Antiholomorphic function

In mathematics, antiholomorphic functions (also called antianalytic functions) 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 ${\bar {z}}$ exists in the neighbourhood of each and every point in that set, where ${\bar {z}}$ is the complex conjugate of $z$.

A definition of antiholomorphic function follows:

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

One can show that if $f(z)$ is a holomorphic function on an open set $D$, then $f({\bar {z}})$ is an antiholomorphic function on ${\bar {D}}$, where ${\bar {D}}$ is the reflection of $D$ across the real axis; in other words, ${\bar {D}}$ 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 ${\bar {z}}$ 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 ${\overline {f(z)}}$ is holomorphic on $D$.

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