# 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  $b_{1},\,b_{2},\,\ldots,\,b_{t}$  with the multiplicities$\tau_{1},\,\tau_{2},\,\ldots,\,\tau_{t}$ ($t\geq 1$), and the numerator has not common zeros, then $R(z)$ can be decomposed uniquely as the sum

 $R(z)\;=\;H(z)+\sum_{j=1}^{t}\left(\frac{A_{j1}}{z-b_{j}}+\frac{A_{j2}}{(z-b_{j% })^{2}}+\ldots+\frac{A_{j\tau_{j}}}{(z-b_{j})^{\tau_{j}}}\right),$

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  $p^{2}<4q$.  Hence the above decomposition leads to the unique decomposition of the form

 $\displaystyle R(x)\;=$ $\displaystyle H(x)+\sum_{i=1}^{m}\left(\frac{A_{i1}}{x-b_{i}}+\frac{A_{i2}}{(x% -b_{i})^{2}}+\ldots+\frac{A_{i\mu_{i}}}{(x-b_{i})^{\mu_{i}}}\right)$ $\displaystyle+\sum_{j=1}^{n}\left(\frac{B_{j1}x+C_{j1}}{x^{2}+p_{j}x+q_{j}}+% \frac{B_{j2}x+C_{j2}}{(x^{2}+p_{j}x+q_{j})^{2}}+\ldots+\frac{B_{j\nu_{j}}x+C_{% j\nu_{j}}}{(x^{2}+p_{j}x+q_{j})^{\nu_{j}}}\right),$

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