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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.
E-Book Content
Alexander M. Kytmanov
The Bochner-Martinelli Integral and Its Applications Translated from the Russian by Harold P. Boas
Birkhauser Verlag Basel· Boston· Berlin
Author: Alexander M. Kytmanov Krasnoyarsk State University Institute of Physics Akademgorodok Krasnoyarsk 660036 Russia Originally published in Russian under the title «Integral Bochnera-Martinelli i evo primeneniya» by Nauka, Novosibirsk branch, 1992.
A CIP catalogue record for this book is available from the Library of Congress, Washington D.C., USA Deutsche Bibliothek Cataloging-in-Publication Data
Kytmanov, Aleksandr M.: The Bochner-Martinelli integral and its applications / Alexander M. Kytmanov. Trans!. from the Russian by Harold P. Boas. - Basel; Boston; Berlin: Birkhiiuser, 1995 Einheitssacht.: Integral Bochnera-Martinelli i ego primenija ISBN-13: 978-3-0348-9904-8 e-ISBN-13: 978-3-0348-9094-6 DOl: 10.1007/978-3-0348-9094-6 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use whatsoever, permission from the copyright owner must be obtained. © 1995 for the English edition: Birkhiiuser Verlag, P.O. Box 133, CH- 4010 Basel, Switzerland Softcover reprint of the hardcover 1st edition 1995 Printed on acid-free paper produced of chlorine-free pulp 00 ISBN-13: 978-3-0348-9904-8
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Contents Preface . . . . . . . . . . . . .
IX
Preface to the English Edition
XI
1 The Bochner-Martinelli Integral 1 The Bochner-Martinelli integral representation . . . . . . . . 1.1 Green's formula in complex form . . . . . . . . . . . . 1.2 The Bochner-Martinelli formula for smooth functions 1.3 The Bochner-Martinelli representation for holomorphic functions . . . . 1.4 Some integral representations . . . . . . . 2 Boundary behavior . . . . . . . . . . . . . . . . . 2.1 The SokhotskiY-Plemelj formula for functions satisfying a Holder condition . . . . . . . . . . . . . . . 2.2 Analogue of Privalov's theorem for integrable functions 2.3 Further results . . . . . . . . . . . . Jump theorems . . . . . . . . .