|
Showing 1 - 3 of
3 matches in All Departments
The monograph is devoted to integral representations for
holomorphic functions in several complex variables, such as
Bochner-Martinelli, Cauchy-Fantappie, Koppelman, multidimensional
logarithmic residue etc., and their boundary properties. The
applications considered are problems of analytic continuation of
functions from the boundary of a bounded domain in C^n. In contrast
to the well-known Hartogs-Bochner theorem, this book investigates
functions with the one-dimensional property of holomorphic
extension along complex lines, and includes the problems of
receiving multidimensional boundary analogs of the Morera theorem.
This book is a valuable resource for specialists in complex
analysis, theoretical physics, as well as graduate and postgraduate
students with an understanding of standard university courses in
complex, real and functional analysis, as well as algebra and
geometry.
The monograph is devoted to integral representations for
holomorphic functions in several complex variables, such as
Bochner-Martinelli, Cauchy-Fantappie, Koppelman, multidimensional
logarithmic residue etc., and their boundary properties. The
applications considered are problems of analytic continuation of
functions from the boundary of a bounded domain in C^n. In contrast
to the well-known Hartogs-Bochner theorem, this book investigates
functions with the one-dimensional property of holomorphic
extension along complex lines, and includes the problems of
receiving multidimensional boundary analogs of the Morera theorem.
This book is a valuable resource for specialists in complex
analysis, theoretical physics, as well as graduate and postgraduate
students with an understanding of standard university courses in
complex, real and functional analysis, as well as algebra and
geometry.
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."
|
|