---
tag: 04R3CY41VJ
kind: statement
role: theorem
name: Euler's formula
msc: [33B10]
uses: [09FTAET5Z4]
sources:
  - url: https://en.wikipedia.org/wiki/Euler%27s_formula
    locator: lead section
same_as:
  wikidata: Q184871
  wikipedia: Euler's formula
---

## Statement

For every complex number $z$,
$$e^{i z} = \cos z + i \sin z.$$
Here $e^{w} = \sum_{n=0}^{\infty} w^n / n!$ is the complex exponential function, and $\cos$ and $\sin$ are given by their power series.

## Remarks

The main case is a real number $z = x$. Then $\cos x$ and $\sin x$ are the real and imaginary parts of $e^{i x}$, and $|e^{i x}| = 1$.
For $x = \pi$, the formula gives [[Euler's identity|09285XX7AE]].
A proof compares the power series: the even terms of $e^{i z}$ give $\cos z$ and the odd terms give $i \sin z$.

## Explanations

### L1 human

As the real number $x$ grows, the point $e^{i x}$ moves around the unit circle at constant speed.
Its coordinates are $\cos x$ and $\sin x$, as in trigonometry.
The formula links the exponential function with the trigonometric functions.