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