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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク De Moivre's formula ： ウィキペディア英語版 De Moivre's formula In mathematics, de Moivre's formula (also known as de Moivre's theorem and de Moivre's identity), named after Abraham de Moivre, states that for any complex number (and, in particular, for any real number) ''x'' and integer ''n'' it holds that :$(\backslash cos\; x\; +\; i\; \backslash sin\; x)^n\; =\; \backslash cos\; (nx)\; +\; i\; \backslash sin\; (nx),$ where ''i'' is the imaginary unit (''i''''2'' = −1). While the formula was named after de Moivre, he never stated it in his works. The expression cos ''x'' + ''i'' sin ''x'' is sometimes abbreviated to cis ''x''. The formula is important because it connects complex numbers and trigonometry. By expanding the left hand side and then comparing the real and imaginary parts under the assumption that ''x'' is real, it is possible to derive useful expressions for cos(''nx'') and sin(''nx'') in terms of cos ''x'' and sin ''x''. As written, the formula is not valid for non-integer powers ''n''. However, there are generalizations of this formula valid for other exponents. These can be used to give explicit expressions for the ''n''th roots of unity, that is, complex numbers ''z'' such that ''zn'' = 1. ==Derivation from Euler's formula== Although historically proven earlier, de Moivre's formula can easily be derived from Euler's formula :$e^\; =\; \backslash cos\; x\; +\; i\backslash sin\; x\backslash ,$ and the exponential law for integer powers :$\backslash left(\; e^\; \backslash right)^n\; =\; e^\; .$ Then, by Euler's formula, :$e^\; =\; \backslash cos\; (nx)\; +\; i\backslash sin\; (nx).\backslash ,$ A more elementary motivation of the theorem comes from calculating : &= \cos(2x) + i \sin (2x)\end where the final equality follows from the trigonometric identities :$\backslash cos^2\; x\; -\; \backslash sin^2\; x\; =\; \backslash cos(2x)$ :$2\backslash sin\; x\; \backslash cos\; x\; =\; \backslash sin(2x)\; .$ This proves the theorem for the case ''n'' = 2. Larger values of ''n'' correspond to trigonometric identities for the triple angle, quadruple angle, etc. 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「De Moivre'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.