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 Enzo Martinelli (1938) and Salomon Bochner (1943).

History

Formula (53) of the present paper and a proof of theorem 5 based on it have just been published by Enzo Martinelli (...).^[1] 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.

— Salomon Bochner, (Bochner 1943, p. 652, footnote 1).

However this author's claim in loc. cit. footnote 1,^[2] that he might have been familiar with the general shape of the formula before Martinelli, was wholly unjustified and is hereby being retracted.

— Salomon Bochner, (Bochner 1947, p. 15, footnote *).

Bochner–Martinelli kernel

For ζ, z in ${\displaystyle \mathbb {C} ^{n}}$ the Bochner–Martinelli kernel ω(ζ,z) is a differential form in ζ of bidegree (n,n−1) defined by

${\displaystyle \omega (\zeta ,z)={\frac {(n-1)!}{(2\pi i)^{n}}}{\frac {1}{|z-\zeta |^{2n}}}\sum _{1\leq j\leq n}({\overline {\zeta }}_{j}-{\overline {z}}_{j})\,d{\overline {\zeta }}_{1}\land d\zeta _{1}\land \cdots \land d\zeta _{j}\land \cdots \land d{\overline {\zeta }}_{n}\land d\zeta _{n}}$

(where the term dζ_j is omitted).

Suppose that f is a continuously differentiable function on the closure of a domain D in ${\displaystyle \mathbb {C} }$ⁿ with piecewise smooth boundary ∂D. Then the Bochner–Martinelli formula states that if z is in the domain D then

${\displaystyle \displaystyle f(z)=\int _{\partial D}f(\zeta )\omega (\zeta ,z)-\int _{D}{\overline {\partial }}f(\zeta )\land \omega (\zeta ,z).}$

In particular if f is holomorphic the second term vanishes, so

${\displaystyle \displaystyle f(z)=\int _{\partial D}f(\zeta )\omega (\zeta ,z).}$

See also

• Bergman–Weil formula