On the Cauchy transform of complex powers of the identity function

TL;DR

This paper generalizes the analysis of a complex integral involving powers of the identity function, providing explicit solutions using hypergeometric functions for complex exponents and deriving related differential equations.

Contribution

It extends previous work by deriving explicit hypergeometric solutions for the integral with complex powers and establishing a differential equation framework.

Findings

Explicit hypergeometric solutions for the integral when |α| ≠ 1 and β ∈ ℂ.

Reduction of the integral to finite sums for rational β.

Derivation of a differential equation in α with properties similar to the hypergeometric equation.

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.