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{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)F0*$)F)\"
\"%F+F,F0F,F+,.F1F,F-F0F4F,F)F0*$)F)\"\"$F+F,F0F,!\"\"" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "numer(f); # numerator" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#,.*$)%\"xG\"\"'\"\"\"\"\"\"*$)F&\"\"#F(!\"$*$)F&
\"\"&F(F)F&F-*$)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)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "g:=normal(f); # r
emove gcd of numerator and denominator" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%\"gG*&,&*$)%\"xG\"\"%\"\"\"\"\"\"!\"$F,F+,&*$)F)\"\"$F+F,F-F,!
\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 262 "The command normal puts \+
rational functions in their normal form, which means the form A/B wher
e A and B are polynomials with no common factors, so the gcd(numer(g),
 denom(g)) will be 1. It also produces 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.*$)F'\"\"%F)F*F.F*F),.F/F*F+F.F2F*
F'F.*$)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
*,.F0F+F,F/F3F+F(F/*$)F(\"\"$F*F+F/F+!\"\"F+*&F(F**$),&F(F+F/F+\"\"$F*
F:!#7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "normal(%);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,2*$)%\"xG\"\"(\"\"\"\"\"\"*$)F'\"\"
'F)!\"**$)F'\"\"&F)\"#F*$)F'\"\"%F)!#R*$)F'\"\"$F)!\"$*$)F'\"\"#F)F2F'
!#X\"#\")F*F)*&),&F'F*F:F*\"\"$F),&F7F*F:F*\"\"\"!\"\"" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "f - g;" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#,&*&,.*$)%\"xG\"\"'\"\"\"\"\"\"*$)F(\"\"#F*!\"$*$)F(\"\"&F*F+F(F
/*$)F(\"\"%F*F+F/F+F*,.F0F+F,F/F3F+F(F/*$)F(\"\"$F*F+F/F+!\"\"F+*&,&F3
F+F/F+F*,&F7F+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 a
lso use normal." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "sqrfree(
f);  # Note: this gives an error in Maple 5, but not in 6 or 7." }}
{PARA 8 "" 1 "" {TEXT -1 57 "Error, (in sqrfree) argument must be a po
lynomial in, \{x\}" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 127 "As you can
 see, a positive multiplicity means that the factor is in the numerato
r, and negative means 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',(*$)F&\"\"#\"\"\"F'F&!\"\"F'F'F'F,F&!\"
\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "factors(x^2+1/x);" }}
{PARA 8 "" 1 "" {TEXT -1 80 "Error, (in factors) argument must be a po
lynomial over an algebraic number field" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 237 "If a rational function has a pole at a point alpha of mu
ltiplicity e  (so with squarefree you'd see a factor with multiplicity
 -e), then the derivative has a pole of order e+1  (so if you then do \+
sqrfree you'd see a multiplicity -(e+1))." }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 29 "f:=3/(x-2)^3+x/(x^2+1)^2+1/x;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"fG,(*&\"\"\"F'*$),&%\"xG\"\"\"!\"#F,\"\"$F'!\"\"\"
\"$*&F+F'*$),&*$)F+\"\"#F'F,F,F,\"\"#F'F/F,*&F'F'F+F/F," }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "f:=normal(f);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"fG*&,2*$)%\"xG\"\"&\"\"\"\"#=*$)F)\"\"$F+\"#VF)\"#:
*$)F)\"\"%F+!#E*$)F)\"\"#F+!#I*$)F)\"\"(F+\"\"\"*$)F)\"\"'F+!\"'!\")F=
F+*(),&F)F=!\"#F=\"\"$F+),&F6F=F=F=\"\"#F+F)\"\"\"!\"\"" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "diff(f,x);" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#,**&,0*$)%\"xG\"\"%\"\"\"\"#!**$)F(\"\"#F*\"$H\"\"#:\"
\"\"*$)F(\"\"$F*!$/\"F(!#g*$)F(\"\"'F*\"\"(*$)F(\"\"&F*!#OF**(),&F(F1!
\"#F1\"\"$F*),&F,F1F1F1\"\"#F*F(\"\"\"!\"\"F1*&,2F;\"#=F2\"#VF(F0F&!#E
F,!#I*$)F(F:F*F1F7!\"'!\")F1F**()FA\"\"%F*)FE\"\"#F*F(\"\"\"FH!\"$*&FJ
F**&)FA\"\"$F*)FE\"\"$F*FH!\"%*&FJF**()FA\"\"$F*)FE\"\"#F*)F(\"\"#F*FH
!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "normal(%);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&,8\"#;\"\"\"%\"xG!#K*$)F(\"\"'\"\"
\"\"$*=*$)F(\"\"#F-\"#l*$)F(\"\"%F-\"$s\"*$)F(\"\"&F-!$3#*$)F(\"\"$F-!
#s*$)F(\"\"*F-!\")*$)F(\"\")F-\"#R*$)F(\"\"(F-!#!)*$)F(\"#5F-F'F-*(),&
F(F'!\"#F'\"\"%F-),&F/F'F'F'\"\"$F-)F(\"\"#F-!\"\"!\"\"" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "normal(diff(%,x));" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#,$*&,>!#K\"\"\"%\"xG\"#!)*$)F(\"\"'\"\"\"!$g$*$)F(
\"\"#F-!$3#*$)F(\"\"%F-!$I$*$)F(\"\"&F-\"$L#*$)F(\"\"$F-\"$y$*$)F(\"\"
*F-\"$_&*$)F(\"\")F-!$K)*$)F(\"\"(F-\"$7**$)F(\"#5F-!$!=*$)F(\"#6F-\"#
o*$)F(\"#7F-!#5*$)F(\"#8F-F'F-*(),&F(F'!\"#F'\"\"&F-),&F/F'F'F'\"\"%F-
)F(\"\"$F-!\"\"F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "normal
(diff(%,x));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,$*&,D%\"xG!$#>*$)F&\"
\"'\"\"\"\"%\"p#*$)F&\"\"#F+\"$g&*$)F&\"\"%F+\"%e?*$)F&\"\"&F+!%;J*$)F
&\"\"$F+!%?6*$)F&\"\"*F+!%C9*$)F&\"\")F+!$:%*$)F&\"\"(F+!#g*$)F&\"#5F+
\"%wL*$)F&\"#6F+!%sC*$)F&\"#7F+\"%!G\"*$)F&\"#8F+!$S$\"#k\"\"\"*$)F&\"
#9F+\"$0\"*$)F&\"#:F+!#7*$)F&\"#;F+FZF+*(),&F&FZ!\"#FZ\"\"'F+),&F-FZFZ
FZ\"\"&F+)F&\"\"%F+!\"\"!\"'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 162 "
Because of that, the derivative of a rational function can not have po
le order 1, if a derivative of a rational function has a pole, the pol
e order is at least 2." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "f
:=1/x^5 + 1/x + 1/(x-2)^2 + 1/(x-3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#>%\"fG,**&\"\"\"F'*$)%\"xG\"\"&F'!\"\"\"\"\"*&F'F'F*F,F-*&F'F'*$),&F
*F-!\"#F-\"\"#F'F,F-*&F'F',&F*F-!\"$F-F,F-" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 9 "int(f,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**&\"
\"\"F%*$)%\"xG\"\"%F%!\"\"#!\"\"\"\"%-%#lnG6#F(\"\"\"*&F%F%,&F(F1!\"#F
1F*F,-F/6#,&F(F1!\"$F1F1" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 119 "Pole
s of order 1 can not come from derivatives of rational 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*$)%\"xG\"\"$\"\"\"\"\"\"*$)F'\"\"#F)!\"(F'\"#;!#7F**$)F'\"\"(F)F-*$
)F'\"\"'F)!#5*$)F'\"\"&F)\"#<*$)F'\"\"%F)F0F)*()F'\"\"&F)),&F'F*!\"#F*
\"\"#F),&F'F*!\"$F*\"\"\"!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 22 "g:=normal(f,expanded);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"
gG*&,2*$)%\"xG\"\"$\"\"\"\"\"\"*$)F)\"\"#F+!\"(F)\"#;!#7F,*$)F)\"\"(F+
F/*$)F)\"\"'F+!#5*$)F)\"\"&F+\"#<*$)F)\"\"%F+F2F+,**$)F)\"\")F+F,F3F0F
6F1F:F2!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 269 "Now f was easy t
o integrate because it was a sum in which each term had the form (...)
/(x-alpha)^e. The function g is the same rational function, but looks \+
more complicated. Maple can convert between these forms, see the help \+
page ?convert,parfrac for more information:" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 21 "convert(g,parfrac,x);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,**&\"\"\"F%*$)%\"xG\"\"&F%!\"\"\"\"\"*&F%F%F(F*F+*&F%F
%*$),&F(F+!\"#F+\"\"#F%F*F+*&F%F%,&F(F+!\"$F+F*F+" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 2 "g;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,2*$)
%\"xG\"\"$\"\"\"\"\"\"*$)F'\"\"#F)!\"(F'\"#;!#7F**$)F'\"\"(F)F-*$)F'\"
\"'F)!#5*$)F'\"\"&F)\"#<*$)F'\"\"%F)F0F),**$)F'\"\")F)F*F1F.F4F/F8F0!
\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 152 "To integrate this ration
al function, Maple will first compute the form given by convert(g,parf
rac,x), and after that integration has become much easier:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "int(g,x);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,**&\"\"\"F%*$)%\"xG\"\"%F%!\"\"#!\"\"\"\"%-%#lnG6#F(\"
\"\"*&F%F%,&F(F1!\"#F1F*F,-F/6#,&F(F1!\"$F1F1" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "33 1" 0 }{VIEWOPTS 1 1 0 1 1 1803 }