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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 Omega constant ： ウィキペディア英語版 Omega constant The omega constant is a mathematical constant defined by :$\backslash Omega\backslash ,e^=1.\backslash ,$ It is the value of ''W''(1) where ''W'' is Lambert's W function. The name is derived from the alternate name for Lambert's ''W'' function, the ''omega function''. The value of Ω is approximately 0.5671432904097838729999686622... . It has properties that :$e^=\backslash Omega,\backslash ,$ or equivalently, :$\backslash ln\; \backslash Omega\; =\; -\; \backslash Omega.\backslash ,$ One can calculate Ω iteratively, by starting with an initial guess Ω0, and considering the sequence :$\backslash Omega\_=e^.\backslash ,$ This sequence will converge towards Ω as ''n''→∞. This convergence is because Ω is an attractive fixed point of the function ''e''−''x''. It is much more efficient to use the iteration :$\backslash Omega\_\; =\; \backslash frac,$ has the same fixed point but features a zero derivative at this fixed point, therefore the convergence is quadratic (the number of correct digits is roughly doubled with each iteration). A beautiful identity due to Victor Adamchik is given by the relationship :$\backslash Omega=\backslash frac\backslash frac\}-1\; .$ ==Irrationality and transcendence== Ω can be proven irrational from the fact that e is transcendental; if Ω were rational, then there would exist integers ''p'' and ''q'' such that : $\backslash frac\; =\; \backslash Omega$ so that : $1\; =\; \backslash frac\; \backslash right)\}\}$ : $e\; =\; \backslash left(\; \backslash frac\; \backslash right)^\; \backslash right)\}\; =\; \backslash sqrt()\}$ and ''e'' would therefore be algebraic of degree ''p''. However ''e'' is transcendental, so Ω must be irrational. Ω is in fact transcendental as the direct consequence of Lindemann–Weierstrass theorem. If Ω were algebraic, e-Ω would be transcendental; but Ω=exp(-Ω), so these cannot both be true. 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「Omega constant」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.