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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 [[Function (mathematics)|function]]s closely related to but distinct from [[holomorphic function]]s.

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 space|connected component]] of its domain.

==References==
{{Reflist}}

{{DEFAULTSORT:Antiholomorphic Function}}
[[Category:Complex analysis]]
[[Category:Types of functions]]

{{mathanalysis-stub}}

Antiholomorphic function - Revision history

imported>Jacobolus: Undid revision 1320709939 by LoneVector (talk) – this doesn't seem to be a "reliable source" by wikipedia standards; see WP:RS

imported>Prime Entelechy: Change to use proper mathematical formatting.