# 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