A First Course in Complex Analysis · Problem 47

Integration — Beck, Marchesi, Pixton & Sabalka

Mathematics — Complex Analysis · complex-logarithm

Problem

Let

\mathcal{L}\!\operatorname{og}

be a branch of the logarithm on

G

, and let

H := \{\mathcal{L}\!\operatorname{og}(z) : z \in G\}

, the image of

\mathcal{L}\!\operatorname{og}

. Show that

\mathcal{L}\!\operatorname{og} : G \to H

is a bijection whose inverse map is

f(z) : H \to G

given by

f(z) = \exp(z)

(i.e.,

f

is the exponential function restricted to

H