partial fractions of expressions

Let  R(z)=P(z)Q(z)  be a fractional expression, i.e., a quotient of the polynomialsPlanetmathPlanetmath 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 numbersMathworldPlanetmathPlanetmath.

If the distinct complex zeros of the denominator are  b1,b2,,bt  with the multiplicitiesτ1,τ2,,τt (t1), and the numerator has not common zeros, then R(z) can be decomposed uniquely as the sum


where H(z) is a polynomial and the Ajk’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 conjugatesMathworldPlanetmath, with same multiplicities, and the corresponding linear factors ( of Q(z) may be pairwise multiplied to quadratic polynomials of the form  z2+pz+q  with real p’s and q’s and  p2<4q.  Hence the above decomposition leads to the unique decomposition of the form

R(x)= H(x)+i=1m(Ai1x-bi+Ai2(x-bi)2++Aiμi(x-bi)μi)

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)=P(x)Q(x).  The coefficients Aik, Bjk and Cjk are uniquely determined real numbers.

Cf. the partial fractions of fractional numbers.


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