The Bochner–Martinelli Formula

The main result

Dear Reader, this week let us close the cycle of briefs on the variants of the Cauchy integral formula — the analytical and the smooth versions, and then the multi-dimensional one. Today we dive into the Bochner–Martinelli formula, which recovers $f(z)$ from the boundary data modulo a term measuring the failure of $f$ to be holomorphic.

The shape of the result should look familiar. As in the smooth version on the plane, a boundary integral is corrected by a volume integral built from $\bar{\partial} f$, and that correction vanishes exactly when $f$ is holomorphic, leaving a pure boundary formula.

What is worth noticing is the trade-off with the previous brief. In one variable the kernel $1/(\zeta - z)$ is both universal and holomorphic. In several variables one has to give up one of the two: the Cauchy–Leray–Fantappiè kernel stays holomorphic but must be rebuilt from a defining function of each domain, whereas the Bochner–Martinelli kernel is a single universal expression depending only on $\zeta - z$ — at the price of no longer being holomorphic in $z$ once $n > 1$. Please enjoy this beautiful formula below.