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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 Cauchy principal value ： ウィキペディア英語版 Cauchy principal value In mathematics, the Cauchy principal value, named after Augustin Louis Cauchy, is a method for assigning values to certain improper integrals which would otherwise be undefined. ==Formulation== Depending on the type of singularity in the integrand ''f'', the Cauchy principal value is defined as one of the following: ;1) The finite number ::$\backslash lim\_\; \backslash left(f(x)\backslash ,\backslash mathrmx+\backslash int\_^c\; f(x)\backslash ,\backslash mathrmx\backslash right)$ :where ''b'' is a point at which the behavior of the function ''f'' is such that ::$\backslash int\_a^b\; f(x)\backslash ,\backslash mathrmx=\backslash pm\backslash infty$ for any ''a'' < ''b'' and ::$\backslash int\_b^c\; f(x)\backslash ,\backslash mathrmx=\backslash mp\backslash infty$ for any ''c'' > ''b'' ::(see plus or minus for precise usage of notations ±, ∓). ;2) The infinite number ::$\backslash lim\_\backslash int\_^a\; f(x)\backslash ,\backslash mathrmx$ ::where $\backslash int\_^0\; f(x)\backslash ,\backslash mathrmx=\backslash pm\backslash infty$ ::and $\backslash int\_0^\backslash infty\; f(x)\backslash ,\backslash mathrmx=\backslash mp\backslash infty$. :In some cases it is necessary to deal simultaneously with singularities both at a finite number ''b'' and at infinity. This is usually done by a limit of the form ::$\backslash lim\_\; \backslash left(f(x)\backslash ,\backslash mathrmx+\backslash int\_^\}f(x)\backslash ,\backslash mathrmx\; \backslash right).$ ;3) In terms of contour integrals of a complex-valued function ''f''(''z''); ''z'' = ''x'' + ''iy'', with a pole on the contour. The pole is enclosed with a circle of radius ''ε'' and the portion of the path outside this circle is denoted ''L''(''ε''). Provided the function ''f''(''z'') is integrable over ''L(ε)'' no matter how small ε becomes, then the Cauchy principal value is the limit: ::$\backslash mathrm\; \backslash int\_\; f(z)\; \backslash \; \backslash mathrmz\; =\; \backslash int\_L^$ * f(z)\ \mathrmz = \lim_ \int_ f(z)\ \mathrmz, :where two of the common notations for the Cauchy principal value appear on the left of this equation. In the case of Lebesgue-integrable functions, that is, functions which are integrable in absolute value, these definitions coincide with the standard definition of the integral. Principal value integrals play a central role in the discussion of Hilbert transforms. 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「Cauchy principal value」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.