Antiholomorphic function

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In mathematics, antiholomorphic functions (also called antianalytic functionsEncyclopedia 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}}.) are a family of functions closely related to but distinct from holomorphic functions.

A function of the complex variable $z$ defined on an open set in the complex plane is said to be antiholomorphic if its derivative with respect to $\backslash bar\; z$ exists in the neighbourhood of each and every point in that set, where $\backslash bar\; z$ is the complex conjugate of $z$.

A definition of antiholomorphic function follows:

"[a] function $f(z)\; =\; u\; +\; i\; v$ of one or more complex variables $z\; =\; \backslash left(z\_1,\; \backslash dots,\; z\_n\backslash right)\; \backslash in\; \backslash Complex^n$ [is said to be anti-holomorphic if (and only if) it] is the complex conjugate of a holomorphic function $\backslash overline\{f\; \backslash left(z\backslash right)\}\; =\; u\; -\; i\; v$."

One can show that if $f(z)$ is a holomorphic function on an open set $D$, then $f(\backslash bar\; z)$ is an antiholomorphic function on $\backslash bar\; D$, where $\backslash bar\; D$ is the reflection of $D$ across the real axis; in other words, $\backslash bar\; D$ is the set of complex conjugates of elements of $D$. 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 $\backslash bar\; z$ in a neighborhood of each point in its domain. Also, a function $f(z)$ is antiholomorphic on an open set $D$ if and only if the function $\backslash overline\{f(z)\}$ is holomorphic on $D$.

If a function is both holomorphic and antiholomorphic, then it is constant on any connected component of its domain.