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 defined on an open set in the complex plane is called antiholomorphic if its derivative with respect to "z"* exists at all points in that set, where "z"* is the complex conjugate.

One can show that if "f"("z") is a holomorphic function on an open set "D", then "f"("z"*) is an antiholomorphic function on "D"*, where "D"* is the reflection against the "x"-axis of "D", or in other words, "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 "z"* in a neighborhood of each point in its domain.

If a function is both holomorphic and antiholomorphic, then it is constant on any connected component of its domain. A function which depends on both "z" and "z"* is neither holomorphic nor antiholomorphic.