==#REDIRECT [[Complex analysis ==]]▼{{mergeto|Complex analysis|discuss=Talk:Complex analysis#Merger proposal|date=December 2018}}
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[[Image:Exponentials_of_complex_number_within_unit_circle.svg|thumb|right|320px|An [[exponentiation|exponential]] function {{math|''A''<sup>''n''</sup>}} of a discrete ([[integer]]) variable {{mvar|n}}, similar to [[geometric progression]]]]<!-- I detest clueless Fr.wikipedia with their plain-ISO-8859-text-only notation: can somebody find an image of the same, but with a good symbolic notation? -->
In mathematics, a '''complex-valued function''' (not to be confused with '''complex variable function''') is a [[function (mathematics)|function]] whose [[value (mathematics)| values]] are [[complex number]]s. Its ___domain does not necessarily have any [[mathematical structure|structure]] related to complex numbers. Most important uses of such functions [[#Complex analysis|in complex analysis]] and [[#Functional analysis|in functional analysis]] are explained below.
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A [[vector space]] and a [[commutative algebra]] of functions over complex numbers can be defined [[real-valued function#In general|in the same way as for real-valued functions]]. Also, any complex-valued function {{mvar|f}} on an arbitrary [[set (mathematics)|set]] {{mvar|X}} can be considered as an [[ordered pair]] of two [[real-valued function]]s: {{math|([[Real part|Re]]''f'', [[Imaginary part|Im]]''f'')}} or, alternatively, as a real-valued function {{mvar|φ}} on {{math|''X'' [[Cartesian product|×]] [[finite set|{0, 1}]]}} (the [[disjoint union]] of [[2 (number)|two]] copies of {{mvar|X}}) such that for any {{mvar|x}}:
[[Image:Euler's formula.svg|thumb|right|[[Euler's formula]] features a complex-valued function of a ''[[real number|real]]'' variable {{mvar|φ}}]]
:{{math|1=Re ''f''(''x'') = ''F''(''x'', 0)|size=120%}}
:{{math|1=Im''f''(''x'') = ''F''(''x'', 1)|size=120%}}
Some properties of complex-valued functions (such as [[measurable function|measurability]] and [[continuous function|continuity]]) are nothing more than [[real-valued function#Measurable|corresponding properties of real-valued functions]].
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Complex analysis considers [[holomorphic function]]s on [[complex manifold]]s, such as [[Riemann surface]]s. The property of [[analytic continuation]] makes them very dissimilar from [[smooth function]]s, for example. Namely, if a function defined in a [[neighborhood (mathematics)|neighborhood]] can be continued to a wider [[___domain (mathematical analysis)|___domain]], then this continuation is [[unique (mathematics)|unique]].
As real functions, any holomorphic function is infinitely smooth and [[analytic function|analytic]]. But there is much less freedom in construction of a holomorphic function than in one of a smooth function.
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== Functional analysis ==
Complex-valued [[Lp space|L<sup>2</sup> spaces]] on [[measure (mathematics)|sets with a measure]] have a particular importance because they are [[Hilbert space]]s. 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 function]]s.
The sets on which the complex-valued L<sup>2</sup> is constructed have the potential to be more exotic than their real-valued analog. For example, complex-valued [[function space]]s are used in some branches of [[p-adic analysis|{{mvar|p}}-adic analysis]] for algebraic reasons: complex numbers form an [[algebraically closed field]] (which facilitates operator theory), whereas neither real numbers nor {{mvar|p}}-adic numbers are not.
Also, complex-valued [[continuous function]]s are an important example in the theory of [[C*-algebra]]s: see [[Gelfand representation]].
==See also==
* [[Function of a complex variable]], the dual concept
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==External links==
{{MathWorld |title=Complex Function |id=ComplexFunction}}
[[Category:Complex analysis]]
[[Category:Types of functions]]
[[Category:Functional analysis]]
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