Definition:Reduction Formula

Definition

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

Let:

$\ds I_n = \int \map f {x, n} \rd x$

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

A reduction formula is a recurrence relation of the form:

$\ds I_k = \map g {I_n}$

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 $\map f {x, 0}$ or $\map f {x, 1}$ can be integrated directly.