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欧拉公式的另一种证明途径

发表时间:2010-01-15  浏览量:1860  下载量:680
全部作者: 龚谊承,柳光明,李德宜
作者单位: 武汉科技大学理学院;武汉科技大学计算机学院
摘 要: 讨论以更简洁的方式证明通过虚数将指数函数与三角函数联系起来的欧拉公式。通过构建恰当的辅助函数,利用Lagrange微分中值定理,给出了欧拉公式的一种新的证明途径。其中,3个相关复函数的求导方法是利用类比方式给出的。通过证明过程可以发现:利用该方法证明欧拉公式,比利用Taylor公式或变上限积分以及其他现有方法更简洁,也更易理解。
关 键 词: 应用数学;欧拉公式;复函数;导数;类比; Lagrange中值定理
Title: Another way to prove the Euler formula
Author: GONG Yicheng, LIU Guangming, LI Deyi
Organization: Science College, Wuhan University of Science and Technology;Computer College, Wuhan University of Science and Technology
Abstract: The research object of this article is the proof methods of the Euler formula, which connects the exponential function and trigonemetric function via the mystrous imaginary number. By constructing a suitable fuction and then basing on the Lagrange mean-value theorem, a new proof method of the Euler formula is put up with, where the derivatives of the three involving complex functions are analogic to the real fuctions. It can be seen from the proof process that the new proof method is simpler and easier to be undestood than the known proof methods, such as the Taylor Seriers method and the intgral method etc.
Key words: applied mathematics; Euler formula; complex function; derivative; analogy; Lagrange mean-value theorem
发表期数: 2010年1月第1期
引用格式: 龚谊承,柳光明,李德宜. 欧拉公式的另一种证明途径[J]. 中国科技论文在线精品论文,2010,3(1):58-60.
 
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