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In mathematics, a complex-valued function (sometimes referred to as complex function) is a function whose values are complex numbers. In other words, it is a function that assigns a complex number to each member of its domain. This domain does not necessarily have any structure related to complex numbers. Most important uses of such functions in complex analysis and in functional analysis are explicated below. A vector space and a commutative algebra of functions over complex numbers can be defined in the same way as for real-valued functions. Also, any complex-valued function on an arbitrary set can be considered as an ordered pair of two real-valued functions: or, alternatively, as a real-valued function on (the disjoint union of two copies of ) such that for any : : : Some properties of complex-valued functions (such as measurability and continuity) are nothing more than corresponding properties of real-valued functions. == Complex analysis == (詳細はholomorphic functions on complex manifolds, such as Riemann surfaces. The property of analytic continuation makes them very dissimilar from smooth functions, for example. Namely, if a function defined in a neighborhood can be continued to a wider domain, then this continuation is unique. As real functions, any holomorphic function is infinitely smooth and analytic. But there is much less freedom in construction of a holomorphic function than in one of a smooth function. 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「Complex-valued function」の詳細全文を読む スポンサード リンク
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