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 if 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 both on z and z* cannot be either holomorphic or antiholomorphic.

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