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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 Proof of Wallis product ： ウィキペディア英語版 Wallis product In mathematics, Wallis' product for π, written down in 1655 by John Wallis, states that :$$ \prod_^ \left(\frac \cdot \frac\right) = \frac \cdot \frac \cdot \frac \cdot \frac \cdot \frac \cdot \frac \cdot \frac \cdot \frac \cdots = \frac ==Derivation== Wallis derived this infinite product as it is done in calculus books today, by comparing $\backslash scriptstyle\; \backslash int\_0^\backslash pi\; \backslash sin^nx\; dx$ for even and odd values of ''n'', and noting that for large ''n'', increasing ''n'' by 1 results in a change that becomes ever smaller as ''n'' increases. Since infinitesimal calculus as we know it did not yet exist then, and the mathematical analysis of the time was inadequate to discuss the convergence issues, this was a hard piece of research, and tentative as well. Wallis' product is, in retrospect, an easy corollary of the later Euler formula for the sine function. In 2015 researchers from the University of Rochester, in a surprise discovery, found the same formula in quantum mechanical calculations of the energy levels of a hydrogen atom.〔(T. Friedmann ; C.R. Hagen. "Quantum mechanical derivation of the Wallis formula for pi." Journal of Mathmatical Physics Vol. 56 (2015) )〕〔("Quantum mechanical derivation of the Wallis formula for pi." Journal of Mathmatical Physics Vol. 56 (arxiv) )〕〔("Discovery of classic pi formula a ‘cunning piece of magic'." University of Rochester (November 10, 2015 )〕〔("New derivation of pi links quantum physics and pure math." American Institute of Physics (November 10, 2015) )〕〔("New derivation of pi links quantum physics and pure math." Phys.org (November 10, 2015) )〕〔("Revealing the hidden connection between pi and Bohr's hydrogen model." Physics World (November 17, 2015) )〕 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「Wallis product」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.