Let \(\Omega\subseteq\CC\) be open and let \(f\) be meromorphic on \(\Omega\) and not identically zero on any connected component of \(\Omega\). The logarithmic derivative of \(f\) is the meromorphic function on \(\Omega\) \[ \partial_{\log}f\coloneqq\frac{f'}{f}. \]
Let \(f\) be as in the definition and let \(z_0\in\Omega\) be a zero of order $m$ or a pole of order $m$ of \(f\). Then \(f'/f\) has a simple pole at \(z_0\) with residue \(m\) in the first case and \(-m\) in the second. At every other point of \(\Omega\), \(f'/f\) is holomorphic.
Near \(z_0\) write \(f(z)=(z-z_0)^{k}h(z)\) with \(h\) holomorphic near \(z_0\) and \(h(z_0)\neq0\), where \(k=m\) at a zero and \(k=-m\) at a pole. Then, for \(z\neq z_0\) near \(z_0\), \[ \frac{f'(z)}{f(z)}=\frac{k}{z-z_0}+\frac{h'(z)}{h(z)}, \] and \(h'/h\) is holomorphic near \(z_0\). At a point where \(f\) is holomorphic and nonzero, \(f'/f\) is holomorphic.