# Complex-valued function

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An exponential function An of a discrete (integer) variable n, similar to geometric progression

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 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 (the disjoint union of two copies of X) such that for any x:

Euler's formula features a complex-valued function of a real (or angular) variable φ
Re f(x) = F(x, 0)
Imf(x) = F(x, 1)

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

## Complex analysis

Main article: Complex analysis

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.

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.

## Functional analysis

Complex-valued L2 spaces on sets with a measure have a particular importance because they are Hilbert spaces. They often appear in functional analysis (for example, in relation with Fourier transform) and operator theory. A major user of such spaces is quantum mechanics, as wave functions.

The sets on which the complex-valued L2 is constructed have the potential to be more exotic than their real-valued analog. For example, complex-valued function spaces are used in some branches of p-adic analysis for algebraic reasons: complex numbers form an algebraically closed field (which facilitates operator theory), whereas neither real numbers nor p-adic numbers are not.

Also, complex-valued continuous functions are an important example in the theory of C*-algebras: see Gelfand representation.