Antiholomorphic function

In mathematics, antiholomorphic functions (also called antianalytic functions¹) are a family of functions closely related to but distinct from holomorphic functions.

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

A definition of antiholomorphic function follows:¹

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

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

If a function is both holomorphic and antiholomorphic, then it is constant on any connected component of its domain.²