{{Short description|Function family in complex analysis}} {{More references|date=December 2009}} In 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 functions closely related to but distinct from holomorphic functions.
A function of the complex variable <math>z</math> defined on an open set in the complex plane is said to be '''antiholomorphic''' if its 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 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 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 it can be expanded in a 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 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== {{Reflist}}
{{DEFAULTSORT:Antiholomorphic Function}} Category:Complex analysis Category:Types of functions
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