---
tag: 0G2YJN0H5S
kind: statement
role: proposition
name: The complex exponential is entire
msc: [33B10]
uses: [054VXHMCA3, 0419794R0T]
sources:
  - url: https://en.wikipedia.org/wiki/Exponential_function
    locator: Complex exponential
  - url: https://en.wikipedia.org/wiki/Entire_function
    locator: lead section
---

## Statement

The [[complex exponential|054VXHMCA3]] $\exp : \mathbb{C} \to \mathbb{C}$ is an [[entire function|0419794R0T]]: it is complex differentiable at each $z \in \mathbb{C}$.

## Remarks

Proof: $\exp$ is given by a power series with infinite radius of convergence, and [[power series are holomorphic in their disc of convergence|0YMP521ZR5]].
Its [[derivative|019H1Z6419]] is $\exp$ itself.

## Explanations

### L1 human

The exponential function has a complex derivative at every point of the plane.
So it is a holomorphic function without any exceptional points.