# 歐拉恆等式

${\displaystyle {{{e}^{{i}\,{\pi }}}+{1}}=0}$

## 證明

${\displaystyle e^{ix}=\cos x+i\sin x\,\!}$歐拉公式
${\displaystyle e^{i\pi }=\cos \pi +i\sin \pi \,}$（代入${\displaystyle x=\pi \,}$
${\displaystyle {{e}^{{i}\,{\pi }}}=-1}$（因${\displaystyle \cos \pi =-1}$${\displaystyle \sin \pi =0}$
${\displaystyle {{{e}^{{i}\,{\pi }}}+{1}}=0}$

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