论文标题
在身份函数的复杂力量的凯奇变换上
On the Cauchy transform of complex powers of the identity function
论文作者
论文摘要
$β= \ frac {1} {1} {1} {2} $的积分$ \ int_ {| z | = 1} \ frac {z^β} {z-α} dz $已由Mortini和Rupp全面研究了教学目的。我们出于类似的目的而撰写，以更一般的考虑$β\ in \ mathbb {c} $详细说明他们的工作。这最终以$ |α|的高几何函数为顶点。 \ neq 1 $和\ Mathbb {C} $中的任何$β\。对于有理$β$，积分将减少为有限的总和。为此积分衍生出$α$中的微分方程，我们显示的属性与超几何方程相似。
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.
