# Definition:Reduction Formula

## Definition

Let $f: \R \times \Z_{\ge 0}: \R$ be an integrable real-valued function on $\R \times \N$.

Let:

$\displaystyle I_n = \int f \left({x, n}\right) \ \mathrm d x$

for some $n \in \Z_{\ge 0}$.

A reduction formula is a recurrence relation of the form:

$\displaystyle I_k = g \left({\displaystyle I_n}\right)$

such that $k < n$.

That is, it is a technique to reduce the integer parameter in the integrand in order to allow evaluation of the integral at, usually, $n = 0$ or $n = 1$.

This technique relies upon the supposition that $f \left({x, 0}\right)$ or $f \left({x, 1}\right)$ can be integrated directly.