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. 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.