查看“︁歐拉恆等式”︁的源代码

{{NoteTA|G1=Math}}[[File:EulerIdentity2.svg|right|frame|从{{計算結果|e^0}}开始，以相对速度''i''，走π长时间，加1，则到达原点]]

'''歐拉恆等式'''是指下列的[[恆等式|關係式]]：

: {{計算結果|e^(i⋅pi)+1}}

其中<math>e\,</math>是[[e (數學常數)|自然對數的底]]，<math>i \,</math>是[[虛數]]單位，<math>\pi \,</math>是[[圓周率]]。

這條恆等式第一次出現於1748年，瑞士數學、物理學家[[萊昂哈德·歐拉]]（{{lang|de|Leonhard Euler}}）在[[洛桑]]出版的書《[[无穷小分析引论]]》（{{lang|la|''Introductio in analysin infinitorum''}}）。這是[[複分析]]的[[歐拉公式]]之特殊情況。

== 證明 ==
: <math>e^{ix} = \cos x + i \sin x \,\!</math>（[[歐拉公式]]）

: <math>e^{i \pi}=\cos \pi+ i \sin \pi\,</math>（代入<math>x=\pi \,</math>）

: {{計算結果|e^(i⋅pi)}}（因{{計算結果|cos(pi)}}和{{計算結果|sin(pi)}}）

: {{計算結果|e^(i⋅pi)+1}}

== 與歐拉恆等式有關的文學作品 ==
《[[博士熱愛的算式]]》（{{lang|ja|博士の愛した数式}}），[[小川洋子]]著，臺灣版本由王蘊潔翻譯，二版，麥田出版社，2008年，ISBN 978-986-173-408-8。

== 参见 ==
* [[欧拉公式]] 

== 參考文獻 ==
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# {{tsl|en|Robert P. Crease||Crease, Robert P.}} (10&nbsp;May 2004), "[http://physicsworld.com/cws/article/print/2004/may/10/the-greatest-equations-ever The greatest equations ever] {{Wayback|url=http://physicsworld.com/cws/article/print/2004/may/10/the-greatest-equations-ever |date=20131224043710 }}", ''{{tsl|en|Physics World||Physics World}}'' [registration required]
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# Euler, Leonhard (1922), ''[http://gallica.bnf.fr/ark:/12148/bpt6k69587.image.r=%22has+celeberrimas+formulas%22.f169.langEN Leonhardi Euleri opera omnia. 1, Opera mathematica. Volumen VIII, Leonhardi Euleri introductio in analysin infinitorum. Tomus primus] {{Wayback|url=http://gallica.bnf.fr/ark:/12148/bpt6k69587.image.r=%22has+celeberrimas+formulas%22.f169.langEN |date=20210402030957 }}'', Leipzig: B. G. Teubneri
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#{{Citation | last1= Zeki | first1= S. | last2=  Romaya | first2=  J. P. | last3= Benincasa | first3= D. M. T. | last4=  Atiyah | first4= M. F. | authorlink1= Semir Zeki | authorlink4=  Michael Atiyah | title= The experience of mathematical beauty and its neural correlates | journal= Frontiers in Human Neuroscience | volume= 8 | year= 2014 | doi= 10.3389/fnhum.2014.00068}}
{{refend}}

{{莱昂哈德·欧拉}}

[[Category:指数]]
[[Category:数学恒等式]]
[[Category:E (数学常数)]]
[[Category:复分析定理]]
[[Category:莱昂哈德·欧拉]]