PlanetMath: partial fractions of expressions

(more info)

Math for the people, by the people.

Main Menu

sections Encyclopædia
Papers
Books
Expositions

meta Requests (232)
Orphanage
Unclass'd
Unproven (519)
Corrections (22)
Classification

talkback Polls
Forums
Feedback
Bug Reports

downloads Snapshots
PM Book

information News
Docs
Wiki
ChangeLog
TODO List
Legalese
About

Entry average rating: No information on entry rating

partial fractions of expressions

(Definition)

Let $R(z) = \frac{P(z)}{Q(z)}$ be a fractional expression, i.e., a quotient of the polynomials and such that is not divisible by . Let's restrict to the case that the coefficients are real or complex numbers.

If the distinct complex zeros of the denominator are $b_1,\,b_2,\,\ldots,\,b_t$ with the multiplicities $\tau_1,\,\tau_2,\,\ldots,\,\tau_t$ ( $t \ge 1$ ), and the numerator has not common zeros, then can be decomposed uniquely as the sum

$\displaystyle R(z) = H(z)+ \sum_{j=1}^t\left(\frac{A_{j1}}{z-b_j}+\frac{A_{j2}}{(z-b_j)^2}+\cdots +\frac{A_{j\tau_j}}{(z-b_j)^{\tau_j}}\right),$

where

is a polynomial and the $A_{jk}$ 's are certain complex numbers.

Let us now take the special case that all coefficients of and are real. Then the imaginary (i.e. non-real) zeros of are pairwise complex conjugates, with same multiplicities, and the corresponding linear factors of may be pairwise multiplied to quadratic polynomials of the form $z^2\!+\!pz\!+\!q$ with real 's and 's and . Hence the above decomposition leads to the unique decomposition of the form

$\displaystyle R(x) = \quad$	$\displaystyle H(x)+ \sum_{i=1}^m\left(\frac{A_{i1}}{x-b_i}+\frac{A_{i2}}{(x-b_i)^2}+\cdots +\frac{A_{i\mu_i}}{(x-b_i)^{\mu_i}}\right)$
	$\displaystyle +\sum_{j=1}^n\left(\frac{B_{j1}x+C_{j1}}{x^2+p_jx+q_j}+ \frac{B_{... ...+q_j)^2}+\cdots +\frac{B_{j\nu_j}x+C_{j\nu_j}}{( x^2+p_jx+q_j)^{\nu_j}}\right),$

where

is the number of the distinct real zeros and

the number of the distinct imaginary zeros of the denominator

of the fractional expression $R(x) = \frac{P(x)}{Q(x)}$ . The coefficients $A_{ik}$ , $B_{jk}$ and $C_{jk}$ are uniquely determined real numbers.

Cf. the partial fractions of fractional numbers.

Example.

$\displaystyle \frac{-x^5\!+\!6x^4\!-\!7x^3\!+\!15x^2\!-\!4x\!+\!3} {(x\!-\!1)^3... ...\frac{3}{(x\!-\!1)^3}\!+ \frac{x}{x^2\!+\!1}\!+\!\frac{2x\!-\!1}{(x^2\!+\!1)^2}$