---
tag: 02P2PYH3X3
kind: statement
role: proposition
name: The complex exponential is 2πi-periodic
aliases: [periodicity-of-the-exponential]
msc: [33B10]
uses: [054VXHMCA3, 0WM7DJCM6G, 04R3CY41VJ]
sources:
  - url: https://en.wikipedia.org/wiki/Exponential_function
    locator: Complex exponential
---

## Statement

For each $z \in \mathbb{C}$,
$$e^{z + 2 \pi i} = e^{z},$$
where $e^z$ is the [[complex exponential|054VXHMCA3]] of $z$.

## Remarks

Proof: by [[Euler's formula|04R3CY41VJ]], $e^{2 \pi i} = \cos 2\pi + i \sin 2\pi = 1$. By the [[product of exponentials|0WM7DJCM6G]], $e^{z + 2\pi i} = e^z e^{2\pi i} = e^z$.
The periods of $\exp$ are exactly the numbers $2 \pi i k$ with $k \in \mathbb{Z}$, by [[exp z = 1 iff z is in 2πiZ|013AHC4DRT]].
So $\exp$ is not injective on $\mathbb{C}$. It is injective on each horizontal strip $\{z : a < \operatorname{Im} z \le a + 2\pi\}$ with $a \in \mathbb{R}$.

## Explanations

### L1 human

Adding $2 \pi i$ to $z$ adds a full turn to the direction of $e^z$ and does not change its size.
So $e^z$ takes the same value again: the exponential repeats itself along each vertical line, with step $2\pi$.