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:

$f(z_0)=\frac{1}{2\pi }\underset{0}{\overset{2\pi }{\int }}f(z_0+R{e}^{it})dt$

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.