---
tag: 23ASRP0MWR
kind: concept
name: Order of a zero or pole
msc: [30D30, 30C15]
sources:
  - url: https://en.wikipedia.org/wiki/Zeros_and_poles
---

## Summary

The order of a [[meromorphic function|0ZZ7ZYDR8W]] at a point $a$ is the integer $n$ such that it behaves like $c (z - a)^n$ near $a$, with $c \neq 0$.
See the [[definition|K2S5TAW2BF]].
At a zero it is the [[order of the zero|0QX19C7K1C]].
At a pole it is minus the [[order of the pole|073MWACF1K]].
This single integer turns products of meromorphic functions into sums, like a logarithm.

## Explanations

### L1 human

Near $0$, the function $z^3$ vanishes like the third power of $z$, and $1/z^2$ blows up like the second power of $1/z$.
The order records both at once with a sign: $3$ for $z^3$ and $-2$ for $1/z^2$.
At ordinary points, where the function is neither $0$ nor infinite, the order is $0$.
When you multiply two functions, their orders add, just as exponents add when you multiply powers.