On the Cauchy transform of complex powers of the identity function

Abstract

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.