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.
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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.