The logarithmic derivative of a meromorphic function

Let $\mathrm{\Omega }\subseteq \mathbb{C}$ be a complex domain, let $f$ be meromorphic on $\mathrm{\Omega }$, let $Z(f)$ be the zero set of $f$, and let $P(f)$ be its pole set. On the open set

$\mathrm{\Omega }\setminus (Z(f)\cup P(f))$

the function $f$ is holomorphic and nonzero, so the quotient

$\frac{f^{\mathrm{\prime }}}{f}$

is holomorphic there. This quotient is the logarithmic derivative of $f$.

The logarithmic derivative is not defined at a zero or a pole of $f$ by the displayed quotient itself. At every zero of finite positive order and at every pole, the next lemma identifies its principal part and shows that the resulting singularity is simple. The identically zero function is excluded from that local conclusion because its quotient $f^{\mathrm{\prime }}\mathrm{/}f$ is defined nowhere.