Antiholomorphic function explained

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

A definition of antiholomorphic function follows:

"[a] function

f(z) = u + i v of one or more complex variables z = \left(z _{1,} ... , z _{n\right)} \in \Complex ^{n} [is said to be anti-holomorphic if (and only if) it] is the complex conjugate of a holomorphic function \overline{f \left(z\right)} = u - i v ."

One can show that if

f(z) is a holomorphic function on an open set D , then f(\bar z) is an antiholomorphic function on \bar D , where \bar D is the reflection of D across the real axis; in other words, \bar 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 \bar 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 \overline{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.

Notes and References

1. Encyclopedia of Mathematics, Springer and The European Mathematical Society, https://encyclopediaofmath.org/wiki/Anti-holomorphic_function, As of 11 September 2020, This article was adapted from an original article by E. D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics, .