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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT 256 28 "Rational functions in Map
le." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{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 9 "numer(f);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "denom(f);" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 13 "g:=normal(f);" }}}{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 th
is 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 -1 164 "Commands like sqr
free and gcd do not accept rational functions as input. The command fa
ctor does, it normalizes and then just factors the numerator and denom
inator." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "factor(x^2+1/x);
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 114 "If a rational function has a
 pole at a point alpha of multiplicity e, then the derivative has a po
le of order 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 -1 131 "Because of that, the derivative of a rat
ional function can not have pole order 1, if there is 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 55 "Pole
s of order 1 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 whic
h each term had the form (...)/(x-alpha)^e. The function g is the same
 rational function, but looks more complicated. Maple can convert betw
een these forms, see the help page ?convert,parfrac for more informati
on:" }}}{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 "" {MPLTEXT 1 0 9 "int(g,x);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}}{MARK "33" 0 }{VIEWOPTS 1 1 0 1 1 1803 }