antiholomorphic

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 antiholomorphic in $D$ .

The following conditions are equivalent (http://planetmath.org/Equivalent3):

•

$f(z)$ is antiholomorphic in $D$ .
•

$\overline{f(z)}$ is holomorphic in $D$ .
•

$f(\overline{z})$ is holomorphic in $\overline{D}\,:=\,\{\overline{z}\;\vdots\;\,z\in D\}$ .
•

$f(z)$ may be to a power series $\sum_{n=0}^{\infty}a_{n}(\overline{z}-u)^{n}$ at each $u\in D$ .
•

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 of minus; cf. the Cauchy–Riemann equations (http://planetmath.org/CauchyRiemannEquations).

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}

and thus

\frac{\partial u}{\partial x}\;=\;\frac{y^{2}\!-\!x^{2}}{(x^{2}\!+\!y^{2})^{2}% },\qquad\frac{\partial v}{\partial y}\;=\;\frac{x^{2}\!-\!y^{2}}{(x^{2}\!+\!y^% {2})^{2}},\qquad\frac{\partial u}{\partial y}\;=\;-\frac{2xy}{(x^{2}\!+\!y^{2}% )^{2}},\qquad\frac{\partial v}{\partial x}\;=\;-\frac{2xy}{(x^{2}\!+\!y^{2})^{% 2}}.