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 z exists in the neighbourhood of each and every point in that set, where z is the complex conjugate.

A definition of antiholomorphic function follows:

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

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. Also, a function f(z) is antiholomorphic on an open set D if and only if the function 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.