{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 "" 0 1 0 0 0 0 0 1 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 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 0 1 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 0 } {CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Tim es" 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 "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 }{PSTYLE "Error" -1 8 1 {CSTYLE "" -1 -1 "Courie r" 1 10 255 0 255 1 2 2 2 2 2 1 1 1 3 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 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 Outpu t" -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 }{PSTYLE "Title" -1 18 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 1 2 2 2 1 1 1 1 }3 1 0 0 12 12 1 0 1 0 2 2 19 1 }{PSTYLE "Author" -1 19 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 8 8 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 17 "Computer Analysis" }} {PARA 19 "" 0 "" {TEXT -1 15 "Jacques Carette" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 15 "Examples of Use " }}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "int(x*BesselJ(nu,x)^2,x); " "6#-%$intG6$*&%\"xG\"\"\"*$-%(BesselJG6$%#nuGF'\"\"#F(F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&)%\"xG\"\"#\"\"\",&*$)-%(BesselJG6$%#nuG F&F'F(F(*&-F-6$,&F/F(F(!\"\"F&F(-F-6$,&F/F(F(F(F&F(F4F(#F(F'" }}} {EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "solve(\{y^2-x^2\},\{y\});" "6# -%&solveG6$<#,&*$%\"yG\"\"#\"\"\"*$%\"xGF*!\"\"<#F)" }}{PARA 11 "" 1 " " {XPPMATH 20 "6$<#/%\"yG,$%\"xG!\"\"<#/F%F'" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "int(exp(-t*x)*ln(x^2+1),x = 0 .. infinity);" "6#-% $intG6$*&-%$expG6#,$*&%\"tG\"\"\"%\"xGF-!\"\"F--%#lnG6#,&*$F.\"\"#F-F- F-F-/F.;\"\"!%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&limitG6 $,$*&,,*&-%$expG6#,$*&%\"xG\"\"\"%\"tGF0!\"\"F0-%#lnG6#,&*$)F/\"\"#F0F 0F0F0F0F0*&-F+6#*&F1F0^#F0F0F0-%#EiG6$F0,&F.F0F=F0F0F0*&-F+6#*&^#F2F0F 1F0F0-F@6$F0,&F.F0FFF0F0F0*&F;F0-F@6$F0F=F0F2*&FDF0-F@6$F0FFF0F2F0F1F2 F2/F/%)infinityG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "inttran s[laplace](ln(x^2+1),x,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&%\"t G!\"\",&*&-%#CiG6#F%\"\"\"-%$cosGF+F,F&*&-%$SsiGF+F,-%$sinGF+F,F&F,\" \"#" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "sum( x^n/n,n=1..infini ty);" "6#-%$sumG6$*&)%\"xG%\"nG\"\"\"F)!\"\"/F);F*%)infinityG" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,$-%#lnG6#,&\"\"\"F(%\"xG!\"\"F*" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "?pdsolve,system" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "restart; PDEtools[declare](); ON;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%1Nothing~declaredG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 368 "sys := subs(Z=lambda,[diff(diff(f( r,x,phi),r),r) =\n((-2*Z*r+l^2+l)*n^2+Z^2*r^2)/r^2/n^2*f(r,x,phi),\ndi ff(diff(f(r,x,phi),r),x) =\ndiff(f(r,x,phi),r)*diff(f(r,x,phi),x)/f(r, x,phi),\ndiff(diff(f(r,x,phi),x),x) =\n(2*x-2*x^3)/(x^4+1-2*x^2)*diff( f(r,x,phi),x)+((l+l^2)*x^2-l-l^2+m^2)/(\nx^4+1-2*x^2)*f(r,x,phi), diff (f(r,x,phi),phi) = I*f(r,x,phi)*m]): PDEtools[declare](sys);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(-%\"fG6%%\"rG%\"xG%$phiG\"\"\"%9will~now~b e~displayed~asGF*F%F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "sys ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&/&%\"fG6$%\"rGF(**,&*&,(*(\"\"# \"\"\"%'lambdaGF/F(F/!\"\"*$)%\"lGF.F/F/F4F/F/)%\"nGF.F/F/*&)F0F.F/)F( F.F/F/F/F(!\"#F6F:F&F//&F&6$F(%\"xG*(&F&6#F(F/&F&6#F>F/F&F1/&F&6$F>F>, &*(,&*&F.F/F>F/F/*&F.F/)F>\"\"$F/F1F/,(*$)F>\"\"%F/F/F/F/*&F.F/)F>F.F/ F1F1FBF/F/*(,**&,&F4F/F2F/F/FSF/F/F4F1F2F1*$)%\"mGF.F/F/F/FNF1F&F/F//& F&6#%$phiG*(F&F/FZF/^#F/F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "pdsolve(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<#/%\"fG*(,&*&%$_C2 G\"\"\"-%*LegendrePG6%%\"lG%\"mG%\"xGF*F**&%$_C3GF*-%*LegendreQGF-F*F* F*,&-%+WhittakerMG6%%\"nG,&F.F*#F*\"\"#F*,$**F " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 23 "Analysis is not Algebra" }{TEXT 258 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 20 "Rational 'functions'" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "x/x;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "x-x;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "1/( x-Pi) - 1/(x-Pi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 7 "Solving" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "solve( \{y^ 2-x^2\}, \{y\} );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$<#/%\"yG,$%\"xG! \"\"<#/F%F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {XPPEDIT 18 0 "sqrt(x^2);" "6#-%%sqrtG6#*$%\"xG\"\"# " }{TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "Long long ago, \+ this used to " }{TEXT 259 8 "simplify" }{TEXT -1 5 " to x" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "simplify(sqrt(x^2));" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#*&-%%csgnG6#%\"xG\"\"\"F'F(" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 72 "But it is still not clear that the whole system re ally understands this:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "solve(sqr t(x^2)-x = 0, x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%\"xG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 8 "And also" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "solve(sqrt(x^2)+x = 0, x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#% \"xG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "Though Maple knows better :" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "solve(abs(x)-x=0, x);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%*RealRangeG6$\"\"!%)infinityG" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 15 "Differentiation" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "e1 := ln(exp(Pi*I*x))/(Pi*I);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%#e1G*(^#!\"\"\"\"\"-%#lnG6#-%$expG6#*(%#PiGF(%\"xGF(^#F(F(F(F0F'" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "e2 := evalc(e1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#e2G*&-%'arctanG6$-%$sinG6#*&%#PiG\"\"\"% \"xGF.-%$cosGF+F.F-!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "diff(e1,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "diff(e2,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(,&%#PiG\"\"\"*(-%$sinG6#*&F%F&%\"xGF&\"\"#-%$cosGF*! \"#F%F&F&F&,&F&F&*&F(F-F.F0F&!\"\"F%F3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\" \"" }}}{EXCHG }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "plot(e1,x=-3. .3,discont=true);" }}{PARA 13 "" 1 "" {GLPLOT2D 459 459 459 {PLOTDATA 2 "6%-%'CURVESG6)7S7$$!+%*******H!\"*$!+T********!#57$$!+SG?yHF*$!+,%G ?y*F-7$$!+`sBfHF*$!+IDP#f*F-7$$!+n%3z$HF*$!+sY3z$*F-7$$!++$Qk\"HF*$!+- IQk\"*F-7$$!+w,2&*GF*$!+jf_(GF*$!+8>f_()F-7$$!+jguaGF*$ !+L1YZ&)F-7$$!+c8`LGF*$!+iNJN$)F-7$$!+'o%Q7GF*$!+jo%Q7)F-7$$!+OFj!z#F* $!+htK1zF-7$$!+1OZrFF*$!+hgt9xF-7$$!+k\\!*\\FF*$!+U'\\!*\\(F-7$$!+gxCG FF*$!+-wZ#G(F-7$$!+IqP2FF*$!+-.xtqF-7$$!+3WU)o#F*$!+#3WU)oF-7$$!+2z)em #F*$!+s!z)emF-7$$!+NlzYEF*$!+_`'zY'F-7$$!+I()eCEF*$!+.t)eC'F-7$$!+g5$ \\g#F*$!+-1J\\gF-7$$!+\"[jLe#F*$!+9[jLeF-7$$!+Xg#Gc#F*$!+^/EGcF-7$$!+& Q(RTDF*$!+aQ(RT&F-7$$!+'=><_#F*$!+i=><_F-7$$!+!f$\\+DF*$!+.f$\\+&F-7$$ !+hhWyCF*$!+8;Y%y%F-7$$!+4QDfCF*$!+%4QDf%F-7$$!+Vb_QCF*$!+JaD&Q%F-7$$! +;76&Q`BF*$!+#)>&Q`$F-7$$!+kA;LBF*$!+UEiJLF-7$$!+l*p:J#F*$!+^'*p:JF-7$$ !+LT+#H#F*$!+K8/?HF-7$$!+5NhqAF*$!+-^81FF-7$$!+-m[]AF*$!+Bg'[]#F-7$$!+ $=[%HAF*$!+K=[%H#F-7$$!+$Gz)3AF*$!+KGz)3#F-7$$!+^bM(=#F*$!+8bXt=F-7$$! +3ggm@F*$!+$3ggm\"F-7$$!+HoRX@F*$!+$HoRX\"F-7$$!+tKOC@F*$!+LFjV7F-7$$! +?c.0@F*$!+-iN]5F-7$$!+[R)G3#F*$!+A[R)G)!#67$$!+H72j?F*$!+MH72jFdy7$$! +Qn%>/#F*$!+EQn%>%Fdy7$$!+/zs@?F*$!+R/zs@Fdy7$$!+1+++?F*$!+qN.4j!#=7S7 $$!+'*******>F*$\"+&y#)o(QFiz7$$!+UG?y>F*$\"+fdrz@Fdy7$$!+bsBf>F*$\"+m WFwSFdy7$$!+p%3z$>F*$\"+%3`\"4iFdy7$$!+,$Qk\">F*$\"+#))phN)Fdy7$$!+y,2 &*=F*$\"+<#)H\\5F-7$$!+#>f_(=F*$\"+x!3uC\"F-7$$!+kgua=F*$\"+c$RDX\"F-7 $$!+e8`L=F*$\"+=kok;F-7$$!+(o%Q7=F*$\"+FJ:w=F-7$$!+PFj!z\"F*$\"+EEn$4# F-7$$!+2OZr<_\"F*$\"+P\"3Gy%F-7$$!+!f$\\+:F*$\"+& 4k]*\\F-7$$!+ghWy9F*$\"+&RQb@&F-7$$!+3QDf9F*$\"+:>Y2aF-7$$!+Ub_Q9F*$\" +xXu9cF-7$$!+;76<9F*$\"+Oy))GeF-7$$!+u9;'R\"F*$\"+a_QQgF-7$$!+B:*eP\"F *$\"+nZ3TiF-7$$!+(>&Q`8F*$\"+E![hY'F-7$$!+jA;L8F*$\"+mtPomF-7$$!+k*p:J \"F*$\"+c.I%)oF-7$$!+KT+#H\"F*$\"+w'e*zqF-7$$!+4Nhq7F*$\"+1\\'QH(F-7$$ !++m[]7F*$\"+(*R8&\\(F-7$$!+\"=[%H7F*$\"+&==bq(F-7$$!+\"Gz)37F*$\"+&=2 7\"zF-7$$!+\\bM(=\"F*$\"+2XaE\")F-7$$!+1ggm6F*$\"+M*RRL)F-7$$!+FoRX6F* $\"+C<.Y&)F-7$$!+rKOC6F*$\"+%Gnjv)F-7$$!+=c.06F*$\"+:Qk\\*)F-7$$!+YR)G 3\"F*$\"+N0;r\"*F-7$$!+F72j5F*$\"+FxGp$*F-7$$!+Nn%>/\"F*$\"+WE`!e*F-7$ $!+-zs@5F*$\"+x4s#y*F-7$$!+/+++5F*$\"+a********F-7S7$$!+!)********F-Fb jl7$$!+R%G?y*F-Fejl7$$!+nDP#f*F-$!+lDP#f*F-7$$!+.Z3z$*F-$!++Z3z$*F-7$$ !+HIQk\"*F-$!+FIQk\"*F-7$$!+&z,2&*)F-Fg[m7$$!+P>f_()F-$!+N>f_()F-7$$!+ b1YZ&)F-$!+`1YZ&)F-7$$!+*e8`L)F-$!+)e8`L)F-7$$!+\")o%Q7)F-$!+zo%Q7)F-7 $$!+!QFj!zF-F^]m7$$!+&3OZr(F-$!+$3OZr(F-7$$!+l'\\!*\\(F-$!+k'\\!*\\(F- 7$$!+;wZ#G(F-$!+:wZ#G(F-7$$!+<.xtqF-$!+:.xtqF-7$$!+&4WU)oF-Fe^m7$$!+&3 z)emF-Fh^m7$$!+h`'zY'F-$!+f`'zY'F-7$$!+9t)eC'F-$!+7t)eC'F-7$$!+71J\\gF -$!+61J\\gF-7$$!+:[jLeF-Fjq7$$!+a/EGcF-F]`m7$$!+`Q(RT&F-Fdr7$$!+g=><_F -$!+f=><_F-7$$!+'*e$\\+&F-$!+%*e$\\+&F-7$$!+.;Y%y%F-$!+/;Y%y%F-7$$!+$3 QDf%F-$!+\"3QDf%F-7$$!+BaD&Q%F-$!+AaD&Q%F-7$$!+`@6rTF-F\\bm7$$!+UZhhRF -$!+TZhhRF-7$$!+C_\"*ePF-$!+A_\"*ePF-7$$!+o>&Q`$F-$!+n>&Q`$F-7$$!+BEiJ LF-F^cm7$$!+L'*p:JF-Facm7$$!+58/?HF-Fdcm7$$!+%3Nhq#F-Fgcm7$$!+&*f'[]#F -Fjcm7$$!+0=[%H#F-F]dm7$$!+.Gz)3#F-F`dm7$$!+\"[bM(=F-Fcdm7$$!+Z+1m;F-F fdm7$$!+g#oRX\"F-Fidm7$$!++FjV7F-F\\em7$$!+mhN]5F-F_em7$$!*Y%R)G)F-$!+ fWR)G)Fdy7$$!*^ArI'F-$!+6D72jFdy7$$!*PtY>%F-$!+rLn%>%Fdy7$$!*+!zs@F-$! +****ys@Fdy7$$!+++++?FizFffm7S7$$\"+++++?FizFjfm7$$\"+3crz@FdyF]gm7$$ \"+IVFwSFdy$\"+HVFwSFdy7$$\"+nH:4iFdyFegm7$$\"+3(phN)Fdy$\"+2(phN)Fdy7 $$\"+0#)H\\5F-F]hm7$$\"+j!3uC\"F-F`hm7$$\"+X$RDX\"F-Fchm7$$\"+6kok;F-F fhm7$$\"+>J:w=F-Fihm7$$\"+?En$4#F-$\"+>En$4#F-7$$\"+:RE&G#F-Faim7$$\"+ N.&4]#F-Fdim7$$\"+%QAvr#F-Fgim7$$\"+$oHi#HF-Fjim7$$\"+0fv:JF-F]jm7$$\" +:47TLF-$\"+947TLF-7$$\"+RY.KNF-$\"+SY.KNF-7$$\"+'o7Tv$F-Fh`l7$F]alF]a l7$FbalFbal7$$\"+Y&RY2aF-$\"+=> Y2aF-7$FeclFecl7$$\"+Zy))GeF-$\"+Yy))GeF-7$$\"+e_QQgF-F]]n7$$\"+wZ3TiF -$\"+xZ3TiF-7$$\"+K![hY'F-Fe]n7$$\"+xtPomF-$\"+wtPomF-7$$\"+n.I%)oF-$ \"+m.I%)oF-7$$\"+!pe*zqF-$\"+*oe*zqF-7$$\"+;\\'QH(F-$\"+:\\'QH(F-7$$\" +0S8&\\(F-$\"+/S8&\\(F-7$$\"+&>=bq(F-Fa_n7$$\"+(>27\"zF-$\"+&>27\"zF-7 $$\"+>XaE\")F-$\"+f_()F-7$$\"+ORDX6F*$!+P1YZ&)F-7 $$\"+U'ok;\"F*$!+vNJN$)F-7$$\"+8`h(=\"F*$!+mo%Q7)F-7$$\"+jsO47F*$!+ntK 1zF-7$$\"+$RE&G7F*$!+ngt9xF-7$$\"+M]4]7F*$!+b'\\!*\\(F-7$$\"+RAvr7F*$! +0wZ#G(F-7$$\"+pHi#H\"F*$!+0.xtqF-7$$\"+\"fv:J\"F*$!+&3WU)oF-7$$\"+#47 TL\"F*$!+v!z)emF-7$$\"+lM?`8F*$!+Y`'zY'F-7$$\"+p7Tv8F*$!+1t)eC'F-7$$\" +R*o]R\"F*$!+01J\\gF-7$$\"+>lj;9F*$!+3[jLeF-7$$\"+bR<_F-7$$\"+5k]*\\\"F*$!+(*e$\\+ &F-7$$\"+SQb@:F*$!+(fhWy%F-7$$\"+#>Y2a\"F*$!+y!QDf%F-7$$\"+eWZh:F*$!+: aD&Q%F-7$$\"+%y))Ge\"F*$!+c@6rTF-7$$\"+E&QQg\"F*$!+QZhhRF-7$$\"+x%3Ti \"F*$!+D_\"*ePF-7$$\"+.[hY;F*F[cm7$$\"+Px$om\"F*$!+EEiJLF-7$$\"+O+V)o \"F*$!+P'*p:JF-7$$\"+oe*zq\"F*$!+<8/?HF-7$$\"+\"\\'QH=bq27\"z\"F*$!+3Gz) 3#F-7$$\"+^Wl7=F*$!+&[bM(=F-7$$\"+%*RRL=F*$!+e+1m;F-7$$\"+tJga=F*$!+o# oRX\"F-7$$\"+Hnjv=F*$!+4FjV7F-7$$\"+#Qk\\*=F*$!+yhN]5F-7$$\"+ag6<>F*$! +xXR)G)Fdy7$$\"+t(Gp$>F*$!+gE72jFdy7$$\"+lK0e>F*$!+'[tY>%Fdy7$$\"+)4s# y>F*$!+k,zs@Fdy7$$\"+'*******>F*$!+&y#)o(QFiz7S7$$\"+1+++?F*$\"+qN.4jF iz7$$\"+grz@?F*$\"+Mgrz@Fdy7$$\"+ZFwS?F*$\"+VZFwSFdy7$$\"+L:4i?F*$\"+E L:4iFdy7$$\"++lj;CF*$\"+\">lj;%F-7$$\"+bRY2a#F*$\"+6>Y2aF-7$$\"+dWZhDF*$\"+uXu9cF-7$$\"+%y) )Ge#F*$\"+Vy))GeF-7$$\"+D&QQg#F*$\"+^_QQgF-7$$\"+w%3Ti#F*$\"+kZ3TiF-7$ $\"+-[hYEF*$\"+B![hY'F-7$$\"+Ox$om#F*$\"+jtPomF-7$$\"+N+V)o#F*$\"+`.I% )oF-7$$\"+ne*zq#F*$\"+t'e*zqF-7$$\"+!\\'QHFF*$\"+-\\'QH(F-7$$\"+)R8&\\ FF*$\"+\")R8&\\(F-7$$\"+<=bqFF*$\"+t\"=bq(F-7$$\"+<27\"z#F*$\"+sr?6zF- 7$$\"+\\Wl7GF*$\"+\"\\Wl7)F-7$$\"+#*RRLGF*$\"+A*RRL)F-7$$\"+rJgaGF*$\" +7<.Y&)F-7$$\"+FnjvGF*$\"+ssOc()F-7$$\"+!Qk\\*GF*$\"+.Qk\\*)F-7$$\"+_g 6 " 0 "" {MPLTEXT 1 0 19 "diff(ln(exp(x)),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "But it sort-of works for another case" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "diff(sqrt(x^2),x);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#*&*$)%\"xG\"\"#\"\"\"#!\"\"F'F&F(" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%%csgnG6#%\"xG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Summation" } }{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "Convergence, schmonvergence:" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "sum(z^n*(-1)^(n-1)/n,n=1..infinity) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%#lnG6#,&\"\"\"F'%\"zGF'" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "r := Sum(z^n/n!, n=0..infini ty);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG-%$SumG6$*&)%\"zG%\"nG\" \"\"-%*factorialG6#F+!\"\"/F+;\"\"!%)infinityG" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 15 "rr := value(r);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#rrG-%$expG6#%\"zG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "eval(r, z=0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$\"\"!/% \"nG;F&%)infinityG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value( %);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "eval(rr, z=0);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "What is going on? Ma ple 'knows' that" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "0^0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "But forgot it in the above..." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 44 "Of course, it is also amazing in other wa ys:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "sum(1/binomial(2*n,n)*z^n,n= 0..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&,&*&\"\"%!\"\"%\" zG\"\"\"F*F*F(F(,&*&#F*\"\"#F**&*&F)F*,&F*F**&F'F(F)F*F(F(F--%'arcsinG 6#,$*&F.F(F)F-F*F*F*F*F*F*F*F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "sum((-1)^n*binomial(2*n,n)^2/binomial(n-k,n)*(1/z)^n/(n!)^2,n= 0..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%*hypergeomG6%7$#\" \"\"\"\"#F'7%F(F(,&F(F(%\"kG!\"\",$*&\"#;F(%\"zGF-F-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "Though the region of validity is hard to determ ine." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 13 "Zero-divisors" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "Re minder: a Ring " }{TEXT 256 1 "R" }{TEXT -1 56 " is said to have zero- divisors if there exists elements " }{XPPEDIT 18 0 "a,b" "6$%\"aG%\"bG " }{TEXT -1 4 " in " }{TEXT 257 1 "R" }{TEXT -1 11 " such that " } {XPPEDIT 18 0 "a.b=0" "6#/-%\".G6$%\"aG%\"bG\"\"!" }{TEXT -1 27 " but \+ neither a nor b are 0." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "(x-abs(x) )*(x+abs(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&%\"xG\"\"\"-%$abs G6#F%!\"\"F&,&F%F&F'F&F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "Consi dered as a real-valued function:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "simplify(%) assuming x::real;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\" !" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "simplify( (x-sqrt(x^2) )*(x+sqrt(x^2)) ) assuming x::real;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Divergence" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "Consider the following function " }{XPPEDIT 18 0 "h(a)" " 6#-%\"hG6#%\"aG" }{TEXT -1 1 ":" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 46 " h := a-> int(exp(-a*x^4)*ln(x),x=0..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hGj+6#%\"aG6\"6$%)operatorG%&arrowGF(-%$intG6$*&-%$ expG6#,$*&9$\"\"\")%\"xG\"\"%F6!\"\"F6-%#lnG6#F8F6/F8;\"\"!%)infinityG F(F(F(6#F@" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "Now h converges for all a such that " }{XPPEDIT 18 0 "Re(a)>0" "6#2\"\"!-%#ReG6#%\"aG" } {TEXT -1 1 "." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "hh := h(a+1)+h(1-a );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#hhG,&*&#\"\"\"\"#KF(*(,***\" \"#F(%#PiGF(F-#F(F--%#lnG6#,&%\"aGF(F(F(F(!\"\"**F-F(F.F(F-F/%&gammaGF (F5**\"\"'F(F.F(F-F/-F16#F-F(F5*&)F.F-F(F-F/F5F(F3#F5\"\"%-%&GAMMAG6## \"\"$F?F5F(F(*&F'F(*(,***F-F(F.F(F-F/-F16#,&F(F(F4F5F(F5**F-F(F.F(F-F/ F7F(F5**F9F(F.F(F-F/F:F(F5FF@F5F(F(" }}}{EXCHG }{EXCHG {PARA 0 "" 0 "" {TEXT -1 108 "This makes no sense as there is no region of t he complex plane where both integrals converge simultaneously!" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "But it do es produce pretty formulas" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "serie s(hh,a=0,4);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#+-%\"aG,$*&#\"\"\"\"#; F(*&,(*&)%#PiG\"\"#F(F/#F(F/!\"\"**F/F(F.F(F/F0%&gammaGF(F1**\"\"'F(F. F(F/F0-%#lnG6#F/F(F1F(-%&GAMMAG6##\"\"$\"\"%F1F(F(\"\"!,&*&#F(\"#KF(*& ,**(F>F1F.F/F/F0F(**F/F1F.F(F/F0F3F(F(*&#F=F/F(*(F.F(F/F0F6F(F(F(*(F/F (F.F(F/F0F1F(F9F1F(F(*&FBF(*&,**(F>F1F.F/F/F0F1**F/F1F.F(F/F0F3F(F1*&# F=F/F(FJF(F1*(F/F(F.F(F/F0F(F(F9F1F(F(F(,$*&F'F(*&,***\"\"&F(FCF1F.F/F /F0F1*,FYF(F)F1F.F(F/F0F3F(F1*&#\"#:F)F(FJF(F1**F=F(F/F1F.F(F/F0F(F(F9 F1F(F(F/,&*&FBF(*&,***\"#fF(\"#[F1F.F(F/F0F(**FgnF(\"$G\"F1F.F/F/F0F1* ,FgnF(\"#kF1F.F(F/F0F3F(F1*&#\"#XFcoF(FJF(F1F(F9F1F(F(*&FBF(*&,***F^oF (F_oF1F.F(F/F0F1**FgnF(FaoF1F.F/F/F0F(*,FgnF(FcoF1F.F(F/F0F3F(F(*&#Ffo FcoF(FJF(F(F(F9F1F(F(F=-%\"OG6#F(F>" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 12 "And more fun" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "map(simplify , %);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+)%\"aG,$*&#\"\"\"\"#;F(**,(* &\"\"#F(%&gammaGF(F(*&\"\"'F(-%#lnG6#F-F(F(%#PiGF(F(F4F(F-#F(F--%&GAMM AG6##\"\"$\"\"%!\"\"F(F<\"\"!,$*&#F(\"$7&F(**F4F(F-F5,**&\"\"&F(F4F(F( *&\"#5F(F.F(F(*&\"#IF(F1F(F(\"#[F " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 19 "Change of variables" }}{PARA 0 "" 0 "" {TEXT -1 31 "Consider th e following integral" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "f := Int(g(sin(x)), x=0..Pi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG-%$ IntG6$-%\"gG6#-%$sinG6#%\"xG/F.;\"\"!%#PiG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "And now let's make the " }{TEXT 260 7 "obvious" }{TEXT -1 20 " change of variables" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "ff : = student[changevar](sin(x)=t, f, t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ffG-%$IntG6$,$*&-%\"gG6#%\"tG\"\"\",&F.F.*$)F-\"\"#F.!\"\"#F3F2F 3/F-;\"\"!F7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "value(ff); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "So all such integrals are 0 ?" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "value(eval(f, g=(x->x)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "I guess n ot..." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 260 "The problem is related to a side-condition in the change-of-variables theorem which is not checked before it is applied. Note that it is a lso a side-condition which is frequently not present in all first-year calculus textbooks! [Injectivity of the function]" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 17 "Risch in tegration" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "Risch integration is \+ based on differential " }{TEXT 261 7 "algebra" }{TEXT -1 14 ", which g ives:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "f := ln(x) + ln(1/x);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG,&-%#lnG6#%\"xG\"\"\"-F'6#*&F*F* F)!\"\"F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "diff(f,x);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 85 "Which makes some sense, as f is actually piecewise consta nt wherever it is defined..." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "int( f,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&%\"xG\"\"\"-%#lnG6#F%F&F& *&-F(6#*&F&F&F%!\"\"F&F%F&F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "Bu t" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "`int/risch`(f,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "B ack to a previously seen function:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "f1 := int(ln(exp(I*x))/I,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% #f1G,$*&#\"\"\"\"\"#F(*$)-%#lnG6#-%$expG6#*&%\"xGF(^#F(F(F)F(F(!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "f2 := `int/risch`(ln(exp( x*I))/I,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#f2G,&*(^#!\"\"\"\"\" %\"xGF)-%#lnG6#-%$expG6#*&F*F)^#F)F)F)F)*&\"\"#F(F*F4F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "plot([f1,f2],x=-15..15,color=[red,g reen]);" }}{PARA 13 "" 1 "" {GLPLOT2D 475 475 475 {PLOTDATA 2 "6&-%'CU RVESG6$7[w7$$!#:\"\"!$\"+%*fFhH!\"*7$$!+M@l$[\"!\")$\"+g@zwDF-7$$!+pUI n9F1$\"+=O.>AF-7$$!+/k&4X\"F1$\"+Y.+))=F-7$$!+Q&3YV\"F1$\"+CBp$e\"F-7$ $!+d,;19F1$\"+1y&y6\"F-7$$!+wS[&Q!#77$$!+ FiOd7F1$\"++@TeE!#97$$!+9\\J\\7F1$\"+a@p!o#Fgo7$$!+o=IT7F1$\"+*ySe<\"F gn7$$!+@))GL7F1$\"+c;qDFFgn7$$!+udFD7F1$\"+]18F-7$$!+[Iiz5F1$\"+Z!)pm:F-7$$!+,#QU1\"F1$\"+ug'3&=F-7$ $!+rrK[5F1$\"+A0lp@F-7$$!+ThTK5F1$\"+e5v8DF-7$$!+6^];5F1$\"+!onJ)GF-7$ $!+\"3%f+5F1$\"+!R+zF$F-7$$!+!ySt%)*F-$\"+>;c'p$F-7$$!+X2u)o*F-$\"+KoP STF-7$$!+:29I&*F-$\"++gM4YF-7$$!+\"oS:P*F-$\"+e%p*oZF-7$$!+b5S3#*F-$\" +'eV%yUF-7$$!+I9EX!*F-$\"+aA`9QF-7$$!++=7#)))F-$\"+[aBxLF-7$$!+v@)*=() F-$\"+\">`l'HF-7$$!++()Gv&)F-$\"+#)*ooi#F-7$$!+I_fJ%)F-$\"+)fKyI#F-7$$ !+g=\"F-7$$!+SQ\"*ewF-$\"+&)\\9j%*FM7$$!+D!\\r\\( F-$\"+X&R&otFM7$$!+]4HsrF-$\"+?Ua_RFM7$$!+vGVZoF-$\"+*ew=f\"FM7$$!++C! 4p'F-$\"+!Rh;J)Fgn7$$!+D>PMlF-$\"+]etaJFgn7$$!+!p1hX'F-$\"+K-4&\\\"Fgn 7$$!+]9%yP'F-$\"+(o#*)zWFgo7$$!+5id*H'F-$\"+w%3LM\"!#87$$!+v4J@iF-$\"+ #\\;U\">Fgo7$$!+S'Q-:'F-$\"+g\"3u$))Fgo7$$!+5j;zgF-$\"+hv=\"3#Fgn7$$!+ vR43gF-$\"+_?w$y$Fgn7$$!+S;-PfF-$\"+?UY\"*fFgn7$$!+vp([z&F-$\"+O^A#>\" FM7$$!+1Bt_cF-$\"+M\\N()>FM7$$!+&z%o9`F-$\"+5i'**o%FM7$$!*GPm(\\F1$\"+ N(Q``)FM7$$!+&)>XL[F-$\"+_L'30\"F-7$$!+!pm-p%F-$\"+g[po7F-7$$!+&R\"3ZX F-$\"+'RGq]\"F-7$$!*5'*QS%F1$\"+iR'ew\"F-7$$!+lvLPUF-$\"+#\\YF4#F-7$$! +I!z22%F-$\"+o2PZCF-7$$!+&\\?U!RF-$\"+)yO(HGF-7$$!*'>mPPF1$\"+bX%)RKF- 7$$!+K%H-f$F-$\"+sc+EOF-7$$!+0pzUMF-$\"+PJ!R.%F-7$$!+yVO&H$F-$\"+_p`jW F-7$$!*&=$z9$F1$\"+?r!\\\"\\F-7$$!+-]<')HF-$\"+s0ieWF-7$$!+b\"=W#GF-$ \"+d*o'))RF-7$$!+38miEF-$\"+?E)[a$F-7$$!*Y/4]#F1$\"+f:EFJF-7$$!+)otoM# F-$\"+a!3Rv#F-7$$!+:H%G>#F-$\"+D+G/CF-7$$!+U@\")Q?F-$\"+vuPy?F-7$$!*P \"y%)=F1$\"+1/?wC\"F1$\"+0e%=r(FM7$$!++d[n%*FM$\"+tUm\"[ %FM7$$!)ev:lF1$\"+#o`F7#FM7$$!+vg$Q#\\FM$\"+%33A@\"FM7$$!+]j\">L$FM$\" +!GL3b&Fgn7$$!+)[cf`#FM$\"+aw`:KFgn7$$!+Dm**RFFgn7$$\"++v.fJFM$ \"+j*e(*)\\Fgn7$$\"++Zf7[FM$\"+)Q`!e6FM7$$\")>:mkF1$\"+?gb!4#FM7$$\"+] Z+X$*FM$\"+*obkO%FM7$$\"*w&QA7F1$\"+IZ8ruFM7$$\"+bx%yP\"F-$\"+!=AB\\*F M7$$\"+](4L`\"F-$\"+%R>b<\"F-7$$\"+XT[+#F-$\"+-Tp4?F-7$$\"+X'[a;#F-$\"+MF-$\"+99>,TF-7$$\"+-(H7d$F- $\"+-;NxOF-7$$\"*MaKs$F1$\"+yKiwKF-7$$\"+#y^?*QF-$\"+n(f(eGF-7$$\"+D# \\31%F-$\"+I))QpCF-7$$\"+omkHUF-$\"+r/^3@F-7$$\"*6W%)R%F1$\"+!pChx\"F- 7$$\"+?h6]XF-$\"+/Ww,:F-7$$\"+I\")y,ZF-$\"+;&3/D\"F-7$$\"+S,Y`[F-$\"+E q0A5F-7$$\"*:K^+&F1$\"+g$*4n\")FM7$$\"+Nm,H`F-$\"+v#*=_XFM7$$\"*7,Hl&F 1$\"+$y!H')>FM7$$\"+g[k*z&F-$\"+?o0p6FM7$$\"+0')QYfF-$\"+I4grcFgn7$$\" +![g(>gF-$\"+!)>jpMFgn7$$\"+]B8$4'F-$\"+Fj+1=Fgn7$$\"+?U]mhF-$\"+b!Qs! oFgo7$$\"*4w)RiF1$\"+l9Wy$*Ff\\l7$$\"+D44?jF-$\"+NJ75oFf\\l7$$\"+gdI+k F-$\"+I-geoFgo7$$\"++1_!['F-$\"+*)31Z>Fgn7$$\"+NatglF-$\"+sOq^QFgn7$$ \"+5^;@nF-$\"+&Q:8f*Fgn7$$\"*y%f\")oF1$\"+'[p/z\"FM7$$\"+5%)\\$=(F-$\" +;%=G0%FM7$$\"*/-a[(F1$\"+l`iEsFM7$$\"+&=!>VwF-$\"+Sj1[#*FM7$$\"+I$y4! yF-$\"+Iy%=:\"F-7$$\"+vkwezF-$\"+Ehy.9F-7$$\"*ial6)F1$\"+>:i!o\"F-7$$ \"+q8#3F)F-$\"+g&[`(>F-7$$\"+?\")3D%)F-$\"+IQ(QH#F-7$$\"+q[Nz&)F-$\"+J t>OEF-7$$\"*i@Ot)F1$\"+i!>B+$F-7$$\"+5Y7&*))F-$\"+iM66MF-7$$\"+0wic!*F -$\"+!4\"*f%QF-7$$\"++18=#*F-$\"+O>&pI%F-7$$\"*fL'z$*F1$\"+%)f*Rz%F-7$ $\"+q,=N&*F-$\"+ax0%f%F-7$$\"+Xns!p*F-$\"+IQmMTF-7$$\"+?LFY)*F-$\"+WYY *p$F-7$$\"+!*>=+5F1$\"+#=g%)G$F-7$$\"+u))3;5F1$\"+B:<$*GF-7$$\"+ed*>. \"F1$\"+gd=BDF-7$$\"+UE!z/\"F1$\"+#*G]y@F-7$$\"+E&4Q1\"F1$\"+?H7f=F-7$ $\"+$p%ez5F1$\"+eqPn:F-7$$\"+g)f`4\"F1$\"+%y;0I\"F-7$$\"+F]866F1$\"++@ ae5F-7$$\"+%>5p7\"F1$\"+l+`9%)FM7$$\"+u;!f:\"F1$\"+N\"4Q2&FM7$$\"+bJ*[ =\"F1$\"+bSftDFM7$$\"+4p],7F1$\"+aom>:FM7$$\"+j17=7F1$\"+SZc9F1$\"+WkpG8F-7$$\"+1j\"[V\"F1$\" +g@R(e\"F-7$$\"+IA6^9F1$\"+g&H5*=F-7$$\"+`\"3uY\"F1$\"+_EA@AF-7$$\"+wS q$[\"F1$\"+c9(zd#F-7$$\"#:F*F+-%'COLOURG6&%$RGBG$\"*++++\"F1$F*F*Fhgn- F$6$7ip7$F($!+@fb*f(F17$F?$!+\"G9tt(F17$FI$!+*3)QAyF17$FT$!+A\")QzyF17 $F_p$!+]aT&*yF17$F]r$!+#yu,(yF17$Fgr$!+3D15yF17$F[t$!+6pf5xF17$Fet$!+9 %3Vk(F17$F_u$!+#[$*yc(F17$Fdu$!+ft-EvF17$Fiu$!+Pek\"[(F17$$!+I2W4'*F-$ !+.=^euF17$F^v$!+?*[ZV(F17$$!+52\\!\\*F-$!+h;jAuF17$$!++2%3X*F-$!+)=d. T(F17$$!+&p:5V*F-$!+(*4;/uF17$$!+!p!>6%*F-$!+'***p%[\"F17$$!+&ol8R*F-$ !+=;)3\\\"F17$Fcv$!+NR-(\\\"F17$F]w$!+bwY#f\"F17$Fgw$!+hbEx;F17$F[y$!+ `0v+=F17$F_z$!+%[N-!>F17$Fiz$!+9@+e>F17$Fh\\l$!+e%HP(>F17$Ff^l$!+Jt/a> F17$F`_l$!+$\\n&))=F17$Fd`l$!+%[Mtz\"F17$Fhal$!+Dk$*\\;F17$Fbbl$!+m0`q :F17$F\\cl$!+o,V#[\"F17$$!+WArFJF-$\"+T>H\"*[F-7$$!+QE\\2JF-$\"+Z_DG[F -7$$!+KIF(3$F-$\"+(QFcw%F-7$$!+EM0nIF-$\"+f$3Mq%F-7$$!+9UhEIF-$\"++o>! e%F-7$Facl$\"+r0ieWF-7$$!+ylH0HF-$\"+.TP?UF-Fecl7$F[dl$\"+@E)[a$F-F_dl Fidl7$Fdel$\"+2/?wgb!4#FMFg[mF[]m7$F`^m$\"+0yu\"4$F-7$Fj^m$\"+*[ZD#RF-Fc_m7$$\"+q) eT8$F-$\"+6fZ6\\F-7$$\"+]>;`JF-$!+p&oS[\"F17$$\"+I];sJF-$!+A&)***[\"F1 7$$\"+4\"o6>$F-$!+iB*e\\\"F17$$\"+oU:F17$$ \"+YF>VLF-$!+nIuT:F17$F^`m$!+R)ps\"F17$F\\bm$!+6%3jz\"F17$F`cm$!+())\\A*=F17$Fjcm$!+tz 0a>F17$Fhem$!+'4FQ(>F17$Ffgm$!+'=;g&>F17$F`hm$!+EYl,>F17$Fdim$!+G(ee!= F17$Fhjm$!+u*)ot;F17$Fb[n$!+r " 0 "" {MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 20 "Residue computations" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 11 "As expected" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "residue ( gamma^2/(x-Pi) + x, x=Pi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$)%&g ammaG\"\"#\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "But ask a stu pid question" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "residue(1/sqrt(z^2) ,z=0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 13 "or even worse" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "residue(1/(csgn(z)*z),z=0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*& \"\"\"F$-%%csgnG6#%\"zG!\"\"" }}}{EXCHG }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 6 "Series" }} {EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "The previous issue can be traced t o" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "series(sqrt(x^2),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+%%\"xG\"\"\"F%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 162 "which is because while series used to be power series va lid in an open set around a point, but then got changed to sometimes m eans one-sided asymptotic expansion." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "series(1/sqrt(x^2),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+%%\"x G\"\"\"!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 57 "and then got chan ged 'back' to encoding a complete series" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "series(1/(csgn(x)*x),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+%%\"xG*&\"\"\"F&-%%csgnG6#F$!\"\"F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "or perhaps clearer?" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "series(ln(x),x=-2);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#+1,&%\"xG\"\"\"\"\"#F&,&-%#lnG6#F'F&*(^#!\"\"F&-%%csgnG6#*&F%F&^#F&F &F&%#PiGF&F&\"\"!#F.F'F&#F.\"\")F'#F.\"#C\"\"$#F.\"#k\"\"%#F.\"$g\"\" \"&-%\"OG6#F&\"\"'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {SECT 0 {PARA 4 "" 0 "" {TEXT -1 12 "Generic-ness" }}{PARA 0 "" 0 "" {TEXT -1 44 "This is a huge topic, so only a few examples" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "assume(x,real);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "a := (x-abs(x))*(x+abs(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG*&,&%#x|irG\"\"\"-%$absG6#F'!\"\"F(,&F'F(F)F (F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "sin(1/a);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$sinG6#*&\"\"\"F'*&,&%#x|irGF'-%$absG6#F*! \"\"F',&F*F'F+F'F'F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "sim plify(cos(x-Pi/2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$sinG6#%#x|ir G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "simplify(cos(1/a-Pi/2) - sin(1/a));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "simplify(sin(1/a));" }}{PARA 8 "" 1 "" {TEXT -1 61 "Error, (in simplify/abs) numeric exception: division by zero\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "simplify(cos( 1/a-Pi/2)/sin(1/a));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "unassign('a'):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "And solve too" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "solve(a*z^2+b*z+c=0,z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$,$*(\"\"#!\"\",&%\"bG\"\"\"*$,&*$)F(F%F)F)*(\"\"%F)%\"aGF)%\"cGF )F&#F)F%F&F)F0F&F&,$*(F%F&,&F(F)F*F)F)F0F&F&" }}}{EXCHG }{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "Implicitly a=0 is discarded." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "int(1/(x-a),x=0..1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%#lnG6#*&,&\"\"\"!\"\"%\"aGF(F(F*F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "which is quite wrong if a is in 0..1." }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 64 "Let's try t o compute the derivative of ln(x) from the definition" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "limit( (ln(z+h)-ln(z))/h, h=0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&\"\"\"F$%\"zG!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 6 "Right?" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "limit( eval(( ln(z+h)-ln(z))/h, z=-2), h=0), eval(1/z, z=-2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$#!\"\"\"\"#F#" }}}{EXCHG }{EXCHG {PARA 0 "" 0 "" {TEXT -1 7 "Oops..." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "limit( eval((ln(z+ I*h)-ln(z))/(I*h), z=-2), h=0), eval(1/z, z=-2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%*undefinedG#!\"\"\"\"#" }}}{EXCHG }{EXCHG {PARA 0 "" 0 "" {TEXT -1 52 "Since ln(z) has a branch cut on the negative axis... " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 11 "Denotations" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 65 "Part o f the above issues are related to ``what does a particular " }{TEXT 262 29 "expression mean as a function" }{TEXT -1 2 "''" }}{PARA 0 "" 0 "" {TEXT -1 37 "Consider the following 3 expressions:" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 37 "(x^2-1)/(x-1), x*sin(1/x), signum(x);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6%*&,&*$)%\"xG\"\"#\"\"\"F)F)!\"\"F),&F' F)F)F*F**&F'F)-%$sinG6#*&F)F)F'F*F)-%'signumG6#F'" }}}{PARA 0 "" 0 "" {TEXT -1 50 "What do they \"mean\" ? Especially at x=0 and x=1 ?" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "Contrast: " }}{PARA 0 "" 0 "" {TEXT -1 19 "- Direct evaluation" }}{PARA 0 "" 0 " " {TEXT -1 21 "- algebraic extension" }}{PARA 0 "" 0 "" {TEXT -1 22 "- continuous extension" }}{PARA 0 "" 0 "" {TEXT -1 21 "- analytic exten sion." }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 15 "The Dark Side ?" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "f1 := Sum(n!*x^n,n=0..infini ty);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#f1G-%$SumG6$*&-%*factorialG 6#%\"nG\"\"\")%\"xGF,F-/F,;\"\"!%)infinityG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "f2 := hypergeom([1,1],[],x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#f2G-%*hypergeomG6%7$\"\"\"F)7\"%\"xG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "series(f2,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+1%\"xG\"\"\"\"\"!F%F%\"\"#F'\"\"'\"\"$\"#C\"\"%\"$?\" \"\"&-%\"OG6#F%F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "f3 := \+ -Ei(1,-1/x)*exp(-1/x)/x;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#f3G,$*( -%#EiG6$\"\"\",$*&F*F*%\"xG!\"\"F.F*-%$expG6#F+F*F-F.F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "series(eval(asympt(eval(f3,x=1/x),x ,7),x=1/x),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+1%\"xG\"\"\"\"\"!F% F%\"\"#F'\"\"'\"\"$\"#C\"\"%\"$?\"\"\"&-%\"OG6#F%F(" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "de := x^2*diff(y(x),x,x)+(3*x-1)*diff(y(x), x)+y(x);" "6#>%#deG,(*&%\"xG\"\"#-%%diffG6%-%\"yG6#F'F'F'\"\"\"F/*&,&* &\"\"$F/F'F/F/F/!\"\"F/-F*6$-F-6#F'F'F/F/-F-6#F'F/" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#deG,(*&)%\"xG\"\"#\"\"\"-%%diffG6$-%\"yG6#F(-%\"$G 6$F(F)F*F**&,&F(\"\"$F*!\"\"F*-F,6$F.F(F*F*F.F*" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 11 "dsolve(de);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/-%\"yG6#%\"xG*(,&*&%$_C1G\"\"\"-%#EiG6$F,,$*&F,F,F'!\"\"F2F,F,%$_C2G F,F,-%$expG6#F1F2F'F2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "Sl ode[hypergeom_formal_sol](de,y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #7$-%$SumG6$*&-%&GAMMAG6#,&%\"nG\"\"\"F-F-F-)%\"xGF,F-/F,;\"\"!%)infin ityG*&-%$expG6#,$*&F-F-F/!\"\"F:F-F/F:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "Resummation (Ecalle + many others) saves the day..." }} {PARA 0 "" 0 "" {TEXT -1 23 "Very recent development" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "Exists many more exampl es of this kind!" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{MARK "1 " 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }