On the Cauchy transform of complex powers of the identity function

Abstract

The integral ${\int }_{|z|=1}\frac{z^2}{z-\hat{I}\pm }dz$ for ${\hat{I}}^2=\frac{1}{2}$ 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 ${\hat{I}}^2\in \mathbb{C}$. This culminates in an explicit solution in terms of the hypergeometric function for $|\hat{I}\pm |\ne 1$ and any ${\hat{I}}^2\in \mathbb{C}$. For rational ${\hat{I}}^2$, the integral is reduced to a finite sum. A differential equation in $\hat{I}\pm$ is derived for this integral, which we show has similar properties to the hypergeometric equation.

18 pages, 3 figures