A First Course in Complex Analysis · Problem 29

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

Mathematics — Complex Analysis · cauchys-theorem

Problem

Show that

\int_{C[0,2]} \frac{dz}{z^3 + 1} = 0

by arguing that this integral does not change if we replace

C[0,2]

C[0,r]

for any

r > 1

, then use Proposition 4.1.8 Item d to obtain an upper bound for

\left|\int_{C[0,r]} \frac{dz}{z^3+1}\right|

that goes to

0

r \to \infty