On the Cauchy Transform of the Complex Power Function

Abstract

The integral ∫_{|z|=1} (zᵝ / (z−α)) dz for β = ½ 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 β ∈ ℂ. This culminates in an explicit solution in terms of the hypergeometric function for |α| ≠ 1 and any β ∈ ℂ. For rational β, the integral is reduced to a finite sum. A differential equation in α is derived for this integral, which we show has similar properties to the hypergeometric equation.