Antiholomorphic function

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Published on February 28, 2022 5:43:10 PM ● Video Link: https://www.youtube.com/watch?v=AuAkznoPn0Q

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In mathematics, antiholomorphic functions (also called antianalytic functions) 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.

According to, "[a] function



f
(
z
)
=
u
+
i
v


{\displaystyle f(z)=u+iv}
of one or more complex variables



z
=

(


z

1


,
…
,

z

n



)

∈


C


n




{\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






f

(
z
)


¯


=
u
−
i
v


{\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.


== References ==

Source: https://en.wikipedia.org/wiki/Antiholomorphic_function
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