Order of a zero or pole

Let $f$ be meromorphic near $a$ and not identically zero. There is a unique integer $m\in\mathbb Z$ and a holomorphic function $u$ near $a$ , with $u(a)\ne0$ , such that

f(z)=(z-a)^m u(z).

The integer $\operatorname{ord}_a(f)=m$ is the order of $f$ at $a$ : $m>0$ is a zero of multiplicity $m$ , $m<0$ is a pole of order $-m$ , and $m=0$ means $f$ is holomorphic and nonzero at $a$ .

Arithmetic

Whenever the expressions are defined,

\operatorname{ord}_a(fg)=\operatorname{ord}_a(f)+\operatorname{ord}_a(g), \qquad \operatorname{ord}_a(f/g)=\operatorname{ord}_a(f)-\operatorname{ord}_a(g).

Thus order behaves as a local additive valuation on nonzero meromorphic germs.

Logarithmic derivative

The logarithmic derivative gives, for a small positively oriented circle around $a$ ,

\operatorname{ord}_a(f) =\frac{1}{2\pi i}\int\frac{f'(z)}{f(z)}\,dz.

This local identity is the building block of the argument principle.

References

Otto Forster, Lectures on Riemann Surfaces, Springer, 1981. Publisher record. Relevant: Chapter 1, §§8–9.