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Antiholomorphic function

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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 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 of .

A definition of antiholomorphic function follows:[1]

"[a] function of one or more complex variables [is said to be anti-holomorphic if (and only if) it] is the complex conjugate of a holomorphic function ."

One can show that if is a holomorphic function on an open set , then is an antiholomorphic function on , where is the reflection of across the real axis; in other words, is the set of complex conjugates of elements of . 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 is antiholomorphic on an open set if and only if the function is holomorphic on .

If a function is both holomorphic and antiholomorphic, then it is constant on any connected component of its domain.

References

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  1. ^ a b 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, ISBN 1402006098.