Euler relation

Euler’s relation (also known as Euler’s formula) is considered the first between the fields of algebra and geometry, as it relates the exponential function to the trigonometric sine and cosine functions.

Euler’s relation states that

e^{ix}=\cos{x}+i\sin{x}

Start by noting that

i^{k}=\begin{cases}1&\mbox{if\; }k\equiv 0\!\!\pmod{4}\\ i&\mbox{if\; }k\equiv 1\!\!\pmod{4}\\ -1&\mbox{if\; }k\equiv 2\!\!\pmod{4}\\ -i&\mbox{if\; }k\equiv 3\!\!\pmod{4}\end{cases}

Using the Taylor series expansions of $e^{x}$ , $\sin x$ and $\cos x$ (see the entries on the complex exponential function and the complex sine and cosine), it follows that

	$\displaystyle e^{ix}$	$\displaystyle=\sum_{n=0}^{\infty}\frac{i^{n}x^{n}}{n!}$
		$\displaystyle=\sum_{n=0}^{\infty}\left(\frac{x^{4n}}{(4n)!}+\frac{ix^{4n+1}}{(% 4n+1)!}-\frac{x^{4n+2}}{(4n+2)!}-\frac{ix^{4n+3}}{(4n+3)!}\right)$

Because the series expansion above is absolutely convergent for all $x$ , we can rearrange the terms of the series as

	$\displaystyle e^{ix}$	$\displaystyle=\sum_{n=0}^{\infty}(-1)^{n}\frac{x^{2n}}{(2n)!}+i\sum_{n=0}^{% \infty}(-1)^{n}\frac{x^{2n+1}}{(2n+1)!}$
		$\displaystyle=\cos{x}+i\sin{x}$

As a special case, we get the beautiful and well-known identity, often called Euler’s identity:

e^{i\pi}=-1

Title	Euler relation
Canonical name	EulerRelation
Date of creation	2013-03-22 11:57:05
Last modified on	2013-03-22 11:57:05
Owner	rm50 (10146)
Last modified by	rm50 (10146)
Numerical id	17
Author	rm50 (10146)
Entry type	Definition
Classification	msc 30B10
Synonym	Euler’s formula
Related topic	TaylorSeries
Related topic	DeMoivreIdentity
Related topic	ComplexSineAndCosine
Defines	Euler identity
Defines	Euler’s identity