Partial Fraction - Basic Principles

Basic Principles

The basic principles involved are quite simple; it is the algorithmic aspects that require attention in particular cases. On the other hand, the existence of a decomposition of a certain kind is an assumption in practical cases, and the principles should explain which assumptions are justified.

Assume a rational function R(x) = ƒ(x)/g(x) in one indeterminate x has a denominator that factors as

over a field K (we can take this to be real numbers, or complex numbers). If P and Q have no common factor, then R may be written as

for some polynomials A(x) and B(x) over K. The existence of such a decomposition is a consequence of the fact that the polynomial ring over K is a principal ideal domain, so that

for some polynomials C(x) and D(x) (see Bézout's identity).

Using this idea inductively we can write R(x) as a sum with denominators powers of irreducible polynomials. To take this further, if required, write:

as a sum with denominators powers of F and numerators of degree less than F, plus a possible extra polynomial. This can be done by the Euclidean algorithm, polynomial case. The result is the following theorem:

Let ƒ and g be nonzero polynomials over a field K. Write g as a product of powers of distinct irreducible polynomials :

There are (unique) polynomials b and a _ij with deg a _ij < deg p _i such that

If deg ƒ < deg g, then b = 0.

Therefore when the field K is the complex numbers, we can assume that each p_i has degree 1 (by the fundamental theorem of algebra) the numerators will be constant. When K is the real numbers, some of the p_i might be quadratic, so in the partial fraction decomposition a quotient of a linear polynomial by a power of a quadratic will occur.

In the preceding theorem, one may replace "distinct irreducible polynomials" by "pairwise coprime polynomials that are coprime with their derivative". For example, the p_i may be the factors of the square-free factorization of g. When K is the field of the rational numbers, as it is typically the case in computer algebra, this allows to replace factorization by greatest common divisor to compute the partial fraction decomposition.