antiholomorphicAntiholomorphic22009-01-07 13:42:282009-01-07 15:57:31Definitioncomplex function% this is the default PlanetMath preamble. as your knowledge
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A complex function \,$f\!: D \to \mathbb{C}$,\, where $D$ is a domain of the complex plane, having the derivative
$$\frac{df}{d \overline{z}}$$
in each point $z$ of $D$, is said to be {\em antiholomorphic} in $D$.\\
The following conditions are \PMlinkname{equivalent}{Equivalent3}:
\begin{itemize}
\item $f(z)$ is antiholomorphic in $D$.
\item \, $\overline{f(z)}$\, is holomorphic in $D$.
\item $f(\overline{z})$ is holomorphic in\, $\overline{D} \,:=\, \{\overline{z}\;\vdots\;\, z \in D\}$.
\item $f(z)$ may be \PMlinkescapetext{expanded} to a power series $\sum_{n=0}^\infty a_n(\overline{z}-u)^n$ at each\, $u \in D$.
\item The real part \,$u(x,\,y)$\, and the imaginary part \,$v(x,\,y)$\, of the function $f$ satisfy the equations
$$\frac{\partial u}{\partial x} \;=\; -\frac{\partial v}{\partial y}, \qquad
\frac{\partial u}{\partial y} \;=\; \frac{\partial v}{\partial x}.$$
N.B. the \PMlinkescapetext{place} of minus; cf. the \PMlinkname{Cauchy--Riemann equations}{CauchyRiemannEquations}.\\
\end{itemize}
\textbf{Example.}\, The function\, $\displaystyle z \mapsto \frac{1}{\overline{z}}$ is antiholomorphic in\,
$\mathbb{C}\!\smallsetminus\!\{0\}$.\, One has
$$f(z) \,=\, \frac{z}{|z|^2} \,=\, \underbrace{\frac{x}{x^2+y^2}}_{u}+i\underbrace{\frac{y}{x^2+y^2}}_{v},$$
$$\frac{\partial u}{\partial x} \;=\; \frac{y^2\!-\!x^2}{(x^2\!+\!y^2)^2}, \quad
\frac{\partial v}{\partial y} \;=\; \frac{x^2\!-\!y^2}{(x^2\!+\!y^2)^2}, \quad
\frac{\partial u}{\partial y} \;=\; -\frac{2xy}{(x^2\!+\!y^2)^2}, \quad
\frac{\partial v}{\partial x} \;=\; -\frac{2xy}{(x^2\!+\!y^2)^2}.$$