Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 69 / 2 / i

Use the Wirtinger derivatives ${\partial }_z=({\partial }_x-i{\partial }_y)/2$ and ${\partial }_{\bar{z}}=({\partial }_x+i{\partial }_y)/2$, and area measure $dA=dxdy=(i/2)dz\wedge d\bar{z}$. The supplied boundary-integral identity is the planar Generalized Stokes theorem; the usual Poincare lemma is a different local exactness result.

Let $f\in C^1(\overline{D})$ and $z\in D$. Remove a disk of radius $ϵ$ around $z$, and apply Generalized Stokes theorem to the one-form $f(\zeta )d\zeta /(\zeta -z)$ on the punctured domain. Away from the puncture,The outer boundary is counterclockwise and the small inner circle clockwise. Its counterclockwise integral tends to $2\pi if(z)$. Passing to the limit gives the Cauchy-Pompeiu formulaThe weak $1/|\zeta -z|$ singularity is locally integrable. If the boundary term vanishes on expanding $D$ to the whole plane, the formula becomes $f(z)={\pi }^{-1}{\int }_{\mathbb{C}}{f}_{\bar{\zeta }}(\zeta )/(z-\zeta )dA(\zeta )$. In particular it yields the distributional normalization ${\partial }_{\bar{z}}[1/(\pi z)]={\delta }_0$. For holomorphic functions the area term vanishes and one recovers the Cauchy integral formula.