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:

$z\mapsto \frac{f^{\prime }(z)}{f(z)}$

This is indeed a holomorphic function, because $f$ being holomorphic, so is $f^{\prime }$, and by the quotient rule, we see that $f^{\prime }/f$ 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 $g^{\prime }$. 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 $f^{\prime }/f$.