Logarithmic derivative of a nowhere zero holomorphic function

Definition

Suppose $U\subset \mathbb {C}$ is an open subset and $f:U\to \mathbb {C} ^{*}$ is an everywhere nonzero holomorphic function. The logarithmic derivative of $f$ is the function:

$z\mapsto {\frac {f'(z)}{f(z)}}$

This is indeed a holomorphic function, because $f$ being holomorphic, so is $f'$, and by the quotient rule, we see that $f'/f$ is also holomorphic.

If there exists a logarithm of $f$, i.e. a function $g:U\to \mathbb {C}$ such that $e^{g(z)}=f(z)$ for all $z\in U$, then the logarithmic derivative equals $g'$. Conversely, if the logarithmic derivative of $f$ admits a primitive, then $f$ admits a holomorphic logarithm on $U$, with the holomorphic logarithm being a primitive of $f'/f$.