查看“歐拉恆等式”的源代码

{{NoteTA|G1=Math}}[[File:EulerIdentity2.svg|right|frame|从''e''<sup>0</sup> = 1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]]：

: <math>e^{i \pi}+1=\,</math>{{#invoke:Complex Number/Calculate|calculate|

    e^(i*pi)+1

|class=cmath|use math=yes}}

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<math>Introductio \,</math>。這是[[複分析]]的[[歐拉公式]]的特殊情況。

美國物理學家[[理查德·費曼]]（{{lang|en|Richard Phillips Feynman}}）稱這恆等式為「數學最奇妙的公式」，因為它把5個最基本的[[數學常數]]簡潔地連繫起來。

== 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>）

: <math>e^{i \pi}=\,</math>{{#invoke:Complex Number/Calculate|calculate|
   exp(1)^(i*pi)
|class=cmath|use math=yes}}（因<math>\cos \pi = -1  \, \! </math>和<math>\sin \pi = 0\,\!</math>）

: <math>e^{i \pi}+1=\,</math>{{#invoke:Complex Number/Calculate|calculate|
   exp(1)^(i*pi)+1
|class=cmath|use math=yes}}

== 與歐拉恆等式有關的文學作品 ==
《[[博士熱愛的算式]]》，[[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 ==
{{refbegin}}
# [[John Horton Conway|Conway, John H.]], and [[Richard K. Guy|Guy, Richard K.]] (1996), ''[https://books.google.com/books?id=0--3rcO7dMYC&pg=PA254 The Book of Numbers]'', Springer {{ISBN|978-0-387-97993-9}}
# [[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]", ''[[Physics World]]'' [registration required]
# [[William Dunham (mathematician)|Dunham, William]] (1999), ''Euler: The Master of Us All'', [[Mathematical Association of America]] {{ISBN|978-0-88385-328-3}}
# 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]'', Leipzig: B. G. Teubneri
# [[Edward Kasner|Kasner, E.]], and [[James R. Newman|Newman, J.]] (1940), ''[[Mathematics and the Imagination]]'', [[Simon & Schuster]]
# [[Eli Maor|Maor, Eli]] (1998), ''{{mvar|e}}: The Story of a number'', [[Princeton University Press]] {{ISBN|0-691-05854-7}}
# Nahin, Paul J. (2006), ''Dr. Euler's Fabulous Formula: Cures Many Mathematical Ills'', [[Princeton University Press]] {{ISBN|978-0-691-11822-2}}
# [[John Allen Paulos|Paulos, John Allen]] (1992), ''Beyond Numeracy: An Uncommon Dictionary of Mathematics'', [[Penguin Books]]  {{ISBN|0-14-014574-5}}
# Reid, Constance (various editions), ''From Zero to Infinity'', [[Mathematical Association of America]]
# Sandifer, C. Edward (2007), ''[https://books.google.com/books?id=sohHs7ExOsYC&pg=PA4 Euler's Greatest Hits]'', [[Mathematical Association of America]] {{ISBN|978-0-88385-563-8}}
#{{citation| title= A Most Elegant Equation: Euler's formula and the beauty of mathematics | first= David | last= Stipp | year=2017 | publisher= [[Basic Books]]}}
#Wells, David (1990), "Are these the most beautiful?", ''[[The Mathematical Intelligencer]]'', 12: 37–41, {{doi|10.1007/BF03024015}}
#{{citation| first= Robin | last= Wilson | author-link= Robin Wilson (mathematician) | title= Euler's Pioneering Equation: The most beautiful theorem in mathematics | publisher= [[Oxford University Press]] | year= 2018}}
#{{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}}

[[Category:数学恒等式]]
[[Category:指数]]

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