In algebra, the partial fraction decomposition or partial fraction expansion of a rational fraction (that is, a fraction such that the numerator and the denominator are both polynomials) is an operation that consists of expressing the fraction as a sum of a polynomial (possibly zero) and one or several fractions with a simpler denominator.
The importance of the partial fraction decomposition lies in the fact that it provides algorithms for various computations with rational functions, including the explicit computation of antiderivatives, Taylor series expansions, inverse Z-transforms, and inverse Laplace transforms. The concept was discovered independently in 1702 by both Johann Bernoulli and Gottfried Leibniz.
In symbols, the partial fraction decomposition of a rational fraction of the form where and are polynomials, is its expression as
where
is a polynomial, and, for each , the denominator is a power of an irreducible polynomial (that is not factorable into polynomials of positive degrees), and the numerator is a polynomial of a smaller degree than the degree of this irreducible polynomial.
When explicit computation is involved, a coarser decomposition is often preferred, which consists of replacing "irreducible polynomial" by "square-free polynomial" in the description of the outcome. This allows replacing polynomial factorization by the much easier-to-compute square-free factorization. This is sufficient for most applications, and avoids introducing irrational coefficients when the coefficients of the input polynomials are integers or rational numbers.
Basic principles
Let
be a rational fraction, where and are univariate polynomials in the indeterminate over a field. The existence of the partial fraction can be proved by applying inductively the following reduction steps.
Polynomial part
There exist two polynomials and such that
and
where denotes the degree of the polynomial .
This results immediately from the Euclidean division of by , which asserts the existence of and such that and
This allows supposing in the next steps that
Factors of the denominator
If and
where and are coprime polynomials, then there exist polynomials and such that
and
This can be proved as follows. Bézout's identity asserts the existence of polynomials and such that
(by hypothesis, is a greatest common divisor of and ).
Let with be the Euclidean division of by Setting one gets
It remains to show that By reducing the last sum of fractions to a common denominator, one gets
and thus
Powers in the denominator
Using the preceding decomposition inductively one gets fractions of the form with where is an irreducible polynomial. If , one can decompose further, by using that an irreducible polynomial is a square-free polynomial, that is, is a greatest common divisor of the polynomial and its derivative. If is the derivative of , Bézout's identity provides polynomials and such that and thus Euclidean division of by gives polynomials and such that and Setting one gets
with
Iterating this process with in place of leads eventually to the following theorem.
Statement
The uniqueness can be proved as follows. Let . All together, and the have coefficients. The shape of the decomposition defines a linear map from coefficient vectors to polynomials of degree less than . The existence proof means that this map is surjective. As the two vector spaces have the same dimension, the map is also injective, which means uniqueness of the decomposition. By the way, this proof induces an algorithm for computing the decomposition through linear algebra.
If is field of complex numbers, the fundamental theorem of algebra implies that all have degree one, and all numerators are constants. When is the field of real numbers, some of the may be quadratic, so, in the partial fraction decomposition, quotients of linear polynomials by powers of quadratic polynomials may also occur.
In the preceding theorem, one may replace "distinct irreducible polynomials" by "pairwise coprime polynomials that are coprime with their derivative". For example, the may be the factors of the square-free factorization of . When is the field of rational numbers, as it is typically the case in computer algebra, this allows to replace factorization by greatest common divisor computation for computing a partial fraction decomposition.
Application to symbolic integration
For the purpose of symbolic integration, the preceding result may be refined into