Editing Partial fraction decomposition (section)

== Over the reals ==

Partial fractions are used in [[real number|real-variable]] [[integral calculus]] to find real-valued [[antiderivative]]s of [[rational function]]s. Partial fraction decomposition of real [[rational function]]s is also used to find their [[Inverse Laplace transform]]s. For applications of '''partial fraction decomposition over the reals''', see

* [[#Application to symbolic integration|Application to symbolic integration]], above
* [[Partial fractions in Laplace transforms]]

=== General result ===

Let <math>f(x)</math> be any rational function over the [[real number]]s.  In other words, suppose there exist real polynomials functions <math>p(x)</math> and <math>q(x) \neq 0</math>, such that
<math display="block">f(x) = \frac{p(x)}{q(x)}</math>

By dividing both the numerator and the denominator by the leading coefficient of <math>q(x)</math>, we may assume [[without loss of generality]] that <math>q(x)</math> is [[monic polynomial|monic]]. By the [[fundamental theorem of algebra]], we can write

<math display="block">q(x) = (x-a_1)^{j_1}\cdots(x-a_m)^{j_m}(x^2+b_1x+c_1)^{k_1}\cdots(x^2 + b_n x + c_n)^{k_n}</math>

where <math>a_1, \dots, a_m</math>, <math>b_1, \dots, b_n</math>, <math>c_1, \dots, c_n</math> are real numbers with <math>b_{i}^{2} -4c_{i} < 0</math>, and <math>j_1, \dots, j_m </math>, <math>k_1, \dots, k_n </math> are positive integers.  The terms <math> (x -a_i) </math> are the ''linear factors'' of <math>q(x)</math> which correspond to real roots of <math>q(x)</math>, and the terms <math> ( x_i^2 + b_ix + c_i ) </math> are the ''irreducible quadratic factors'' of <math>q(x)</math> which correspond to pairs of [[complex number|complex]] conjugate roots of  <math>q(x)</math>.

Then the partial fraction decomposition of <math>f(x)</math>  is the following:

<math display="block">f(x) = \frac{p(x)}{q(x)} = P(x) + \sum_{i=1}^m\sum_{r=1}^{j_i} \frac{A_{ir}}{(x-a_i)^r} + \sum_{i=1}^n\sum_{r=1}^{k_i} \frac{B_{ir}x+C_{ir}}{(x^2+b_ix+c_i)^r}</math>

Here, ''P''(''x'') is a (possibly zero) polynomial, and the ''A''<sub>''ir''</sub>, ''B''<sub>''ir''</sub>, and ''C''<sub>''ir''</sub> are real constants.  There are a number of ways the constants can be found.

The most straightforward method is to multiply through by the common denominator ''q''(''x'').  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 ''A''<sub>''ir''</sub>, ''B''<sub>''ir''</sub>, and ''C''<sub>''ir''</sub>.  Since two polynomials are equal if and only if their corresponding coefficients are equal, we can equate the coefficients of like terms.  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]]. It can also be found with [[limit (mathematics)|limits]] (see [[#Example 5 (limit method)|Example 5]]).