I'm attempting to use Green's Theorem to express the area of a region in the complex plane in terms of a contour integral, but I'm a little confused as to how this works. I have a simple closed curve $\gamma$ with interior $D$, and I believe I'm supposed to get $$\mathrm{Area}(D)=\frac{1}{2i} \oint_\gamma \overline{z} \,dz.$$ Can anyone help me justify this?
Using Green's Theorem to compute area in the complex plane
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complex-analysis