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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 Machin-like formulas ： ウィキペディア英語版 Machin-like formula In mathematics, Machin-like formulae are a popular technique for computing π to a large number of digits. They are generalizations of John Machin's formula from 1706: :$\backslash frac\; =\; 4\; \backslash arctan\; \backslash frac\; -\; \backslash arctan\; \backslash frac$ which he used to compute π to 100 decimal places. Machin-like formulas have the form: Where $a\_n$ and $b\_n$ are positive integers such that , $c\_n$ is a signed non-zero integer, and $c\_0$ is a positive integer. These formulae are used in conjunction with the Taylor series expansion for arctangent: ==Derivation== In Angle addition formula we learned the following equations: :$\backslash sin(\backslash alpha\; +\; \backslash beta)\; =\; \backslash sin\; \backslash alpha\; \backslash cos\; \backslash beta\; +\; \backslash cos\; \backslash alpha\; \backslash sin\; \backslash beta$ :$\backslash cos(\backslash alpha\; +\; \backslash beta)\; =\; \backslash cos\; \backslash alpha\; \backslash cos\; \backslash beta\; -\; \backslash sin\; \backslash alpha\; \backslash sin\; \backslash beta$ Simple algebraic manipulations of these equations yield the following: if All of the Machin-like formulae can be derived by repeated application of this equation. As an example, we show the derivation of Machin's original formula: :$2\; \backslash arctan\; \backslash frac$ ::$=\; \backslash arctan\; \backslash frac\; +\; \backslash arctan\; \backslash frac$ ::$=\; \backslash arctan\; \backslash frac$ ::$=\; \backslash arctan\; \backslash frac$ ::$=\; \backslash arctan\; \backslash frac$ :$4\; \backslash arctan\; \backslash frac$ ::$=\; 2\; \backslash arctan\; \backslash frac\; +\; 2\; \backslash arctan\; \backslash frac$ ::$=\; \backslash arctan\; \backslash frac\; +\; \backslash arctan\; \backslash frac$ ::$=\; \backslash arctan\; \backslash frac$ ::$=\; \backslash arctan\; \backslash frac$ :$4\; \backslash arctan\; \backslash frac\; -\; \backslash frac$ ::$=\; 4\; \backslash arctan\; \backslash frac\; -\; \backslash arctan\; \backslash frac$ ::$=\; 4\; \backslash arctan\; \backslash frac\; +\; \backslash arctan\; \backslash frac$ ::$=\; \backslash arctan\; \backslash frac\; +\; \backslash arctan\; \backslash frac$ ::$=\; \backslash arctan\; \backslash frac$ ::$=\; \backslash arctan\; \backslash frac$ :$\backslash frac\; =\; 4\; \backslash arctan\; \backslash frac\; -\; \backslash arctan\; \backslash frac$ An insightful way to visualize equation is to picture what happens when two complex numbers are multiplied together: :$(b\_1\; +\; a\_1\; i)$ *(b_2 + a_2 i) ::$=b\_1\; b\_2\; +\; a\_2\; b\_1\; i\; +\; a\_1\; b\_2\; i\; -\; a\_1\; a\_2$ The angle associated with a complex number $(b\_n\; +\; a\_n\; i)$ is given by: :$\backslash arctan\; \backslash frac$ Thus, in equation , the angle associated with the product is: :$\backslash arctan\; \backslash frac$ Note that this is the same expression as occurs in equation . Thus equation can be interpreted as saying that the act of multiplying two complex numbers is equivalent to adding their associated angles (see multiplication of complex numbers). The expression: :$c\_n\; \backslash arctan\; \backslash frac$ is the angle associated with: :$(b\_n\; +\; a\_n\; i)^$ Equation can be re-written as: :$k$ * (1 + i)^ = \prod_^N (b_n + a_n i)^ Where $k$ is an arbitrary constant that accounts for the difference in magnitude between the vectors on the two sides of the equation. The magnitudes can be ignored, only the angles are significant. 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「Machin-like 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.