歐拉恆等式
从
e
0
=
1
{\displaystyle {{e}^{0}}=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 是指下列的關係式 :
e
i
π π -->
+
1
=
0
{\displaystyle {{{e}^{{i}\,{\pi }}}+{1}}=0}
其中
e
{\displaystyle e\,}
是自然對數的底 ,
i
{\displaystyle i\,}
是虛數 單位,
π π -->
{\displaystyle \pi \,}
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 (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 的歐拉公式 之特殊情況。
證明
e
i
x
=
cos
-->
x
+
i
sin
-->
x
{\displaystyle e^{ix}=\cos x+i\sin x\,\!}
(歐拉公式 )
e
i
π π -->
=
cos
-->
π π -->
+
i
sin
-->
π π -->
{\displaystyle e^{i\pi }=\cos \pi +i\sin \pi \,}
(代入
x
=
π π -->
{\displaystyle x=\pi \,}
)
e
i
π π -->
=
− − -->
1
{\displaystyle {{e}^{{i}\,{\pi }}}=-1}
(因
cos
-->
π π -->
=
− − -->
1
{\displaystyle \cos \pi =-1}
和
sin
-->
π π -->
=
0
{\displaystyle \sin \pi =0}
)
e
i
π π -->
+
1
=
0
{\displaystyle {{{e}^{{i}\,{\pi }}}+{1}}=0}
與歐拉恆等式有關的文學作品
《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
參考文獻
Conway, John H. , and Guy, Richard K. (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
Crease, Robert P. (10 May 2004), "The greatest equations ever (页面存档备份 ,存于互联网档案馆 )", Physics World [registration required]
Dunham, William (1999), Euler: The Master of Us All , Mathematical Association of America ISBN 978-0-88385-328-3
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
Kasner, E. , and Newman, J. (1940), Mathematics and the Imagination , Simon & Schuster
Maor, Eli (1998), 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
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), 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
Stipp, David, A Most Elegant Equation: Euler's formula and the beauty of mathematics, Basic Books , 2017
Wells, David (1990), "Are these the most beautiful?", The Mathematical Intelligencer , 12: 37–41, doi :10.1007/BF03024015
Wilson, Robin , Euler's Pioneering Equation: The most beautiful theorem in mathematics, Oxford University Press , 2018
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