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The BochnerMartinelli Integral and Its Applications,Used
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The BochnerMartinelli integral representation for holomorphic functions or'sev eral complex variables (which has already become classical) appeared in the works of Martinelli and Bochner at the beginning of the 1940's. It was the first essen tially multidimensional representation in which the integration takes place over the whole boundary of the domain. This integral representation has a universal 1 kernel (not depending on the form of the domain), like the Cauchy kernel in e . However, in en when n > 1, the BochnerMartinelli kernel is harmonic, but not holomorphic. For a long time, this circumstance prevented the wide application of the BochnerMartinelli integral in multidimensional complex analysis. Martinelli and Bochner used their representation to prove the theorem of Hartogs (Osgood Brown) on removability of compact singularities of holomorphic functions in en when n > 1. In the 1950's and 1960's, only isolated works appeared that studied the boundary behavior of BochnerMartinelli (type) integrals by analogy with Cauchy (type) integrals. This study was based on the BochnerMartinelli integral being the sum of a doublelayer potential and the tangential derivative of a singlelayer potential. Therefore the BochnerMartinelli integral has a jump that agrees with the integrand, but it behaves like the Cauchy integral under approach to the boundary, that is, somewhat worse than the doublelayer potential. Thus, the BochnerMartinelli integral combines properties of the Cauchy integral and the doublelayer potential.
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