Wirtinger Kalkül

Man definiert für reell differenzierbare komplexe Funktionen die sogenannten Wirtinger Ableitungen durch

$\displaystyle \frac{\partial}{\partial z}$	$\displaystyle =$	$\displaystyle \frac{1}{2}\left(\frac{\partial}{\partial x}+\frac{1}{i}\frac{\partial}{\partial y}\right)$
$\displaystyle \frac{\partial}{\partial\overline{z}}$	$\displaystyle =$	$\displaystyle \frac{1}{2}\left(\frac{\partial}{\partial x}-\frac{1}{i}\frac{\partial}{\partial y}\right)$

Seien nun $A,B\subset\mathbb{C}$ offen und $f:A\rightarrow B,g:B\rightarrow\mathbb{C}$ reell differenzierbar. Sei weiter $a\in A$ und $b:=f\left(a\right)\in B$ . Definiere $\overline{f}:A\rightarrow\mathbb{C}$ durch $\overline{f}\left(z\right):=\overline{f\left(z\right)}$ .

$\displaystyle \overline{\left(\frac{\partial f}{\partial z}\left(a\right)\right)}$	$\displaystyle =$	$\displaystyle \frac{\partial\overline{f}}{\partial\overline{z}}\left(a\right)$
$\displaystyle \overline{\left(\frac{\partial f}{\partial\overline{z}}\left(a\right)\right)}$	$\displaystyle =$	$\displaystyle \frac{\partial\overline{f}}{\partial z}\left(a\right)$
$\displaystyle \frac{\partial\left(g\circ f\right)}{\partial z}\left(a\right)$	$\displaystyle =$	$\displaystyle \frac{\partial g}{\partial\omega}\left(b\right)\frac{\partial f}{... ...ine{\omega}}\left(b\right)\frac{\partial\overline{f}}{\partial z}\left(a\right)$
$\displaystyle \frac{\partial\left(g\circ f\right)}{\partial\overline{z}}\left(a\right)$	$\displaystyle =$	$\displaystyle \frac{\partial g}{\partial\omega}\left(b\right)\frac{\partial f}{... ...}}\left(b\right)\frac{\partial\overline{f}}{\partial\overline{z}}\left(a\right)$
$\displaystyle \Delta=\frac{\partial^{2}}{\partial z\partial\overline{z}}$	$\displaystyle =$	$\displaystyle \frac{1}{4}\left(\frac{\partial^{2}}{\partial x^{2}}+\frac{\partial^{2}}{\partial y^{2}}\right)$