Cauchy–Pompeiu formula

Remarks

Proof idea: apply Green's theorem, in the form $\oint_{\partial \Omega} g \, d\zeta = 2i \iint_\Omega \partial g / \partial \bar \zeta \, dA$, to $g(\zeta) = f(\zeta)/(\zeta - z)$ on $D$ minus a small disc around $z$. Since $1/(\zeta - z)$ is holomorphic there, $\partial g / \partial \bar \zeta = (\partial f / \partial \bar \zeta)/(\zeta - z)$; the small circle contributes $2\pi i f(z)$ in the limit. If $f$ is holomorphic, then $\partial f / \partial \bar \zeta = 0$ and the formula is Cauchy's integral formula. If $f$ has compact support in $\C$, the boundary term vanishes and $f(z) = \frac{1}{\pi} \iint \frac{\partial f / \partial \bar \zeta}{z - \zeta} \, dA(\zeta)$. One deduces from it that, for $g$ of class $C^1$ with compact support, $u(z) = \frac{1}{\pi} \iint \frac{g(\zeta)}{z - \zeta} \, dA(\zeta)$ solves $\partial u / \partial \bar z = g$.

Going deeper

For a student

The formula is the basic tool for the inhomogeneous Cauchy–Riemann equation $\partial u / \partial \bar z = g$. The kernel $\frac{1}{\pi z}$ is a fundamental solution of the operator $\partial / \partial \bar z$. This is the starting point of the $\bar\partial$ method, of the theory of the Beltrami equation through the Cauchy and Beurling transforms, and of generalizations to several complex variables.