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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 Estimation lemma ： ウィキペディア英語版 Estimation lemma In mathematics the estimation lemma, also known as the M-L inequality, gives an upper bound for a contour integral. If is a complex-valued, continuous function on the contour and if its absolute value is bounded by a constant for all on , then :$\backslash left|\backslash int\_\backslash Gamma\; f(z)\; \backslash ,\; dz\backslash right|\; \backslash le\; M\backslash ,\; l(\backslash Gamma),$ where is the arc length of . In particular, we may take the maximum :$M:=\; \backslash max\_|f(z)|$ as upper bound. Intuitively, the lemma is very simple to understand. If a contour is thought of as many smaller contour segments connected together, then there will be a maximum for each segment. Out of all the maximum s for the segments, there will be an overall largest one. Hence, if the overall largest is summed over the entire path then the integral of over the path must be less than or equal to it. Formally, the inequality can be shown to hold using the definition of contour integral, the absolute value inequality for integrals and the formula for the length of a curve as follows: :$\backslash biggl|\backslash int\_\backslash Gamma\; f(z)\backslash ,\; dz\; \backslash biggr|$ = \biggl|\int_\alpha^\beta f(\gamma(t))\gamma'(t)\, dt \biggr| \leq \int_\alpha^\beta \left|f(\gamma(t))\right|\left|\gamma'(t)\right|\, dt \leq M \int_\alpha^\beta \left|\gamma'(t)\right|\, dt = M\, l(\Gamma) The estimation lemma is most commonly used as part of the methods of contour integration with the intent to show that the integral over part of a contour goes to zero as goes to infinity. An example of such a case is shown below. == Example == Problem. Find an upper bound for :$\backslash biggl|\backslash int\_\backslash Gamma\; \backslash frac\; \backslash ,\; dz\backslash biggr|,$ where is the upper half-circle with radius traversed once in the counterclockwise direction. Solution. First observe that the length of the path of integration is half the circumference of a circle with radius , hence :$l(\backslash Gamma)=\backslash frac(2\backslash pi\; a)=\backslash pi\; a.$ Next we seek an upper bound for the integrand when . By the triangle inequality we see that :$|z|^2=|z^2|\; =\; |z^2+1-1|\; \backslash le\; |z^2+1|+1,$ therefore :$|z^2+1|\backslash ge\; |z|^2\; -\; 1\; =\; a^2\; -\; 1>0$ because on . Hence :$\backslash left|\backslash frac\backslash right|\; \backslash le\; \backslash frac.$ Therefore we apply the estimation lemma with . The resulting bound is :$\backslash biggl|\backslash int\_\backslash Gamma\; \backslash frac\backslash ,dz\backslash biggr|\; \backslash le\; \backslash frac.$ 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「Estimation lemma」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.