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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 Residue theorem ： ウィキペディア英語版 Residue theorem In complex analysis, a field in mathematics, the residue theorem, sometimes called Cauchy's residue theorem (one of many things named after Augustin-Louis Cauchy), is a powerful tool to evaluate line integrals of analytic functions over closed curves; it can often be used to compute real integrals as well. It generalizes the Cauchy integral theorem and Cauchy's integral formula. From a geometrical perspective, it is a special case of the generalized Stokes' theorem. The statement is as follows: Suppose ''U'' is a simply connected open subset of the complex plane, and ''a''1,...,''a''''n'' are finitely many points of ''U'' and ''f'' is a function which is defined and holomorphic on ''U'' \ . If γ is a closed rectifiable curve in ''U'' which does not meet any of the ''a''''k'', :$\backslash oint\_\backslash gamma\; f(z)\backslash ,\; dz\; =$ 2\pi i \sum_^n \operatorname(\gamma, a_k) \operatorname( f, a_k ). If γ is a positively oriented simple closed curve, I(γ, ''a''''k'') = 1 if ''a''''k'' is in the interior of γ, and 0 if not, so :$\backslash oint\_\backslash gamma\; f(z)\backslash ,\; dz\; =$ 2\pi i \sum \operatorname( f, a_k ) with the sum over those ''k'' for which ''a''''k'' is inside γ. Here, Res(''f'', ''a''''k'') denotes the residue of ''f'' at ''a''''k'', and I(γ, ''a''''k'') is the winding number of the curve γ about the point ''a''''k''. This winding number is an integer which intuitively measures how many times the curve γ winds around the point ''a''''k''; it is positive if γ moves in a counter clockwise ("mathematically positive") manner around ''a''''k'' and 0 if γ doesn't move around ''a''''k'' at all. The relationship of the residue theorem to Stokes' theorem is given by the Jordan curve theorem. The general plane curve γ must first be reduced to a set of simple closed curves whose total is equivalent to γ for integration purposes; this reduces the problem to finding the integral of ''f'' ''dz'' along a Jordan curve γ''i'' with interior ''V''. The requirement that ''f'' be holomorphic on ''U''0 = ''U'' \ is equivalent to the statement that the exterior derivative ''d''(''f'' ''dz'') = 0 on ''U''0. Thus if two planar regions ''V'' and ''W'' of ''U'' enclose the same subset of , the regions ''V''\''W'' and ''W''\''V'' lie entirely in ''U''0, and hence $\backslash scriptstyle\backslash int\_\; d(f\; \backslash ,\; dz)\; -\; \backslash int\_\; d(f\; \backslash ,\; dz)$ is well-defined and equal to zero. Consequently, the contour integral of ''f'' ''dz'' along γ''j'' = ∂V is equal to the sum of a set of integrals along paths λ''j'', each enclosing an arbitrarily small region around a single ''a''''j''—the residues of ''f'' (up to the conventional factor 2π''i'') at . Summing over , we recover the final expression of the contour integral in terms of the winding numbers . In order to evaluate real integrals, the residue theorem is used in the following manner: the integrand is extended to the complex plane and its residues are computed (which is usually easy), and a part of the real axis is extended to a closed curve by attaching a half-circle in the upper or lower half-plane, forming a semicircle. The integral over this curve can then be computed using the residue theorem. Often, the half-circle part of the integral will tend towards zero as the radius of the half-circle grows, leaving only the real-axis part of the integral, the one we were originally interested in. ==Example== The integral :$\backslash int\_^\backslash infty\; \backslash ,dx$ arises in probability theory when calculating the characteristic function of the Cauchy distribution. It resists the techniques of elementary calculus but can be evaluated by expressing it as a limit of contour integrals. Suppose ''t'' > 0 and define the contour ''C'' that goes along the real line from −''a'' to ''a'' and then counterclockwise along a semicircle centered at 0 from ''a'' to −''a''. Take ''a'' to be greater than 1, so that the imaginary unit ''i'' is enclosed within the curve. The contour integral is :$\backslash int\_C\; \backslash ,dz\; =\backslash int\_C\; \backslash ,dz.$ Since ''e''''itz'' is an entire function (having no singularities at any point in the complex plane), this function has singularities only where the denominator ''z''2 + 1 is zero. Since ''z''2 + 1 = (''z'' + ''i'')(''z'' − ''i''), that happens only where ''z'' = ''i'' or ''z'' = −''i''. Only one of those points is in the region bounded by this contour. Because ''f''(''z'') is : $\backslash begin$ \frac & =\frac\left(\frac-\frac\right) \\ & =\frac -\frac , \end the residue of ''f''(''z'') at ''z'' = ''i'' is :$\backslash operatorname\backslash limits\_f(z)=.$ According to the residue theorem, then, we have :$\backslash int\_C\; f(z)\backslash ,dz=2\backslash pi\; i\backslash cdot\backslash operatorname\backslash limits\_f(z)=2\backslash pi\; i=\backslash pi\; e^.$ The contour ''C'' may be split into a "straight" part and a curved arc, so that :$\backslash int\_\}\; f(z)\backslash ,dz=\backslash pi\; e^\backslash ,$ and thus :$\backslash int\_^a\; f(z)\backslash ,dz\; =\backslash pi\; e^-\backslash int\_\}\backslash ,dz\backslash right|\; \backslash leq\; \backslash int\_\; \backslash over\; z^2+1\}\backslash right|\; dz\; \backslash le\; \backslash int\_dz\backslash leq\; \backslash int\_dz\; =\; \backslash frac.$ and :$\backslash lim\_\; \backslash frac\; =\; 0.$ Note that, since ''t'' > 0 and for complex numbers in the upper halfplane the argument lies between 0 and π, one can estimate :$\backslash left|e^\backslash right|=\backslash left|e^\backslash right|=\backslash left|e^\backslash right|=e^\; \backslash le\; 1.$ Therefore :$\backslash int\_^\backslash infty\; \backslash ,dz=\backslash pi\; e^.$ If ''t'' < 0 then a similar argument with an arc ''C' '' that winds around −''i'' rather than ''i'' shows that :$\backslash int\_^\backslash infty\backslash ,dz=\backslash pi\; e^t,$ and finally we have :$\backslash int\_^\backslash infty\backslash ,dz=\backslash pi\; e^.$ (If ''t'' = 0 then the integral yields immediately to elementary calculus methods and its value is π.) 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「Residue theorem」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. 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