Wirtinger derivative
= Wirtinger derivative
{c}
{title2=$\partial_z,\partial_{\bar z}$}
{wiki=Wirtinger_derivatives}
For $z=x+iy$, the Wirtinger derivatives are $\partial_z=(\partial_x-i\partial_y)/2$ and $\partial_{\bar z}=(\partial_x+i\partial_y)/2$. The <Cauchy-Riemann equations> become $\partial_{\bar z}F=0$ for a <holomorphic function> $F$, and $4\partial_z\partial_{\bar z}=\Delta$. Thus $u_z$ is <holomorphic> whenever $u$ is a twice continuously <differentiable> <harmonic function>.