Complex-valued function: Difference between revisions

Revision as of 12:47, 18 September 2017

Revision as of 12:47, 18 September 2017

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[[Image:Exponentials_of_complex_number_within_unit_circle.svg|thumb|right|320px|An [[exponentiation|exponential]] function {{math|''A''^''n''}} 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''' (should not mix up with '''complex variable function''') is a [[function (mathematics)|function]] whose [[value (mathematics)| valuevalues]] 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 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}}: