[1] 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,[2] Taylor series expansions, inverse Z-transforms, and inverse Laplace transforms.
The concept was discovered independently in 1702 by both Johann Bernoulli and Gottfried Leibniz.
be a rational fraction, where F and G are univariate polynomials in the indeterminate x over a field.
The existence of the partial fraction can be proved by applying inductively the following reduction steps.
denotes the degree of the polynomial P. This results immediately from the Euclidean division of F by G, which asserts the existence of E and F1 such that
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 K is the field of complex numbers, the fundamental theorem of algebra implies that all pi have degree one, and all numerators
When K is the field of real numbers, some of the pi may be quadratic, so, in the partial fraction decomposition, quotients of linear polynomials by powers of quadratic polynomials may also occur.
For the purpose of symbolic integration, the preceding result may be refined into Theorem — Let f and g be nonzero polynomials over a field K. Write g as a product of powers of pairwise coprime polynomials which have no multiple root in an algebraically closed field:
Reducing the sum of fractions in the Theorem to a common denominator, and equating the coefficients of each power of x in the two numerators, one gets a system of linear equations which can be solved to obtain the desired (unique) values for the unknown coefficients.
, where the αn are distinct constants and deg P < n, explicit expressions for partial fractions can be obtained by supposing that
and solving for the ci constants, by substitution, by equating the coefficients of terms involving the powers of x, or otherwise.
A more direct computation, which is strongly related to Lagrange interpolation, consists of writing
This approach does not account for several other cases, but can be modified accordingly: In an example application of this procedure, (3x + 5)/(1 − 2x)2 can be decomposed in the form
Over the complex numbers, suppose f(x) is a rational proper fraction, and can be decomposed into
Partial fractions are used in real-variable integral calculus to find real-valued antiderivatives of rational functions.
Partial fraction decomposition of real rational functions is also used to find their Inverse Laplace transforms.
Here, P(x) is a (possibly zero) polynomial, and the Air, Bir, and Cir are real constants.
We then obtain an equation of polynomials whose left-hand side is simply p(x) and whose right-hand side has coefficients which are linear expressions of the constants Air, Bir, and Cir.
In this way, a system of linear equations is obtained which always has a unique solution.
This solution can be found using any of the standard methods of linear algebra.
Multiplying through by the denominator on the left-hand side gives us the polynomial identity
This example illustrates almost all the "tricks" we might need to use, short of consulting a computer algebra system.
Multiplying through by the denominator on the left-hand side we have the polynomial identity
Alternatively, instead of expanding, one can obtain other linear dependences on the coefficients computing some derivatives at
The partial fraction decomposition of a rational function can be related to Taylor's theorem as follows.
Taylor's theorem (in the real or complex case) then provides a proof of the existence and uniqueness of the partial fraction decomposition, and a characterization of the coefficients.
The above partial fraction decomposition implies, for each 1 ≤ i ≤ r, a polynomial expansion
The idea of partial fractions can be generalized to other integral domains, say the ring of integers where prime numbers take the role of irreducible denominators.