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In mathematics, holomorphic functions are the central objects of study in complex analysis. A holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighborhood of every point in its domain. The existence of a complex derivative in a neighborhood is a very strong condition, for it implies that any holomorphic function is actually infinitely differentiable and equal to its own Taylor series. The term ''analytic function'' is often used interchangeably with "holomorphic function", although the word “analytic” is also used in a broader sense to describe any function (real, complex, or of more general type) that can be written as a convergent power series in a neighborhood of each point in its domain. The fact that all holomorphic functions are complex analytic functions, and vice versa, is a major theorem in complex analysis.〔''(Analytic functions of one complex variable )'', Encyclopedia of Mathematics. (European Mathematical Society ft. Springer, 2015)〕 Holomorphic functions are also sometimes referred to as ''regular functions''〔(Springer Online Reference Books ), (Wolfram MathWorld )〕 or as ''conformal maps''. A holomorphic function whose domain is the whole complex plane is called an entire function. The phrase "holomorphic at a point ''z''0" means not just differentiable at ''z''0, but differentiable everywhere within some neighborhood of ''z''0 in the complex plane. == Definition == Given a complex-valued function ''f'' of a single complex variable, the derivative of ''f'' at a point ''z''0 in its domain is defined by the limit〔Ahlfors, L., ''Complex Analysis, 3 ed.'' (McGraw-Hill, 1979).〕 : This is the same as the definition of the derivative for real functions, except that all of the quantities are complex. In particular, the limit is taken as the complex number ''z'' approaches ''z''0, and must have the same value for any sequence of complex values for ''z'' that approach ''z''0 on the complex plane. If the limit exists, we say that ''f'' is complex-differentiable at the point ''z''0. This concept of complex differentiability shares several properties with real differentiability: it is linear and obeys the product rule, quotient rule, and chain rule.〔Henrici, P., ''Applied and Computational Complex Analysis'' (Wiley). (volumes: 1974, 1977, 1986. )〕 If ''f'' is ''complex differentiable'' at ''every'' point ''z''0 in an open set ''U'', we say that ''f'' is holomorphic on U. We say that ''f'' is holomorphic at the point ''z''0 if it is holomorphic on some neighborhood of ''z''0.〔Peter Ebenfelt, Norbert Hungerbühler, Joseph J. Kohn, Ngaiming Mok, Emil J. Straube (2011) ''Complex Analysis'' Springer Science & Business Media〕 We say that ''f'' is holomorphic on some non-open set ''A'' if it is holomorphic in an open set containing ''A''. The relationship between real differentiability and complex differentiability is the following. If a complex function is holomorphic, then ''u'' and ''v'' have first partial derivatives with respect to ''x'' and ''y'', and satisfy the Cauchy–Riemann equations:〔Markushevich, A.I.,''Theory of Functions of a Complex Variable'' (Prentice-Hall, 1965). (volumes. )〕 : or, equivalently, the Wirtinger derivative of ''f'' with respect to the complex conjugate of ''z'' is zero: : which is to say that, roughly, ''f'' is functionally independent from the complex conjugate of ''z''. If continuity is not a given, the converse is not necessarily true. A simple converse is that if ''u'' and ''v'' have ''continuous'' first partial derivatives and satisfy the Cauchy–Riemann equations, then ''f'' is holomorphic. A more satisfying converse, which is much harder to prove, is the Looman–Menchoff theorem: if ''f'' is continuous, ''u'' and ''v'' have first partial derivatives (but not necessarily continuous), and they satisfy the Cauchy–Riemann equations, then ''f'' is holomorphic.〔.〕 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「Holomorphic function」の詳細全文を読む スポンサード リンク
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