PlanetMath: partial fractions of expressions

(more info)

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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.

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