Cauchy integral formula for constant functions

Statement

Suppose $c\in \mathbb{C}$ is a complex number, and $U$ is a domain in $\mathbb{C}$. Then, if $D$ is a disk centered at $z_0$ with radius $r$, and $z$ is any point in the interion of the disk, we have:

$c=\frac{1}{2\pi i}{\oint }_{\partial D}\frac{c}{\xi -z}d\xi$.

Proof

We use the parametrization of the circle $\partial D$ by angle. In other words, we define $\xi =r{e}^{i\theta }$, with $\theta$ moving from $0$ to $2\pi$. Thus:

${\oint }_{\partial D}\frac{c}{\xi -z}d\xi =\underset{0}{\overset{2\pi }{\int }}\frac{cir{e}^{i\theta }}{r{e}^{i\theta }-z}d\theta$