Let \(U\subseteq\CC\) be open, identify \(\CC\) with \(\RR^2\) by \(z=x+iy\), and let \(F\colon U\to\CC\) be \(C^1\) as a function of \((x,y)\). The Wirtinger derivatives of \(F\) are \[ \frac{\partial F}{\partial z}\coloneqq\frac12\qty{\frac{\partial F}{\partial x}-i\frac{\partial F}{\partial y}},\qquad \frac{\partial F}{\partial\bar z}\coloneqq\frac12\qty{\frac{\partial F}{\partial x}+i\frac{\partial F}{\partial y}}, \] and the operators \(\del\coloneqq\del_z\coloneqq\frac12(\del_x-i\del_y)\) and \(\delbar\coloneqq\del_{\bar z}\coloneqq\frac12(\del_x+i\del_y)\) are the del and del-bar operators.
Since \(\frac1i=-i\), the same formulas read \(\frac{\partial F}{\partial z}=\frac12\qty{\frac{\partial F}{\partial x}+\frac1i\frac{\partial F}{\partial y}}\) and \(\frac{\partial F}{\partial\bar z}=\frac12\qty{\frac{\partial F}{\partial x}-\frac1i\frac{\partial F}{\partial y}}\).
With \(F\) as in the definition, \(dz=dx+i\,dy\), and \(d\bar z=dx-i\,dy\), the differential of \(F\) is \[ dF=\frac{\partial F}{\partial z}\dz+\frac{\partial F}{\partial\bar z}\dzbar. \]
Substituting \(dz\) and \(d\bar z\), the right side equals \[ \frac12\qty{F_x-iF_y}(dx+i\,dy)+\frac12\qty{F_x+iF_y}(dx-i\,dy)=F_x\dx+F_y\dy=dF. \]