= Cauchy-Pompeiu formula
{c}
{title2=$f(z)=\frac1{2\pi i}\int_{\partial D}\frac{f(\zeta)}{\zeta-z}d\zeta-\frac1\pi\int_D\frac{f_{\bar\zeta}(\zeta)}{\zeta-z}dA$}
= Pompeiu formula
{c}
{synonym}
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 $2\pi i f(z)$. If the boundary term vanishes at infinity, it gives the <Cauchy-Green operator> representation $f(z)=\pi^{-1}\int f_{\bar\zeta}(\zeta)/(z-\zeta)dA$.
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