---
tag: 1TE3QF8EYC
kind: concept
name: Zeros and poles
msc: [30D30, 30C15]
sources:
  - url: https://en.wikipedia.org/wiki/Zeros_and_poles
same_as:
  wikidata: Q899731
  wikipedia: Zeros and poles
---

## Summary

The zeros of a [[meromorphic function|0ZZ7ZYDR8W]] are the points where it takes the value $0$.
Its poles are the points where it tends to $\infty$.
See the [[definition|9DNDF7AX59]].
They are the two kinds of special points of a meromorphic function, and they are exchanged by $f \mapsto 1/f$.
Counting them, each with its [[order|23ASRP0MWR]], is the subject of the [[argument principle|0PN2BXRS8G]] and of [[Rouché's theorem|0D430QR02W]].

## Explanations

### L1 human

Picture the graph of $\abs{f}$ over the complex plane as a landscape.
The zeros are the points where the landscape touches the ground.
The poles are the points where it shoots up to infinity, like tent poles holding up a tent.
For a rational function such as $(z - 1)/(z^2 + 1)$, the zeros are the roots of the numerator and the poles are the roots of the denominator.