{VERSION 4 0 "IBM INTEL LINUX22" "4.0" } {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 Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 1 24 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{PSTYLE "Normal " -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }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 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 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 256 65 "Computer A lgebra, week 1, lecture 3:\nRational functions in Maple." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "Example:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "f:=(x^6-3*x^2+x^5-3*x+x^4-3) /(x^5-3*x^2+x^4-3*x+x^3-3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG* &,.*$)%\"xG\"\"'\"\"\"F+*&\"\"$F+)F)\"\"#F+!\"\"*$)F)\"\"&F+F+*&F-F+F) F+F0*$)F)\"\"%F+F+F-F0F+,.F1F+*&F-F+F.F+F0F5F+*&F-F+F)F+F0*$)F)F-F+F+F -F0F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "numer(f); # numera tor" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,.*$)%\"xG\"\"'\"\"\"F(*&\"\"$F ()F&\"\"#F(!\"\"*$)F&\"\"&F(F(*&F*F(F&F(F-*$)F&\"\"%F(F(F*F-" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "denom(f); # denominator" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,.*$)%\"xG\"\"&\"\"\"F(*&\"\"$F()F&\" \"#F(!\"\"*$)F&\"\"%F(F(*&F*F(F&F(F-*$)F&F*F(F(F*F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "g:=normal(f); # remove gcd of numerator a nd denominator" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gG*&,&*$)%\"xG\" \"%\"\"\"F+\"\"$!\"\"F+,&*$)F)F,F+F+F,F-F-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 262 "The command normal puts rational functions in their norm al form, which means the form A/B where A and B are polynomials with n o common factors, so the gcd(numer(g), denom(g)) will be 1. It also pr oduces this normal form when you have a sum of rational functions." }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "f;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,.*$)%\"xG\"\"'\"\"\"F)*&\"\"$F))F'\"\"#F)!\"\"*$)F' \"\"&F)F)*&F+F)F'F)F.*$)F'\"\"%F)F)F+F.F),.F/F)*&F+F)F,F)F.F3F)*&F+F)F 'F)F.*$)F'F+F)F)F+F.F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "f -12*x/(x-3)^3;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&,.*$)%\"xG\"\"' \"\"\"F**&\"\"$F*)F(\"\"#F*!\"\"*$)F(\"\"&F*F**&F,F*F(F*F/*$)F(\"\"%F* F*F,F/F*,.F0F**&F,F*F-F*F/F4F**&F,F*F(F*F/*$)F(F,F*F*F,F/F/F**&*&\"#7F *F(F*F**$),&F(F*F,F/F,F*F/F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "normal(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,2*$)%\"xG\"\"( \"\"\"F)*&\"\"*F))F'\"\"'F)!\"\"*&\"#FF))F'\"\"&F)F)*&\"#RF))F'\"\"%F) F.*&\"\"$F))F'F8F)F.*&F0F))F'\"\"#F)F)*&\"#XF)F'F)F.\"#\")F)F)*&),&F'F )F8F.F8F),&*$F9F)F)F8F.F)F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "f - g;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&,.*$)%\"xG\"\"'\"\"\" F**&\"\"$F*)F(\"\"#F*!\"\"*$)F(\"\"&F*F**&F,F*F(F*F/*$)F(\"\"%F*F*F,F/ F*,.F0F**&F,F*F-F*F/F4F**&F,F*F(F*F/*$)F(F,F*F*F,F/F/F**&,&F4F*F,F/F*, &F:F*F,F/F/F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "normal(%); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 56 "To test if a rational function is 0, we also use normal. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "sqrfree(f); # Note: th is gives an error in Maple 5, but not in 6 or 7." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$\"\"\"7&7$,2*$)%\"xG\"\"&F$\"#=*&\"#VF$)F*\"\"$F$F$*& \"#:F$F*F$F$*&\"#EF$)F*\"\"%F$!\"\"*&\"#IF$)F*\"\"#F$F7*$)F*\"\"(F$F$* &\"\"'F$)F*F@F$F7\"\")F7F$7$,&F*F$F;F7!\"$7$,&*$F:F$F$F$F$!\"#7$F*F7" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 127 "As you can see, a positive mul tiplicity means that the factor is in the numerator, and negative mean s it is in the denominator." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "factor(x^2+1/x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&*&,&%\"xG\" \"\"F'F'F',(*$)F&\"\"#F'F'F&!\"\"F'F'F'F'F&F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "factors(x^2+1/x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$\"\"\"7%7$,(*$)%\"xG\"\"#F$F$F*!\"\"F$F$F$7$,&F*F$F$F$F$7$F*F, " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 237 "If a rational function has a pole at a point alpha of multiplicity e (so with squarefree you'd se e a factor with multiplicity -e), then the derivative has a pole of or der e+1 (so if you then do sqrfree you'd see a multiplicity -(e+1)). " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "f:=3/(x-2)^3+x/(x^2+1)^ 2+1/x;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG,(*&\"\"\"F'*$),&%\"xG F'\"\"#!\"\"\"\"$F'F-F.*&F+F'*$),&*$)F+F,F'F'F'F'F,F'F-F'*&F'F'F+F-F' " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "f:=normal(f);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG*&,2*$)%\"xG\"\"&\"\"\"\"#=*&\"#VF+)F) \"\"$F+F+*&\"#:F+F)F+F+*&\"#EF+)F)\"\"%F+!\"\"*&\"#IF+)F)\"\"#F+F7*$)F )\"\"(F+F+*&\"\"'F+)F)F@F+F7\"\")F7F+*(),&F)F+F;F7F0F+),&*$F:F+F+F+F+F ;F+F)F+F7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "diff(f,x);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#,**&,0*$)%\"xG\"\"%\"\"\"\"#!**&\"$H\" F*)F(\"\"#F*F*\"#:F**&\"$/\"F*)F(\"\"$F*!\"\"*&\"#gF*F(F*F5*&\"\"(F*)F (\"\"'F*F**&\"#OF*)F(\"\"&F*F5F**(),&F(F*F/F5F4F*),&*$F.F*F*F*F*F/F*F( F*F5F**&*&F4F*,2*$F>F*\"#=*&\"#VF*F3F*F**&F0F*F(F*F**&\"#EF*F'F*F5*&\" #IF*F.F*F5*$)F(F9F*F**&F;F*F:F*F5\"\")F5F*F**()FBF)F*FCF*F(F*F5F5*&*&F )F*FHF*F**&FAF*)FDF4F*F5F5*&FHF**(FAF*FCF*F.F*F5F5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "normal(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&,8\"#;\"\"\"*&\"#KF'%\"xGF'!\"\"*&\"$*=F')F*\"\"'F'F'*&\"#lF')F *\"\"#F'F'*&\"$3#F')F*\"\"&F'F+*&\"$s\"F')F*\"\"%F'F'*&\"#sF')F*\"\"$F 'F+*&\"\")F')F*\"\"*F'F+*&\"#RF')F*FAF'F'*&\"#!)F')F*\"\"(F'F+*$)F*\"# 5F'F'F'*(),&F*F'F3F+F;F'),&*$F2F'F'F'F'F?F'F2F'F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "normal(diff(%,x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&,>!#K\"\"\"*&\"#!)F'%\"xGF'F'*&\"$g$F')F*\"\"'F'!\" \"*&\"$3#F')F*\"\"#F'F/*&\"$L#F')F*\"\"&F'F'*&\"$I$F')F*\"\"%F'F/*&\"$ y$F')F*\"\"$F'F'*&\"$_&F')F*\"\"*F'F'*&\"$K)F')F*\"\")F'F/*&\"$7*F')F* \"\"(F'F'*&\"$!=F')F*\"#5F'F/*&\"#oF')F*\"#6F'F'*&FOF')F*\"#7F'F/*$)F* \"#8F'F'F'*(),&F*F'F3F/F7F'),&*$F2F'F'F'F'F;F'F>F'F/F3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "normal(diff(%,x));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,$*&,D\"#k\"\"\"*&\"$#>F'%\"xGF'!\"\"*&\"%\"p#F')F *\"\"'F'F'*&\"$g&F')F*\"\"#F'F'*&\"%;JF')F*\"\"&F'F+*&\"%e?F')F*\"\"%F 'F'*&\"%?6F')F*\"\"$F'F+*&\"%C9F')F*\"\"*F'F+*&\"$:%F')F*\"\")F'F+*&\" #gF')F*\"\"(F'F+*&\"%wLF')F*\"#5F'F'*&\"%sCF')F*\"#6F'F+*&\"%!G\"F')F* \"#7F'F'*&\"$S$F')F*\"#8F'F+*&\"$0\"F')F*\"#9F'F'*&FWF')F*\"#:F'F+*$)F *\"#;F'F'F'*(),&F*F'F3F+F/F'),&*$F2F'F'F'F'F7F'F:F'F+!\"'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 162 "Because of that, the derivative of a rat ional function can not have pole order 1, if a derivative of a rationa l function has a pole, the pole order is at least 2." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "f:=1/x^5 + 1/x + 1/(x-2)^2 + 1/(x-3);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG,**&\"\"\"F'*$)%\"xG\"\"&F'!\"\" F'*&F'F'F*F,F'*&F'F'*$),&F*F'\"\"#F,F2F'F,F'*&F'F',&F*F'\"\"$F,F,F'" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "int(f,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**&\"\"\"F%*$)%\"xG\"\"%F%!\"\"#F*F)-%#lnG6#F(F%*&F %F%,&F(F%\"\"#F*F*F*-F-6#,&F(F%\"\"$F*F%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 119 "Poles of order 1 can not come from derivatives of ration al functions. Such poles lead to logarithms when you integrate." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "normal(f);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,2*$)%\"xG\"\"$\"\"\"F)*&\"\"(F))F'\"\"#F)!\"\"*& \"#;F)F'F)F)\"#7F.*&F-F))F'F+F)F)*&\"#5F))F'\"\"'F)F.*&\"# " 0 "" {MPLTEXT 1 0 22 "g:=normal(f,expanded);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gG*&,2*$)%\"xG\"\"$\"\"\"F+*&\"\"(F+)F)\"\" #F+!\"\"*&\"#;F+F)F+F+\"#7F0*&F/F+)F)F-F+F+*&\"#5F+)F)\"\"'F+F0*&\"# " 0 "" {MPLTEXT 1 0 21 "convert(g,parfrac,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**&\"\"\"F%*$)%\"xG\"\"&F%!\"\"F%*&F%F%F(F*F%*&F%F%*$) ,&F(F%\"\"#F*F0F%F*F%*&F%F%,&F(F%\"\"$F*F*F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "g;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,2*$)%\"x G\"\"$\"\"\"F)*&\"\"(F))F'\"\"#F)!\"\"*&\"#;F)F'F)F)\"#7F.*&F-F))F'F+F )F)*&\"#5F))F'\"\"'F)F.*&\"# " 0 "" {MPLTEXT 1 0 9 "int(g,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**&\"\"\"F%*$)%\" xG\"\"%F%!\"\"#F*F)-%#lnG6#F(F%*&F%F%,&F(F%\"\"#F*F*F*-F-6#,&F(F%\"\"$ F*F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "35" 0 } {VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }