysG%-dalembertsolG%*de2diffop G%+dfieldplotG%*diffop2deG%*eigenringG%6endomorphism_charpolyG%'equinv G%&eta_kG%*eulersolsG%)exactsolG%(expsolsG%/exterior_powerG%+formal_so lG%(gen_expG%,generate_icG%+genhomosolG%-hamilton_eqsG%+indicialeqG%'i nfgenG%/integrate_solsG%*intfactorG%,kovacicsolsG%-leftdivisionG%'lies olG%)line_intG%*linearsolG%)matrixDEG%/matrix_riccatiG%-moser_reduceG% %multG%/newton_polygonG%+odeadvisorG%'odepdeG%.parametricsolG%.phasepo rtraitG%)poincareG%)polysolsG%(ratsolsG%,reduceOrderG%.regular_partsG% *regularspG%/riccati_systemG%+riccatisolG%.rightdivisionG%-separableso lG%-super_reduceG%'symgenG%0symmetric_powerG%2symmetric_productG%(symt estG%)transinvG%*translateG%,untranslateG%)varparamG%%zoomG" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "ratsols(should_be_zero, c(x) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$7\"*&,&\"\"\"F'*$)%\"xG\"\"$\" \"\"F'F,F*!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "%[2];" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&\"\"\"F%*$)%\"xG\"\"$\"\"\"F%F*F(! \"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "subs(c(x)=%, int_te rm2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&*&,&\"\"\"F&*$)%\"xG\"\"$\" \"\"F&F&)-%$expG6#F)\"\"#F+F+F)!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "int_term2:=%;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*i nt_term2G*&*&,&\"\"\"F(*$)%\"xG\"\"$\"\"\"F(F()-%$expG6#F+\"\"#F-F-F+! \"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "Now lets try the other te rm:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "term_2:=coeff(newf,T heta,-2) * Theta^(-2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'term_2G,$ *&\"\"\"F'*$)%&ThetaG\"\"#F'!\"\"!\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "term_2:=usef(term_2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'term_2G,$*&\"\"\"F'*$)-%$expG6#%\"xG\"\"#F'!\"\"!\"#" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "int_term_2:=c(x)*theta^(-2); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%+int_term_2G*&-%\"cG6#%\"xG\"\" \"*$)-%$expGF(\"\"#F*!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "should_be_zero := normal( ( diff(int_term_2,x) - term_2 ) / theta^ (-2) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%/should_be_zeroG,(-%%diff G6$-%\"cG6#%\"xGF,\"\"\"F)!\"#\"\"#F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "ratsols(should_be_zero, c(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$7\"\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "%[2];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "subs(c(x)=%,int_term_2);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#*&\"\"\"F$*$)-%$expG6#%\"xG\"\"#F$!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "int_term_2:=%;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%+int_term_2G*&\"\"\"F&*$)-%$expG6#%\"xG\"\"#F&!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "A4:=int_term2 + int_term_ 2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#A4G,&*&*&,&\"\"\"F)*$)%\"xG\" \"$\"\"\"F)F))-%$expG6#F,\"\"#F.F.F,!\"\"F)*&F.F.*$)F0\"\"#F.F4F)" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "normal(newf - diff(A4,x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&,6*()%&ThetaG\"\"%\"\"\")-%$expG 6#%\"xG\"\"#F*)F/F)F*!\"#*(F'F*F+F*)F/\"\"$F*F2*&F'F*F+F*\"\"\"*(F'F*F +F*F/F7F2*&)F/F0F*F+F*F0*(F4F*)F,F)F*)F(F0F*F0*&F " 0 "" {MPLTEXT 1 0 32 "normal(usef(newf) - diff(A4,x));" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 15 "OK, we're done." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "result:=A1+A2+A3+A4;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'resul tG,,*&,&!\"\"\"\"\"%\"xGF(\"\"\"*$),&*$)-%$expG6#F*\"\"#F+F)F)F)\"\"#F +!\"\"F)*&F+F+F.F6F)-%#lnG6#F.F4*&*&,&F)F)*$)F*\"\"$F+F)F)F0F+F+F*F6F) *&F+F+*$)F1\"\"#F+F6F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "n ormal(diff(result,x)-f);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 125 "(5*exp(x)^3-exp(x)^5+x*exp( x)^3-4*x^3*exp(x)^2+8*x^3*exp(x)-4*x^3+2*x^4*exp(x)\n^2-4*x^4*exp(x)+2 *x^4)/(exp(x)-1)^2/exp(x)^2\n;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,4 *$)-%$expG6#%\"xG\"\"$\"\"\"\"\"&*$)F'F-F,!\"\"*&F*\"\"\"F&F,F2*&)F*F+ F,)F'\"\"#F,!\"%*&F4F,F'F2\"\")*$F4F,F7*&)F*\"\"%F,F5F,F6*&F " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "Assignme nt: integrate this function." }}}}{MARK "87 0 0" 36 }{VIEWOPTS 1 1 0 1 1 1803 }