  The integral $\int_{|z|=1} \frac{z^\beta}{z-\alpha} dz$ for
$\beta=\frac{1}{2}$ has been comprehensively studied by Mortini and Rupp for
pedagogical purposes. We write for a similar purpose, elaborating on their work
with the more general consideration $\beta \in \mathbb{C}$. This culminates in
an explicit solution in terms of the hypergeometric function for $|\alpha| \neq
1$ and any $\beta \in \mathbb{C}$. For rational $\beta$, the integral is
reduced to a finite sum. A differential equation in $\alpha$ is derived for
this integral, which we show has similar properties to the hypergeometric
equation.
