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The Bochner-Martinelli Integral and Its Applications by Alexander M. Kytmanov

The Bochner-Martinelli Integral and Its Applications

by Alexander M. Kytmanov
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Book cover type: Paperback
  • ISBN13: 9783034899048
  • Binding: Paperback
  • Subject: N/A
  • Publisher: Springer
  • Publisher Imprint: Birkhauser
  • Publication Date:
  • Pages: 308
  • Original Price: EUR 49.99
  • Language: English
  • Edition: Softcover Repri
  • Item Weight: 559 grams
  • BISAC Subject(s): Calculus and Mathematical Analysis

The Bochner-Martinelli 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 Bochner-Martinelli kernel is harmonic, but not holomorphic. For a long time, this circumstance prevented the wide application of the Bochner-Martinelli 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 Bochner-Martinelli (type) integrals by analogy with Cauchy (type) integrals. This study was based on the Bochner-Martinelli integral being the sum of a double-layer potential and the tangential derivative of a single-layer potential. Therefore the Bochner-Martinelli 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 double-layer potential. Thus, the Bochner-Martinelli integral combines properties of the Cauchy integral and the double-layer potential.

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