Bochner–Martinelli formula explained

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

Bochner–Martinelli kernel

For, in

\C ^{n} the Bochner–Martinelli kernel is a differential form in of bidegree defined by\omega(\zeta,z) = ((n -1 )!)/(( 2 \pi i) ^{n}) (1)/(|z - \zeta| ^{2n}) \sum _{1\le} (\overline\zeta _{j-\overline} z _{j)} d\overline\zeta _{1} \land d\zeta _{1} \land … \land d\zeta _{j} \land … \land d\overline\zeta _{n} \land d\zeta _{n}

(where the term is omitted).

Suppose that is a continuously differentiable function on the closure of a domain in

C ⁿ with piecewise smooth boundary . Then the Bochner–Martinelli formula states that if is in the domain then\displaystyle f(z) = \int _{\partial} f(\zeta)\omega(\zeta, z) - \int _{D\overline\partial} f(\zeta)\land\omega(\zeta,z).

In particular if is holomorphic the second term vanishes, so

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

See also

• Bergman–Weil formula

References

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• , (ebook).
• . The first paper where the now called Bochner-Martinelli formula is introduced and proved.
• . Available at the SEALS Portal . In this paper Martinelli gives a proof of Hartogs' extension theorem by using the Bochner-Martinelli formula.
• . The notes form a course, published by the Accademia Nazionale dei Lincei, held by Martinelli during his stay at the Accademia as "Professore Linceo".
• . In this article, Martinelli gives another form to the Martinelli–Bochner formula.