Complex-valued function: Difference between revisions

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In mathematics, a '''complex-valued function''' (sometimes referred to as '''complex function''') is a [[function (mathematics)|function]] whose [[codomain|values]] are [[complex number]]s. In other words, it is a function that assigns a complex number to each member of its [[___domain of a function|___domain]]. This ___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 explicated below.

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]]'' (or [[angle|angular]]) variable  {{mvar|φ}}]]

:{{math|1=Re ''f''(''x'') = ''F''(''x'', 0)|size=120%}}

:{{math|1=Im''f''(''x'') = ''F''(''x'', 1)|size=120%}}