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).

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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 \and d\zeta_1 \and \cdots \and d\zeta_j \and \cdots \and d\overline\zeta_n \and d\zeta_n

(where the term dζj is omitted).

Bochner–Martinelli formula

Suppose that f is a continuously differentiable function on the closure of a domain D in Cn 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)\and\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).

See also

References