Antiholomorphic function
In mathematics, antiholomorphic 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 exists in the neighbourhood of each and every point in that set, where is the complex conjugate.
One can show that if f is a holomorphic function on an open set D, then f is an antiholomorphic function on, where is the reflection against the x-axis of D, or in other words, 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 in a neighborhood of each point in its domain. Also, a function f is antiholomorphic on an open set D if and only if the function is holomorphic on D.
If a function is both holomorphic and antiholomorphic, then it is constant on any connected component of its domain.
