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欧拉恒等式

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e0 = 1开始,以相对速度i,走π长时间,加1,则到达原点。

欧拉恒等式是指下列的关系式

其中自然对数的底虚数单位,圆周率

这条恒等式第一次出现于1748年瑞士数学、物理学家莱昂哈德·欧拉Leonhard Euler)在洛桑出版的书。这是复分析欧拉公式的特殊情况。

美国物理学家理查德·费曼Richard Phillips Feynman)称这恒等式为“数学最奇妙的公式”,因为它把5个最基本的数学常数简洁地连系起来。

证明[编辑]

欧拉公式
(代入
(因

与欧拉恒等式有关的文学作品[编辑]

博士热爱的算式》,小川洋子著,台湾版本由王蕴洁翻译,二版,麦田出版社,2008年,ISBN 978-986-173-408-8

参见[编辑]

参考文献[编辑]

  1. Conway, John H., and Guy, Richard K. (1996), The Book of Numbers, Springer ISBN 978-0-387-97993-9
  2. Crease, Robert P. (10 May 2004), "The greatest equations ever", Physics World [registration required]
  3. Dunham, William (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, Leipzig: B. G. Teubneri
  5. Kasner, E., and Newman, J. (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, 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, 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