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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 Leibniz formula for π ： ウィキペディア英語版 Leibniz formula for π :''See List of things named after Gottfried Leibniz for other formulas known under the same name.'' In mathematics, the Leibniz formula for , named after Gottfried Leibniz, states that :$1\; \backslash ,-\backslash ,\; \backslash frac\; \backslash ,+\backslash ,\; \backslash frac\; \backslash ,-\backslash ,\; \backslash frac\; \backslash ,+\backslash ,\; \backslash frac\; \backslash ,-\backslash ,\; \backslash cdots\; \backslash ;=\backslash ;\; \backslash frac.\backslash !$ Using summation notation: :$\backslash sum\_^\backslash infty\; \backslash ,\; \backslash frac\; \backslash ;=\backslash ;\; \backslash frac.\backslash !$ ==Names== The infinite series above has also been called the Leibniz series or Gregory–Leibniz series (after the work of James Gregory), or more recently Madhava-Leibniz series, after the discovery that it is a special case of a more general series expansion for the inverse tangent function, first discovered by an Indian mathematician Madhava of Sangamagrama in 14th century. The series for the inverse tangent function, which is also known as Gregory's series, can be given by: : $\backslash arctan\; x\; =\; x\; -\; \backslash frac\; +\; \backslash frac\; -\; \backslash frac\; +\; \backslash cdots$ The Leibniz formula for can be obtained by plugging ''x'' = 1 into the above inverse-tangent series. It also is the Dirichlet L-series of the non-principal Dirichlet character of modulus 4 evaluated at , and therefore the value of the Dirichlet beta function. 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「Leibniz formula for π」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.