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 $\xi =re^{i\theta }$, with $\theta$ moving from $0$ to $2\pi$. Thus:

$\oint _{\partial D}{\frac {c}{\xi -z}}\,d\xi =\int _{0}^{2\pi }{\frac {cire^{i\theta }}{re^{i\theta }-z}}\,d\theta$