\begin{definition}[Definition:Reduction Formula]
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.
Category:Definitions/Proof Techniques
Category:Definitions/Integral Calculus
\end{definition}