# Antiholomorphic function

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{{Short description|Function family in complex analysis}}
{{More references|date=December 2009}}
In [mathematics](/source/mathematics),  '''antiholomorphic functions''' (also called '''antianalytic functions'''<ref name="math-encyclopedia">Encyclopedia of Mathematics, Springer and The European Mathematical Society, https://encyclopediaofmath.org/wiki/Anti-holomorphic_function, As of 11 September 2020, This article was adapted from an original article by E. D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics, {{ISBN|1402006098}}.</ref>) are a family of  [function](/source/Function_(mathematics))s closely related to but distinct from [holomorphic function](/source/holomorphic_function)s.

A function of the complex variable <math>z</math> defined on an [open set](/source/open_set) in the [complex plane](/source/complex_plane) is said to be '''antiholomorphic''' if its [derivative](/source/derivative) with respect to <math>\bar z</math> exists in the neighbourhood of each and every point in that set, where <math>\bar z</math> is the [complex conjugate](/source/complex_conjugate) of <math>z</math>. 

A definition of antiholomorphic function follows:<ref name="math-encyclopedia" /> <blockquote>"[a] function <math>f(z) = u + i v</math> of one or more complex variables <math>z = \left(z_1, \dots, z_n\right) \in \Complex^n</math> [is said to be anti-holomorphic if (and only if) it] is the complex conjugate of a holomorphic function <math>\overline{f \left(z\right)} = u - i v</math>."</blockquote>

One can show that if <math>f(z)</math> is a [holomorphic function](/source/holomorphic_function) on an open set <math>D</math>, then <math>f(\bar z)</math> is an antiholomorphic function on <math>\bar D</math>, where <math>\bar D</math> is the reflection of <math>D</math> across the real axis; in other words, <math>\bar D</math> is the set of complex conjugates of elements of <math>D</math>. Moreover, any antiholomorphic function can be obtained in this manner from a holomorphic function. This implies that a function is antiholomorphic [if and only if](/source/if_and_only_if) it can be expanded in a [power series](/source/power_series) in <math>\bar z</math> in a neighborhood of each point in its domain. Also, a function <math>f(z)</math> is antiholomorphic on an open set <math>D</math> if and only if the function <math>\overline{f(z)}</math> is holomorphic on <math>D</math>.

If a function is both holomorphic and antiholomorphic, then it is constant on any [connected component](/source/connected_space) of its domain.<ref>{{Cite book |last=Ahlfors |first=Lars |author-link=Lars Ahlfors |title=Complex Analysis: An Introduction to the Theory of Analytic Functions of One Complex Variable |year=1953 |isbn=978-0070006577}}</ref>

==References==
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Category:Complex analysis
Category:Types of functions

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Adapted from the Wikipedia article [Antiholomorphic function](https://en.wikipedia.org/wiki/Antiholomorphic_function) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Antiholomorphic_function?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.