partial fractions of expressions

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

If the distinct complex zeros of the denominator are  $b_1,b_2,\mathrm{\dots },b_t$  with the multiplicities  ${\tau }_1,{\tau }_2,\mathrm{\dots },{\tau }_t$ ($t\ge 1$), and the numerator has not common zeros, then $R(z)$ can be decomposed uniquely as the sum

$$R(z)=H(z)+\sum _{j=1}^t\left(\frac{A_{j1}}{z-b_j}+\frac{A_{j2}}{{(z-b_j)}^2}+\mathrm{\dots }+\frac{A_{j{\tau }_j}}{{(z-b_j)}^{{\tau }_j}}\right),$$

where $H(z)$ is a polynomial and the $A_{jk}$’s are certain complex numbers.

Let us now take the special case that all coefficients of $P(z)$ and $Q(z)$ are real.  Then the (i.e. non-real) zeros of $Q(z)$ are pairwise complex conjugates, with same multiplicities, and the corresponding linear factors (http://planetmath.org/Product) of $Q(z)$ may be pairwise multiplied to quadratic polynomials of the form  $z^2+pz+q$  with real $p$’s and $q$’s and  .  Hence the above decomposition leads to the unique decomposition of the form

$R(x)=$ $H(x)+{\displaystyle \sum _{i=1}^m}\left({\displaystyle \frac{A_{i1}}{x-b_i}}+{\displaystyle \frac{A_{i2}}{{(x-b_i)}^2}}+\mathrm{\dots }+{\displaystyle \frac{A_{i{\mu }_i}}{{(x-b_i)}^{{\mu }_i}}}\right)$

$+{\displaystyle \sum _{j=1}^n}\left({\displaystyle \frac{B_{j1}x+C_{j1}}{x^2+p_{j}x+q_j}}+{\displaystyle \frac{B_{j2}x+C_{j2}}{{(x^2+p_{j}x+q_j)}^2}}+\mathrm{\dots }+{\displaystyle \frac{B_{j{\nu }_j}x+C_{j{\nu }_j}}{{(x^2+p_{j}x+q_j)}^{{\nu }_j}}}\right),$

where $m$ is the number of the distinct real zeros and $2n$ the number of the distinct zeros of the denominator $Q(x)$ 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.

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

Title	partial fractions of expressions
Canonical name	PartialFractionsOfExpressions
Date of creation	2013-03-22 14:20:27
Last modified on	2013-03-22 14:20:27
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	29
Author	pahio (2872)
Entry type	Definition
Classification	msc 26C15
Synonym	partial fractions
Related topic	ALectureOnThePartialFractionDecompositionMethod
Related topic	PartialFractionsForPolynomials
Related topic	ConjugatedRootsOfEquation2
Related topic	MixedFraction
Defines	fractional expression