A First Course in Complex Analysis · Problem 27

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

Mathematics — Complex Analysis · cauchys-theorem

Problem

Fix

a \in \mathbb{C}

. Compute

I(r) := \int_{C[0,r]} \frac{dz}{z - a}.

(You should get different answers for

r < |a|

and

r > |a|

.) Hint. In one case

\gamma_r

is contractible in

\mathbb{C} \setminus \{a\}

. In the other you can combine Exercise 4.5.4 and Exercise 4.5.22.