Mean-value theorem

This article gives the statement, and possibly proof, of a basic fact in complex analysis.
View a complete list of basic facts in complex analysis

Statement

Suppose $U\subset \mathbb {C}$ is an open subset and $f:U\to \mathbb {C}$ is a holomorphic function. Let $R>0$ be such that the circle of radius $R$ centered at $z_{0}$, lies completely inside $U$, then:

$f(z_{0})={\frac {1}{2\pi }}\int _{0}^{2\pi }f(z_{0}+Re^{it})\,dt$

Facts used

• Cauchy integral formula: In fact, the mean-value theorem is a direct consequence of the Cauchy integral formula, with the angle-based parametrization of the circle.