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 $f(z) = u + i v$ of one or more complex variables $z = \left(z_1, \dots, z_n\right) \in \Complex^n$ [is said to be anti-holomorphic if (and only if) it] is the complex conjugate of a holomorphic function $\overline{f \left(z\right)} = u - i v$.'"

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.