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In mathematics, Cauchy's integral formula, named after Augustin-Louis Cauchy, is a central statement in complex analysis. It expresses the fact that a holomorphic function defined on a disk is completely determined by its values on the boundary of the disk, and it provides integral formulas for all derivatives of a holomorphic function. Cauchy's formula shows that, in complex analysis, "differentiation is equivalent to integration": complex differentiation, like integration, behaves well under uniform limits – a result denied in real analysis. ==Theorem== We begin with a theorem that is less general than what can actually be said. Suppose ''U'' is an open subset of the complex plane C, ''f'' : ''U'' → C is a holomorphic function and the closed disk ''D'' = is completely contained in ''U''. Let be the circle forming the boundary of ''D''. Then for every ''a'' in the interior of ''D'': : where the contour integral is taken counter-clockwise. The proof of this statement uses the Cauchy integral theorem and like that theorem it only requires ''f'' to be complex differentiable. Since the reciprocal of the denominator of the integrand in Cauchy's integral formula can be expanded as a power series in the variable (''a'' − ''z''0) (namely, when ''z''0=0, ), it follows that holomorphic functions are analytic. In particular ''f'' is actually infinitely differentiable, with : This formula is sometimes referred to as Cauchy's differentiation formula. The theorem stated above can be generalized. The circle ''γ'' can be replaced by any closed rectifiable curve in ''U'' which has winding number one about ''a''. Moreover, as for the Cauchy integral theorem, it is sufficient to require that ''f'' be holomorphic in the open region enclosed by the path and continuous on its closure. Note that not every continuous function on the boundary can be used to produce a function inside the boundary that fits the given boundary function. For instance, if we put the function, defined for |''z''|=1, into the Cauchy integral formula, we get zero for all points inside the circle. In fact, giving just the real part on the boundary of a holomorphic function is enough to determine the function up to an imaginary constant – there is only one imaginary part on the boundary that corresponds to the given real part, up to addition of a constant. We can use a combination of a Möbius transformation and the Stieltjes inversion formula to construct the holomorphic function from the real part on the boundary. For example, the function has real part . On the unit circle this can be written . Using the Möbius transformation and the Stieltjes formula we construct the function inside the circle. The ''i/z'' term makes no contribution, and we find the function . This has the correct real part on the boundary, and also gives us the corresponding imaginary part, but off by a constant, namely ''i''. 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's integral formula」の詳細全文を読む スポンサード リンク
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