{VERSION 3 0 "SGI MIPS UNIX" "3.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "Rational functions." }}} {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 functio ns in their normal form, which means the form A/B where A and B are po lynomials 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 rationa l 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 ra tional 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 sqrfree and gcd do not accept rational functions as input. The command factor does, it normalizes and then just factors t he numerator and denominator." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "factor(x^2+1/x);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 114 "If a r ational function has a pole at a point alpha of multiplicity e, then t he derivative has a pole 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(di ff(%,x));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "normal(diff(%, x));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 131 "Because of that, the der ivative of a rational 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 "Poles 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 w as a sum in which each term had the form (...)/(x-alpha)^e. The functi on g is the same rational function, but looks more complicated. Maple \+ can convert between these forms, see the help page ?convert,parfrac fo r more information:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "conv ert(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 }