nLab
Euler's formula

Contents

Contents

Euler’s formula is a fundamental relation between the exponential function and the trigonometric functions.

Statement

The exponential function exp:\exp: \mathbb{C} \to \mathbb{C} relates to sine and cosine cos,sin:\cos, \sin \colon \mathbb{R} \to \mathbb{R} via

exp(ix)=cos(x)+isin(x) \exp(i x) = \cos(x) + i \sin(x)

This can be interpreted as either a theorem (especially when xx is restricted to real numbers and cos,sin\cos, \sin have been defined independently of exp\exp), or as an implicit definition of cos,sin\cos, \sin (where more explicitly we have cos(x)12(exp(ix)+exp(ix))\cos(x) \coloneqq \frac1{2}(\exp(i x) + \exp(-i x)) and sin(x)12i(exp(ix)exp(ix))\sin(x) \coloneqq \frac1{2 i}(\exp(i x) - \exp(-i x))).

References

Named after Leonhard Euler.

Last revised on May 10, 2019 at 11:09:15. See the history of this page for a list of all contributions to it.