We can parameterize the circle by letting $z={z}_{0}+r{e}^{i\varphi}$.
Then $dz=ir{e}^{i\varphi}d\varphi $. Using the Cauchy integral formula^{} we can express $f({z}_{0})$ in the following way:

$f({z}_{0})$ 
$=$ 
$\frac{1}{2\pi i}}{\displaystyle {\oint}_{C}}{\displaystyle \frac{f(z)}{z{z}_{0}}}\mathit{d}z$ 



$=$ 
$\frac{1}{2\pi i}}{\displaystyle {\int}_{0}^{2\pi}}{\displaystyle \frac{f({z}_{0}+r{e}^{i\varphi})}{r{e}^{i\varphi}}}ir{e}^{i\varphi}\mathit{d}\varphi $ 



$=$ 
$\frac{1}{2\pi}}{\displaystyle {\int}_{0}^{2\pi}}f({z}_{0}+r{e}^{i\varphi})\mathit{d}\varphi .$ 
