Bochner–Martinelli formula

In mathematics, the Bochner–Martinelli formula is a generalization of the Cauchy integral formula to functions of several complex variables, introduced by and.

History
"Formula (53) of the present paper and a proof of theorem 5 based on it have just been published by Enzo Martinelli (...). The present author may be permitted to state that these results have been presented by him in a Princeton graduate course in Winter 1940/1941 and were subsequently incorporated, in a Princeton doctorate thesis (June 1941) by Donald C. May, entitled: An integral formula for analytic functions of $k$ variables with some applications."

"However this author's claim in loc. cit. footnote 1, that he might have been familiar with the general shape of the formula before Martinelli, was wholly unjustified and is hereby being retracted."

Bochner–Martinelli kernel
For $ζ$, $z$ in $$\C^n$$ the Bochner–Martinelli kernel $ω(ζ,z)$ is a differential form in $ζ$ of bidegree $(n,n−1)$ defined by
 * $$\omega(\zeta,z) = \frac{(n-1)!}{(2\pi i)^n}\frac{1}{|z-\zeta|^{2n}}

\sum_{1\le j\le n}(\overline\zeta_j-\overline z_j) \, d\overline\zeta_1 \land d\zeta_1 \land \cdots \land d\zeta_j \land \cdots \land d\overline\zeta_n \land d\zeta_n$$

(where the term $d\overline{&zeta;}_{j}$ is omitted).

Suppose that $f$ is a continuously differentiable function on the closure of a domain $D$ in $$\mathbb{C}$$n with piecewise smooth boundary $∂D$. Then the Bochner–Martinelli formula states that if $z$ is in the domain $D$ then
 * $$\displaystyle f(z) = \int_{\partial D}f(\zeta)\omega(\zeta, z) - \int_D\overline\partial f(\zeta)\land\omega(\zeta,z).$$

In particular if $f$ is holomorphic the second term vanishes, so
 * $$\displaystyle f(z) = \int_{\partial D}f(\zeta)\omega(\zeta, z). $$