Antiholomorphic function

In mathematics, antiholomorphic functions (also called antianalytic functions[1]) 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:[1]

"[a] function ${\displaystyle f(z)=u+iv}$ of one or more complex variables ${\displaystyle z=\left(z_{1},\dots ,z_{n}\right)\in \mathbb {C} ^{n}}$ [is said to be anti-holomorphic if (and only if) it] is the complex conjugate of a holomorphic function ${\displaystyle {\overline {f\left(z\right)}}=u-iv}$."

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.