A First Course in Complex Analysis · Problem 15

Consequences of Cauchy's Theorem — Beck, Marchesi, Pixton & Sabalka

Mathematics — Complex Analysis · cauchys-theorem

Problem

Taking the previous two exercises one step further, fix

z_0 \in \mathbb{C}

and let

\gamma

be a simple, closed, positively oriented, piecewise smooth path such that

z_0

is inside

\gamma

. Show that, for any integer

n

\int_{\gamma} (z - z_0)^n\, dz = \begin{cases} 2\pi i & \text{if } n = -1, \\ 0 & \text{otherwise.} \end{cases}