Editing Holomorphic function (section)

{{Use American English|date = February 2019}}
{{Short description|Complex-differentiable (mathematical) function}}
{{for|Zariski's theory of holomorphic functions on an algebraic variety|formal holomorphic function}}
{{Redirect-distinguish|Holomorphism|Homomorphism}}
[[Image:Conformal map.svg|right|thumb|A rectangular grid (top) and its image under a [[conformal map]] {{tmath|f}} (bottom).]]
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[[File:Mapping f z equal 1 over z.gif|thumb|Mapping of the function <math>f(z)=\frac{1}{z}</math>. The animation shows different <math>z</math> in blue color with the corresponding <math>f(z)</math> in red color. The point <math>z</math> and <math>f(z)</math> are shown in the <math>\mathbb{C}\tilde{=}\mathbb{R}^2</math>. y-axis represents the imaginary part of the complex number of <math>z</math> and <math>f(z)</math>.]]

In [[mathematics]], a '''holomorphic function''' is a [[complex-valued function]] of one or [[Function of several complex variables|more]] [[complex number|complex]] variables that is [[Differentiable function#Differentiability in complex analysis|complex differentiable]] in a [[neighbourhood (mathematics)|neighbourhood]] of each point in a [[domain (mathematical analysis)|domain]] in [[Function of several complex variables#The complex coordinate space|complex coordinate space]] {{tmath|\C^n}}. The existence of a complex derivative in a neighbourhood is a very strong condition: It implies that a holomorphic function is [[infinitely differentiable]] and locally equal to its own [[Taylor series]] (is ''[[analytic function|analytic]]''). Holomorphic functions are the central objects of study in [[complex analysis]].

Though the term ''[[analytic function]]'' is often used interchangeably with "holomorphic function", the word "analytic" is defined in a broader sense to denote any function (real, complex, or of more general type) that can be written as a convergent [[power series]] in a neighbourhood of each point in its [[domain of a function|domain]]. That all holomorphic functions are complex analytic functions, and vice versa, is a [[Holomorphic functions are analytic|major theorem in complex analysis]].<ref>
{{cite encyclopedia
 |title=Analytic functions of one complex variable
 |year=2015
 |encyclopedia=Encyclopedia of Mathematics
 |publisher=European Mathematical Society / Springer
 |url=https://www.encyclopediaofmath.org/index.php/Analytic_function#Analytic_functions_of_one_complex_variable
 |via=encyclopediaofmath.org
}}
</ref>

Holomorphic functions are also sometimes referred to as ''regular functions''.<ref>{{SpringerEOM|title=Analytic function|access-date=February 26, 2021}}</ref> A holomorphic function whose domain is the whole [[complex plane]] is called an [[entire function]]. The phrase "holomorphic at a point {{tmath|z_0}}" means not just differentiable at {{tmath|z_0}}, but differentiable everywhere within some close neighbourhood of {{tmath|z_0}} in the complex plane.