# prove that for any $f \in C^1(U)$

Guys, I need help with this question. I honestly don't even my take on this one yet because I do not know where or how to start. Please help/direct me.

Have you seen Green's theorem? This is Green's theorem rephrased in terms of complex line integrals and the "conjugate derivative" $$\partial_{\overline{z}}$$. Write $$f$$ as a function of $$x$$ and $$y$$, and convert everything to real integrals.
Well, actually, that should be $$\frac1{2i}\oint_{\gamma} f(z)\,dz$$ on the left. You can verify this yourself with the example $$f(z)=\overline{z}$$ on the unit circle.