{VERSION 6 0 "IBM INTEL NT" "6.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Input" 2 19 "" 0 1 255 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 256 "" 1 16 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Ti mes" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Maple Output" -1 12 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plot " -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 } 3 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT 256 52 "Integrals in Maple: the s o-called branch-cut problem" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 193 "Sometimes Maple gives integration results that have to be taken carefully. In particular, anti-derivatives sometimes can be treacherous in that they are valid only in a limited parameter range." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 143 "We illustrate the problem on a simple example which may help user s to identify and hopefully prevent difficult situations. We can contr ol with " }{TEXT 19 14 "infolevel[int]" }{TEXT -1 83 " how much inform ations is displayed about the progress of the integral evaluations." } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "restart; infolevel[int]:=0 ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%*infolevelG6#%$intG\"\"!" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "Our integrand contains two paramet ers: " }{TEXT 19 3 "eta" }{TEXT -1 14 " is real, and " }{TEXT 19 9 "0 \+ < e < 1" }{TEXT -1 65 ". We were interested in the answer of the defin ite integral from " }{TEXT 19 1 "0" }{TEXT -1 4 " to " }{TEXT 19 4 "2* Pi" }{TEXT -1 101 ". Maple7 and Maple8 gave correct answers for the de finite integral, but was giving a wrong answer of " }{TEXT 19 1 "0" } {TEXT -1 25 " when the antiderivative " }{TEXT 19 1 "F" }{TEXT -1 39 " was obtained first, and evaluated for " }{TEXT 19 12 "F(2*Pi)-F(0)" } {TEXT -1 102 ". Maple9 still gives a wrong answer for the latter, but \+ returns unevaluated for the definite integral." }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 40 "f:=sin(u+eta)*cos(u+eta)/(1+e*cos(u))^4;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG*(-%$sinG6#,&%\"uG\"\"\"%$etaGF+ F+-%$cosGF(F+,&F+F+*&%\"eGF+-F.6#F*F+F+!\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "int(f,u=0..2*Pi) assuming(eta>0,e>0,e<1);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%$intG6$*(-%$sinG6#,&%\"uG\"\"\"%$eta GF,F,-%$cosGF)F,,&F,F,*&%\"eGF,-F/6#F+F,F,!\"%/F+;\"\"!,$*&\"\"#F,%#Pi GF,F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 67 "Let us illustrate the pr oblem for a numerical choice of parameters:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "e:=1/2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"eG# \"\"\"\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "eta:=1;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$etaG\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "f;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(-%$sinG6# ,&%\"uG\"\"\"F)F)F)-%$cosGF&F),&F)F)*&#F)\"\"#F)-F+6#F(F)F)!\"%" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "Definite integration does work now :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "DI:=int(f,u=0..2*Pi); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#DIG,$*&#\"$g\"\"#\")\"\"\"**-%$ sinG6#F*F*-%$cosGF.F*\"\"$#F*\"\"#%#PiGF*F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "evalf(DI);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$ \"+FZw')[!\"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "We can verify th e answer numerically:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "ev alf(Int(f,u=0..2*Pi));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+DZw')[! \"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 53 "For the general parameter \+ case Maple7 or Maple8 gets:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "-5*cos(eta)*sin(eta)*e^2*Pi/(e+1)^3/(-1+e)^3/ (-e^2+1)^(1/2);" "6#,$*2\"\"&\"\"\"-%$cosG6#%$etaGF&-%$sinG6#F*F&%\"eG \"\"#%#PiGF&*$,&F.F&F&F&\"\"$!\"\"*$,&F&F4F.F&F3F4),&*$F.F/F4F&F&*&F&F &F/F4F4F4" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "Maple9.03 has no answer in this case." }}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 23 "Try the antiderivative:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "F:=unapply(int(f,u),u);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"FGf*6#%\"uG6\"6$%)operatorG%&arrow GF(,2*(\"\")\"\"\"-%$cosG6#F/\"\"#,&F3F/-F16#9$F/!\"#F/*&#\"#K\"\"$F/* &F0F3F4!\"$F/!\"\"*&#\"$/(\"#FF/**F0F/-%$sinGF2F/-%$tanG6#,$*&#F/F3F/F 7F/F/F/,&*$)FGF3F/F/FF/ F?*&#\"%CEFCF/**F0F/FEF/FGF/FMF8F/F/*(F.F/FEF3F4F8F?*&#F;FF/F/F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "B:=F(2*Pi); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BG,&*&#\"#S\"#\")\"\"\"*$)-%$c osG6#F*\"\"#F*F*F**&#F(F)F**$)-%$sinGF/F0F*F*!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "A:=F(0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%\"AG,&*&#\"#S\"#\")\"\"\"*$)-%$cosG6#F*\"\"#F*F*F**&#F(F)F**$)-%$sin GF/F0F*F*!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "B-A;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 72 "This answer is definitely wrong. The error is most likely caused by the " }{TEXT 19 11 "arctan(tan)" }{TEXT -1 73 " part. We ve rify that there is a problem by graphing the anti-derivative:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "plot(F(x),x=0..2*Pi,discont= true);" }}{PARA 13 "" 1 "" {GLPLOT2D 1064 342 342 {PLOTDATA 2 "6%-%'CU RVESG6%7Y7$$\"\"!F)$!3MTcS1h/b?!#=7$$\"3'>mc,*R=Zo!#>$!3!efM=s3c*>F,7$ $\"3i7)Ho@)[!G\"F,$!3#pY)eO*Gw%>F,7$$\"3=R>e'o$\\]>F,$!3)43G17)\\)*=F, 7$$\"3$R,/vQS\\i#F,$!3U0_?_D%\\&=F,7$$\"3Y>o)4c\"='H$F,$!3__x.'*=F==F, 7$$\"37.ll`!3&=RF,$!3exo#*e+'4z\"F,7$$\"3a+(e-4\"*Gc%F,$!3G*)*3\\UT-x \"F,7$$\"3c_@^[HJH_F,$!3'[*pQ<%*zdP\")f*od'F,$!3'4AR$H&RZw\"F,7$$\"319,A#eX(yrF,$!31l_'*phm$y \"F,7$$\"3#R&[3&o&GcyF,$!3opp'3F$ygO&)F,$!3%yw))))*[\\ o=F,7$$\"3WnEd#GCA>*F,$!3h;5@N6/L>F,7$$\"3M^vAGoe(y*F,$!3X$f=Crvl+#F,7 $$\"3i\\4(z/`&\\5!#<$!3y(H;fQlS6#F,7$$\"3xMOOh^_46Ffp$!3M_*\\'\\1ZBAF, 7$$\"33tQN')pGz6Ffp$!3=vhA\"QjMP#F,7$$\"3Wsl:A!Q5C\"Ffp$!3\")pU:k0(y_# F,7$$\"3-9[W&p)y38Ffp$!3U\"z/&3P'=s#F,7$$\"3m.Q0eLIt8Ffp$!3D!z7*=pLJHF ,7$$\"3jXPXSwhS9Ffp$!3_#e,/Ffp$!3@kdJoQn?cF,7$$\"3zx2O@TAJ?Ffp$!3!)QaRc?$\\( eF,7$$\"3obEaL3v%4#Ffp$!3UX5ni?r>gF,7$$\"3=E?Ie\"zD;#Ffp$!3GI'fXsew/'F ,7$$\"3r!G0=qTSA#Ffp$!3s@:@^S$*>fF,7$$\"35&[Ri\\O7H#Ffp$!3y_nIX86ibF,7 $$\"3S*G*=h:YaBFfp$!3OmGsH0Jj\\F,7$$\"3vBi`r*\\0U#Ffp$!3$z.QN+sT+%F,7$ $\"3ok>^sM;&[#Ffp$!3)=JLV[Phn#F,7$$\"3=bf'*>y!Gb#Ffp$!3P!='p38I-\")F07 $$\"34\\bGmt&zh#Ffp$\"3GDjaDrD'\\\"F,7$$\"3M/[:(=#e%o#Ffp$\"395jo(yEVS %F,7$$\"3!e6p3Kb1v#Ffp$\"3L48s$***pUyF,7$$\"3KPdw%\\75y#Ffp$\"33]*o&\\ gK,'*F,7$$\"3FfBmo'p8\"GFfp$\"3#yvD)*H%eY6Ffp7$$\"3%[?`dTih%GFfp$\"3Ye KyQdPs8Ffp7$$\"3S]S%G;b4)GFfp$\"30Yt%=mx(4;Ffp7$$\"3r^%41@u?\"HFfp$\"3 ,S\"y/:-1$=Ffp7$$\"3,`[PeK>VHFfp$\"3A&4r-f@!e?Ffp7$$\"3925'Qlsj(HFfp$ \"3+d6S.i.1BFfp7$$\"3DhrM\\?b4IFfp$\"317I.GIydDFfp7$$\"3d!)GUk*38/$Ffp $\"3rPOdLQC+GFfp7$$\"3))*f)\\ze1tIFfp$\"37d`c)['=UIFfp7$$\"3z*H\\<-$>2 JFfp$\"3m?SMqAL*H$Ffp7$$\"39+++k,KTJFfp$\"3)*)y#G?[7^NFfp7[o7$$\"3:+++ 0;!>9$Ffp$!3[*o/LN59L\"Ffp7$$\"39#)))Gm$>!fJFfp$!3PUEx()3&z?\"Ffp7$$\" 37kxdFr8wJFfp$!3+\"H;bhLn3\"Ffp7$$\"3aYm')))[D$>$Ffp$!3O=f!*\\@M!o*F,7 $$\"3`Gb:]EP5KFfp$!3x9/F#Rc6_)F,7$$\"3t'4%H&zg,C$Ffp$!3'p$*y&e'evd'F,7 $$\"3%\\mK/%*[*pKFfp$!3;D3T%zYvt%F,7$$\"3#)He3D)[MI$Ffp$!3hX\"*>R@u/GF ,7$$\"3r%**Q(4([pL$Ffp$!3c`(RKe?y-\"F,7$$\"3#)4KP`1nqLFfp$\"3N;Fc$\\IX %fF07$$\"3\"\\U2qf#R/MFfp$\"3=9\"*[4h'Q/#F,7$$\"3n))*HhEazV$Ffp$\"3]JG 'Ga2?J$F,7$$\"3)Gb__$f^rMFfp$\"3QETnhPQ1WF,7$$\"3lf=s)*=j-NFfp$\"3AR$H V?\"Go_F,7$$\"3)o;\">iyuLNFfp$\"3S]Tlbl\\()fF,7$$\"3e9A#zT&=)f$Ffp$\"3 O;_6*)zg\\qF,7$$\"3:x,3Ch]JOFfp$\"3%ymX,Dv0R(F,7$$\"39S\"Q-$o#[m$Ffp$ \"3(R/ypP(y.wF,7$$\"3?/NN`PV\"o$Ffp$\"3]'HGQaw$Ffp$\"3%pWH+o-Nf(F,7$$\"3U+vRpNe*z $Ffp$\"3K[X7n*[)3uF,7$$\"3H&Rf,qv'HQFfp$\"3G0r>=g7(=(F,7$$\"3g!H@4$ywf QFfp$\"3&=TU%Ffp$\"3Svp5\"H!*pp#F,7$$\"3Qe@ @))QT^UFfp$\"3)RL#eoAV\\>F,7$$\"3lEO[.\\<@VFfp$\"3)f5$*HWIp9\"F,7$$\"3 ?7![DADHQ%Ffp$\"3]WRfB3UY]F07$$\"33F2Y4^n]WFfp$!3(H%fZo#f(G7F07$$\"34` UDB!*=:XFfp$!3wMPZ%>UdX'F07$$\"3M:QMCD]#e%Ffp$!3t\\5#*\\(4g6\"F,7$$\"3 c()[MJtJWYFfp$!3iaf?EHk%[\"F,7$$\"3O/fe^H*4r%Ffp$!3mUd)z*ee>=F,7$$\"3% fju@=]-y%Ffp$!3B^Wfn910@F,7$$\"3y[i*\\!)Q0%[Ffp$!3r\"Gh=:irI#F,7$$\"3k %Q/$p@l0\\Ffp$!33o<^*)fP#[#F,7$$\"3Er>)fk?H(\\Ffp$!3$o?GM\\6?i#F,7$$\" 3SjEvE(H(Q]Ffp$!3OyH\"fVuJs#F,7$$\"3%\\>QiR.C5&Ffp$!3I,OTAC<#z#F,7$$\" 37>%Q,:-J<&Ffp$!3Wjbi!3S-%GF,7$$\"3py=(\\7GmB&Ffp$!3'z\"[b+(p:'GF,7$$ \"3!*HWXicX/`Ffp$!3moZ)4)f0lGF,7$$\"3Ksyc#\\x%zzYp&Ffp$!3F%\\#R/4EQEF,7$$\"3?Pq!\\eG)fdFfp$!3![_-Rrkpd#F,7 $$\"3#eUoCj_k#eFfp$!3$*oZsM7@6DF,7$$\"37^#y#**\\_#*eFfp$!3ZEufK!QSW#F, 7$$\"3gd\"=$\\@'Ffp$!3qhW+&HI\"=@F,7$$\"3?+++3`=$G'Ffp$!3s])oc5Y]0# F,-%'COLOURG6&%$RGBG$\"*++++\"!\")F(F(-%+AXESLABELSG6$Q\"x6\"Q!F^am-%% VIEWG6$;F($\"0ezrI&=$G'!#9%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 258 "It is a good idea to check the antiderivative for discontinuities in the range defined by the limits of integration. Maple's answer for the antiderivative implicitly made a branch cut choice during the cal culation that led to a discontinuous anti-derivative." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 108 "The correct answer ca n only be obtained from an anti-derivative that is continuous over the entire interval." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 137 "We might think that the following test would give us a p ointer: compare the derivative of the anti-derivative with the origina l function:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "Fp:=diff(F(u ),u);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#FpG,8**\"#;\"\"\"-%$cosG6# F(\"\"#,&F,F(-F*6#%\"uGF(!\"$-%$sinGF/F(F(**\"#KF(F)F,F-!\"%F2F(!\"\"* &#\"$/(\"#FF(**F)F(-F3F+F(,&#F(F,F(*&F?F(*$)-%$tanG6#,$*&F,F7F0F(F(F,F (F(F(F(,&FAF(\"\"$F(F7F(F7*&#\"%39F;F(*,F)F(F=F(FCF,FH!\"#F>F(F(F(*&# \"$g\"\"#\")F(**F)F(F=F(F>F(,&F(F(*&#F(FIF(FAF(F(F7F(F(*&#\"$'*)\"\"*F (**F)F(F=F(F>F(FHF1F(F7*&#\"%#z\"FIF(*,F)F(F=F(FCF,FHF6F>F(F(F(*&#\"%C EF;F(**F)F(F=F(F>F(FHFNF(F(*&#\"&'\\5F;F(*,F)F(F=F(FCF,FHF1F>F(F(F7**F 'F(F=F,F-F1F2F(F7**F5F(F=F,F-F6F2F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "This looks different from " }{TEXT 19 1 "f" }{TEXT -1 44 ". However, with some massaging we find that " }{TEXT 19 1 "F" }{TEXT -1 36 " is a legitimate anti-derivative to " }{TEXT 19 1 "f" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "simplify(expand( f-Fp));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 184 "The fact that our anti-derivative as calculated b y Maple has a jump discontinuity is not picked up by the test to compa red the derivative of the indefinite integral with the integrand!" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 88 "We can ca lculate the correct answer by carefully tip-toeing around the disconti nuity in " }{TEXT 19 1 "F" }{TEXT -1 4 " at " }{TEXT 19 4 "x=Pi" } {TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "Ra:=limit( F(x),x=Pi,left)-F(0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#RaG,**&#\" #!)\"#\")\"\"\"**-%$cosG6#F*F*-%$sinGF.F*\"\"$#F*\"\"#%#PiGF*F*F**&#\" $s%F)F**$)F,F3F*F*!\"\"#\"\")F1F**&#\"#SF)F**$)F/F3F*F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "Rb:=-limit(F(x),x=Pi,right)+F(2*Pi) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#RbG,**&#\"#!)\"#\")\"\"\"**-%$ cosG6#F*F*-%$sinGF.F*\"\"$#F*\"\"#%#PiGF*F*F*#\"\")F1!\"\"*&#\"$s%F)F* *$)F,F3F*F*F**&#\"#SF)F**$)F/F3F*F*F7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "Ra+Rb,DI;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$,$*&#\"$g \"\"#\")\"\"\"**-%$cosG6#F(F(-%$sinGF,F(\"\"$#F(\"\"#%#PiGF(F(F(F#" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 84 "This verifies that once one takes care of the jump discontinuity everything is fine." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 273 "It raises the question : how does Maple's definite integration package avoid the problem? It \+ is probably true that Mathematica is better in these matters by making use of other representations of the anti-derivative. When encounterin g inverse trig functions in answers from " }{TEXT 19 3 "int" }{TEXT -1 645 ": beware! These problems with Maple's answers have been around for many years. Sometimes Maple yields complex-valued answers when in tegrating real functions over the real axis. Checking symbolic answers by numerical evaluation for fixed parameter choices provides some ass urance that results are reasonable. It would be useful if the Maple de velopment team took these issues more seriously. It seems that they ac t like mathematicians who are satisfied when it is shown that an answe r can be obtained. The least we should expect is a warning that the ca lculated anti-derivative has potential problems with discontinuities i n particular locations." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 73 "This particular integration problem was encountered \+ by Michael Horbatsch." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}}{MARK "0 0 0" 52 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }