partialfraction

Partialfraction, short for partial fraction decomposition, is a technique for rewriting a rational function as a sum of simpler fractions whose denominators are factors of the original denominator. It is widely used in integration, solving differential equations, and signal processing.

Given a rational function f(x) = P(x)/Q(x), first perform polynomial long division if the degree of P is not less than the degree of Q, writing f(x) = S(x) + R(x)/Q(x) with deg R < deg Q. Then decompose the proper fraction R(x)/Q(x) by factoring Q(x) over the chosen field (usually the real or complex numbers).

The form of the decomposition depends on the factorization of Q(x):

- For distinct linear factors (x − a1)(x − a2)…(x − ak): f(x) = A1/(x − a1) + A2/(x − a2) + … + Ak/(x − ak).

- For linear factors with multiplicities: f(x) = ∑i ∑j=1..mi Ai,j/(x − ai)^j.

- For irreducible quadratic factors over the reals, such as x^2 + px + q: f(x) = (Bx + C)/(x^2 + px + q). If such a quadratic is repeated, include terms (Dx + E)/(x^2 + px + q)^j for j = 1..m.

Procedure: multiply both sides by Q(x) to obtain an identity, then solve for coefficients by equating coefficients or substituting convenient x-values. The decomposition is unique given a fixed factorization and a field.

Example: Decompose (2x+3)/(x^2 − 3x + 2) = (2x+3)/[(x−1)(x−2)] as A/(x−1) + B/(x−2). Solving yields A = −5 and B = 7, so the decomposition is −5/(x−1) + 7/(x−2).