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

 $\frac{\partial u}{\partial x}\;=\;\frac{y^{2}\!-\!x^{2}}{(x^{2}\!+\!y^{2})^{2}% },\qquad\frac{\partial v}{\partial y}\;=\;\frac{x^{2}\!-\!y^{2}}{(x^{2}\!+\!y^% {2})^{2}},\qquad\frac{\partial u}{\partial y}\;=\;-\frac{2xy}{(x^{2}\!+\!y^{2}% )^{2}},\qquad\frac{\partial v}{\partial x}\;=\;-\frac{2xy}{(x^{2}\!+\!y^{2})^{% 2}}.$
Title antiholomorphic Antiholomorphic 2014-11-06 12:07:50 2014-11-06 12:07:50 pahio (2872) pahio (2872) 9 pahio (2872) Definition msc 30A99 antiholomorphic function ComplexConjugate