Cauchy-Pompeiu formula ($f(z)=\frac{1}{2\pi i}{\int }_{\partial D}\frac{f(\zeta )}{\zeta -z}d\zeta -\frac{1}{\pi }{\int }_D\frac{f_{\bar{\zeta }}(\zeta )}{\zeta -z}dA$)

For a continuously differentiable function on a bounded planar domain, the Cauchy-Pompeiu formula supplements the Cauchy integral formula with an area integral of its Wirtinger derivative. Apply the Generalized Stokes theorem to $f(\zeta )d\zeta /(\zeta -z)$ on a punctured domain; the small-circle integral supplies $2\pi if(z)$. If the boundary term vanishes at infinity, it gives the Cauchy-Green operator representation $f(z)={\pi }^{-1}\int f_{\bar{\zeta }}(\zeta )/(z-\zeta )dA$.