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 $i \sin z$ .

Reading path

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Complex numbers object
Euler's formula statement

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Tag	`04R3CY41VJ`
URI	`m2://04R3CY41VJ`
Kind	statement, theorem
Writing	W1 stated
MSC	33B10
Status	This copy of the site has no build data. Run `m2 status 04R3CY41VJ`.
Uses	Complex numbers
Same as	Wikidata Q184871 Wikipedia Euler's formula

Euler's formula

Statement

Remarks

Reading path

Where this leads

Further reading

What this page needs for W2

What links here

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