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In complex analysis, a branch of mathematics, the Casorati–Weierstrass theorem describes the behaviour of holomorphic functions near their essential singularities. It is named for Karl Theodor Wilhelm Weierstrass and Felice Casorati. In Russian literature it is called Sokhotski's theorem. ==Formal statement of the theorem== Start with some open subset ''U'' in the complex plane containing the number , and a function ''f'' that is holomorphic on , but has an essential singularity at . The ''Casorati–Weierstrass theorem'' then states that :if ''V'' is any neighbourhood of contained in ''U'', then is dense in C. This can also be stated as follows: :for any ε > 0, δ >0, and complex number ''w'', there exists a complex number ''z'' in ''U'' with |''z'' − | < δ and |''f''(''z'') − ''w''| < ε . Or in still more descriptive terms: :''f'' comes arbitrarily close to ''any'' complex value in every neighbourhood of . This form of the theorem also applies if ''f'' is only meromorphic. The theorem is considerably strengthened by Picard's great theorem, which states, in the notation above, that ''f'' assumes ''every'' complex value, with one possible exception, infinitely often on ''V''. In the case that ''f'' is an entire function and ''a=∞'', the theorem says that the values ''f(z)'' approach every complex number and ''∞'', as ''z'' tends to infinity. It is remarkable that this does not hold for holomorphic maps in higher dimensions, as the famous example of Pierre Fatou shows. 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「Casorati–Weierstrass theorem」の詳細全文を読む スポンサード リンク
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