In algebra, the partial fraction decomposition or partial fraction expansion is a procedure used to reduce the degree of either the numerator or the denominator of a rational function. The main motivation to decompose a rational function into a sum of simpler fractions is that it makes it simpler to perform linear operations on it.
The problem of computing derivatives, anti-derivatives, integrals, inverse Laplace transforms, power series expansions, Fourier series, residues, and linear functional transformations of rational functions can be reduced, via partial fraction decomposition, to making the computation on each single element used in the decomposition.
Standard Form 1:
p p 1 1
---------------- = ------- ( -------- - ------- )
(x+a)(x+b) b-a x+a x+b
Example:
5 5 1 1 5 1 1
---------------- = --------- ( -------- - ------ ) = ---- ( ------- - -------- )
(x-1)(x+3) 3-(-1) x-1 x+3 4 x-1 x+3
Standard Form 2:
p p 1 1
-------------------- = ------- ( --------- - --------- )
(xn+a)(xn+b) b-a xn+a xn+b
Example:
3 3 1 1 3 1 1
------------------ = --------- ( --------- - --------- ) = ---- ( -------- - --------- )
(x2-2)(x2+3) 3-(-2) x2-2 x2+3 5 x2-2 x2+3
Standard Form 3:
p p 1 x-a
---------------- = ---------- ( -------- - ---------- )
(x+a)(x2+b) b+a2 x+a x2+b
Example:
7 7 1 x+2 1 x+2
----------------- = ------------ ( ------- - --------- ) = ------- - ---------
(x-2)(x2+3) 3+(-2)2 x-2 x2+3 x-2 x2+3
