Antiholomorphic function

Template:More references 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 $\overline{z}$ exists in the neighbourhood of each and every point in that set, where $\overline{z}$ is the complex conjugate of $z$.

A definition of antiholomorphic function follows:

"[a] function

$f(z)=u+iv$

of one or more complex variables

$z=(z_1,\dots ,z_n)\in {ℂ}^n$

[is said to be anti-holomorphic if (and only if) it] is the complex conjugate of a holomorphic function

$\overline{f\left(z\right)}=u-iv$

."

One can show that if $f(z)$ is a holomorphic function on an open set $D$, then $f(\overline{z})$ is an antiholomorphic function on $\overline{D}$, where $\overline{D}$ is the reflection of $D$ across the real axis; in other words, $\overline{D}$ 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 $\overline{z}$ 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.

References

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