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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 9 "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);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
21 "numer(f); # numerator" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
23 "denom(f); # denominator" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
55 "g:=normal(f); # remove gcd of numerator and denominator" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 262 "The command normal puts rational \+
functions in their normal form, which means the form A/B where 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;" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "f-12*x/(x-3)^3;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "normal(%);" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 6 "f - g;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 10 "normal(%);" }}}{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 11 "sqrfree(f);" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 127 "As you can see, a positive multiplicity means that \+
the factor is in the numerator, and negative means it is in the denomi
nator." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "factor(x^2+1/x);
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "factors(x^2+1/x);" }}}
{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 see a f
actor 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
;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "f:=normal(f);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "diff(f,x);" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 10 "normal(%);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 18 "normal(diff(%,x));" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 18 "normal(diff(%,x));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 162 "Because of that, the derivative of a rational function can not
 have pole order 1, if a derivative of a rational 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);" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 9 "int(f,x);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
119 "Poles of order 1 can not come from derivatives of rational functi
ons. Such poles lead to logarithms when you integrate." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "normal(f);" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 22 "g:=normal(f,expanded);" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 269 "Now f was easy to integrate because it was a sum in wh
ich each term had the form (...)/(x-alpha)^e. The function g is the sa
me rational function, but looks more complicated. Maple can convert be
tween these forms, see the help page ?convert,parfrac for more informa
tion:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "convert(g,parfrac,
x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "g;" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 152 "To integrate this rational function, Maple wil
l first compute the form given by convert(g,parfrac,x), and after that
 integration has become much easier:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 9 "int(g,x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 
"" }}}}{MARK "0 2 0" 9 }{VIEWOPTS 1 1 0 3 2 1804 1 1 1 1 }
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