Complex function

In mathematics, a complex-valued function (should not mix up with complex variable function) is a function whose values are complex numbers. Its 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 f on an arbitrary set X can be considered as an ordered pair of two real-valued functions: (Ref, Imf) or, alternatively, as a real-valued function φ on X × {0, 1} (the disjoint union of two copies of X) such that for any x:

Some properties of complex-valued functions (such as measurability and continuity) are nothing more than corresponding properties of real-valued functions.

Complex analysis considers 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.

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