Bochner–Martinelli formula

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

Bochner–Martinelli kernel

For Template:Math, Template:Math in ${ℂ}^n$ the Bochner–Martinelli kernel Template:Math is a differential form in Template:Math of bidegree Template:Math defined by

$\omega (\zeta ,z)=\frac{(n-1)!}{(2\pi i{)}^n}\frac{1}{|z-\zeta {|}^{2n}}\sum _{1\le j\le n}({\bar{\zeta }}_j-{\bar{z}}_j)d{\bar{\zeta }}_1\wedge d{\zeta }_1\wedge \cdots \wedge d{\zeta }_j\wedge \cdots \wedge d{\bar{\zeta }}_n\wedge d{\zeta }_n$

(where the term Template:Math is omitted).

Suppose that Template:Math is a continuously differentiable function on the closure of a domain Template:Math in $ℂ$ⁿ with piecewise smooth boundary Template:Math. Then the Bochner–Martinelli formula states that if Template:Math is in the domain Template:Math then

$f(z)=\underset{\partial D}{\int }f(\zeta )\omega (\zeta ,z)-\underset{D}{\int }\bar{\partial }f(\zeta )\wedge \omega (\zeta ,z).$

In particular if Template:Math is holomorphic the second term vanishes, so

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

See also

• Bergman–Weil formula

References

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