翻訳と辞書 Words near each other ・ Eulima labiosa ・ Eulima lacca ・ Eulima lactea ・ Eulima langleyi ・ Eulima lapazana ・ Eulima latipes ・ Eulima leachi ・ Eulima legrandi ・ Eulima leptostoma ・ Eulima leptozona ・ Eulima lodderae ・ Euler's Disk ・ Euler's equations (rigid body dynamics) ・ Euler's factorization method ・ Euler's flycatcher ・ Euler's formula ・ Euler's four-square identity ・ Euler's identity ・ Euler's laws of motion ・ Euler's pump and turbine equation ・ Euler's rotation theorem ・ Euler's sum of powers conjecture ・ Euler's theorem ・ Euler's theorem (differential geometry) ・ Euler's theorem in geometry ・ Euler's three-body problem ・ Euler's totient function ・ Eulerian matroid ・ Eulerian number ・ Eulerian path Dictionary Lists mini英和辞書 mini和英辞書 Webster 1913 Latin-English FOLDOC Wikipedia English ウィキペディア

翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Euler's formula ： ウィキペディア英語版 Euler's formula Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that, for any real number : :$e^=\backslash cos\; x+i\backslash sin\; x$ where is the base of the natural logarithm, is the imaginary unit, and and are the trigonometric functions cosine and sine respectively, with the argument ''x'' given in radians. This complex exponential function is sometimes denoted ("cosine plus i sine"). The formula is still valid if is a complex number, and so some authors refer to the more general complex version as Euler's formula. Euler's formula is ubiquitous in mathematics, physics, and engineering. The physicist Richard Feynman called the equation "our jewel" and "the most remarkable formula in mathematics." ==History== Johann Bernoulli noted that〔Johann Bernoulli, Solution d'un problème concernant le calcul intégral, avec quelques abrégés par rapport à ce calcul, ''Mémoires de l'Académie Royale des Sciences de Paris'', 197-289 (1702).〕 $\backslash dfrac\; \backslash left(\; \backslash dfrac\; +\backslash dfrac\; \backslash right)$ And since $\backslash int\; \backslash dfrac\; =\backslash dfrac\; \backslash ln\; \backslash left(\; 1+ax\backslash right)\; +C$ the above equation tells us something about complex logarithms. Bernoulli, however, did not evaluate the integral. Bernoulli's correspondence with Euler (who also knew the above equation) shows that Bernoulli did not fully understand complex logarithms. Euler also suggested that the complex logarithms can have infinitely many values. Meanwhile, Roger Cotes, in 1714, discovered that $ix=\backslash ln\; \backslash left(\; \backslash cos\; x+i\backslash sin\; x\backslash right)$ ($\backslash ln$ is the natural logarithm). Cotes missed the fact that a complex logarithm can have infinitely many values, differing by multiples of , due to the periodicity of the trigonometric functions. Around 1740 Euler turned his attention to the exponential function instead of logarithms, and obtained the formula used today that is named after him. It was published in 1748, obtained by comparing the series expansions of the exponential and trigonometric expressions.〔 None of these mathematicians saw the geometrical interpretation of the formula; the view of complex numbers as points in the complex plane was described some 50 years later by Caspar Wessel. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Euler's formula」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.