Mean Value Theorem for Holomorphic Functions

Theorem
Let $D$ be a region.

Let $f : D \to \C$ be a holomorphic function.

Let $z \in D$.

Let $r$ be such that $\map {B_r} z \subseteq D$.

Then:
 * $\ds \map f z = \frac 1 {2 \pi} \int_0^{2 \pi} \map f {z + r e^{i \theta} } \rd \theta$

Proof
By Cauchy's Integral Formula, we have:
 * $\ds \map f z = \frac 1 {2 \pi i} \oint_{\partial \map {B_r} z} \frac {\map f t} {t - z} \rd t$

where $\partial \map {B_r} z$ is the boundary of $\map {B_r} z$.

That is, $\partial \map {B_r} z$ is the circle of radius $r$, centred at $z$.

Note that we can parameterise $\partial \map {B_r} z$ by the function $\gamma : \hointr 0 {2 \pi} \to \partial \map {B_r} z$ defined by:
 * $\map \gamma \theta = z + r e^{i \theta}$

for each $\theta \in \hointr 0 {2 \pi}$.

Then, by the definition of a contour integral, we have: