On the Cauchy transform of complex powers of the identity function

Abstract

The integral $\int_{|z|=1} \frac{z^β}{z-α} dz$ for $β=\frac{1}{2}$ 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 $β\in \mathbb{C}$. This culminates in an explicit solution in terms of the hypergeometric function for $|α| \neq 1$ and any $β\in \mathbb{C}$. 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.

Figures (3)

• Figure 1: Contour for $\theta = \pi$.
• Figure 2: Illustration of the case $\delta \geq \frac{\pi}{2}$.
• Figure 3: Illustration of the case $\delta < \frac{\pi}{2}$.

Theorems & Definitions (9)

• Theorem 1
• Lemma 1
• proof
• Proposition 1
• proof
• Corollary 1
• proof
• Remark 1
• Remark 2