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歐拉恆等式

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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關係式

其中自然對數的底虛數單位,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Leonhard Euler)在洛桑出版的書《无穷小分析引论》(Introductio in analysin infinitorum)。這是複分析歐拉公式之特殊情況。

證明

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歐拉公式
(代入
(因

與歐拉恆等式有關的文學作品

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博士熱愛的算式》(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

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參考文獻

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  1. Conway, John H., and Guy, Richard K.英语Richard K. Guy (1996), The Book of Numbers页面存档备份,存于互联网档案馆, Springer ISBN 978-0-387-97993-9
  2. Crease, Robert P.英语Robert P. Crease (10 May 2004), "The greatest equations ever页面存档备份,存于互联网档案馆)", Physics World英语Physics World [registration required]
  3. Dunham, William英语William Dunham (mathematician) (1999), Euler: The Master of Us All, Mathematical Association of America ISBN 978-0-88385-328-3
  4. Euler, Leonhard (1922), Leonhardi Euleri opera omnia. 1, Opera mathematica. Volumen VIII, Leonhardi Euleri introductio in analysin infinitorum. Tomus primus页面存档备份,存于互联网档案馆, Leipzig: B. G. Teubneri
  5. Kasner, E.英语Edward Kasner, and Newman, J.英语James R. Newman (1940), Mathematics and the Imagination, Simon & Schuster
  6. Maor, Eli (1998), e: The Story of a number, Princeton University Press ISBN 0-691-05854-7
  7. Nahin, Paul J. (2006), Dr. Euler's Fabulous Formula: Cures Many Mathematical Ills, Princeton University Press ISBN 978-0-691-11822-2
  8. Paulos, John Allen (1992), Beyond Numeracy: An Uncommon Dictionary of Mathematics, Penguin Books ISBN 0-14-014574-5
  9. Reid, Constance (various editions), From Zero to Infinity, Mathematical Association of America
  10. Sandifer, C. Edward (2007), Euler's Greatest Hits页面存档备份,存于互联网档案馆, Mathematical Association of America ISBN 978-0-88385-563-8
  11. Stipp, David, A Most Elegant Equation: Euler's formula and the beauty of mathematics, Basic Books, 2017 
  12. Wells, David (1990), "Are these the most beautiful?", The Mathematical Intelligencer英语The Mathematical Intelligencer, 12: 37–41, doi:10.1007/BF03024015
  13. Wilson, Robin, Euler's Pioneering Equation: The most beautiful theorem in mathematics, Oxford University Press, 2018 
  14. Zeki, S.; Romaya, J. P.; Benincasa, D. M. T.; Atiyah, M. F., The experience of mathematical beauty and its neural correlates, Frontiers in Human Neuroscience, 2014, 8, doi:10.3389/fnhum.2014.00068