---
tag: 02QGGD5G87
kind: concept
name: Complex logarithm
aliases: [logarithm-of-a-complex-number]
msc: [33B10]
sources:
  - url: https://en.wikipedia.org/wiki/Complex_logarithm
    locator: lead section
same_as:
  wikidata: Q2520206
  wikipedia: Complex logarithm
---

## Summary

A complex logarithm of a nonzero complex number $z$ is a complex number $w$ with $e^w = z$, see the [[definition|0X2FY491PF]].
Each $z \neq 0$ has infinitely many logarithms, which differ by integer multiples of $2\pi i$.
A complex logarithm function is a continuous choice of a logarithm on an open subset of $\mathbb{C} \setminus \{0\}$.