Holomorphic logarithm of a nowhere zero holomorphic function

Definition

Suppose $U\subset \mathbb {C}$ is an open subset and $f:U\to \mathbb {C} ^{*}$ is a nowhere zero holomorphic function on $U$. A holomorphic logarithm of $f$ is a holomorphic function $g:U\to \mathbb {C}$ such that:

$f(z)=\exp(g(z))\ \forall \ z\in U$

where $\exp$ denotes the complex exponential.

A holomorphic logarithm need not always exist. If there exists a holomorphic logarithm $g$, then any function of the form:

$z\mapsto g(z)+2n\pi i,\qquad n\in \mathbb {Z}$

is also a holomorphic logarithm, and all holomorphic logarithms are of the above form.

Note that we can take $f$ to be the inclusion map of math>U</math> in $\mathbb {C}$, in which case $g$ is a holomorphic branch of the logarithm on $U$. The image $g(U)$ is termed a holomorphic logarithm of the open subset $U$.