Antiholomorphic function

In mathematics, antiholomorphic functions (also called antianalytic functions^[1] ) are a family of functions closely related to but distinct from holomorphic functions.

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

A definition of antiholomorphic function follows:^[1]

"[a] function ${\displaystyle f(z)=u+iv}${\displaystyle f(z)=u+iv} of one or more complex variables ${\displaystyle z=\left(z_{1},\dots ,z_{n}\right)\in \mathbb {C} ^{n}}${\displaystyle z=\left(z_{1},\dots ,z_{n}\right)\in \mathbb {C} ^{n}} [is said to be anti-holomorphic if (and only if) it] is the complex conjugate of a holomorphic function ${\displaystyle {\overline {f\left(z\right)}}=u-iv}${\displaystyle {\overline {f\left(z\right)}}=u-iv}."

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

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

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