PlanetMath: Euler relation

(more info)

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Euler relation

(Definition)

Euler's relation (also known as Euler's formula) is considered the first bridge 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

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

Start by noting that

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

Using the Taylor series expansions of , $\sin x$ 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

, we can rearrange the terms of the series as

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