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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 Liouville's theorem (complex analysis) ： ウィキペディア英語版 Liouville's theorem (complex analysis) In complex analysis, Liouville's theorem, named after Joseph Liouville, states that every bounded entire function must be constant. That is, every holomorphic function ''f'' for which there exists a positive number ''M'' such that $|f(z)|\; \backslash leq\; M$ for all $z$ in $\backslash mathbb$ is constant. Equivalently, holomorphic functions on $\backslash mathbb$ have dense images. The theorem is considerably improved by Picard's little theorem, which says that every entire function whose image omits at least two complex numbers must be constant. == Proof == The theorem follows from the fact that holomorphic functions are analytic. If ''f'' is an entire function, it can be represented by its Taylor series about 0: : $f(z)\; =\; \backslash sum\_^\backslash infty\; a\_k\; z^k$ where (by Cauchy's integral formula) : $$ a_k = \frac = \oint_ \frac \oint_ \frac \, |d\zeta| \le \frac \oint_ \frac \, |d\zeta| = \frac |d\zeta| = \frac, where in the second inequality we have used the fact that |''z''|=''r'' on the circle ''C''''r''. But the choice of ''r'' in the above is an arbitrary positive number. Therefore, letting ''r'' tend to infinity (we let ''r'' tend to infinity since f is analytic on the entire plane) gives ''a''''k'' = 0 for all ''k'' ≥ 1. Thus ''f''(''z'') = ''a''0 and this proves the theorem. 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「Liouville's theorem (complex analysis)」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.