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Separately Analytic Functions
EMS Tracts in Mathematics Vol. 16

Marek Jarnicki (Jagiellonian University, Kraków, Poland)
Peter Pflug (University of Oldenburg, Germany)

Separately Analytic Functions

ISBN print 978-3-03719-098-2, ISBN online 978-3-03719-598-7
DOI 10.4171/098
August 2011, 306 pages, hardcover, 17 x 24 cm.
58.00 Euro

The story of separately holomorphic functions began about 100 years ago. During the second half of the 19th century, it became known that a separately continuous function is not necessarily continuous as a function of all variables. At the beginning of the 20th century, the study of separately holomorphic functions started due to the fundamental work of Osgood and Hartogs.

This book provides the first self-contained and complete presentation of the study of separately holomorphic functions, starting from its birth up to current research. Most of the results presented have never been published before in book form. The text is divided into two parts. A more elementary one deals with separately holomorphic functions “without singularities”, another addresses the situation of existing singularities. A discussion of the classical results related to separately holomorphic functions leads to the most fundamental result, the classical cross theorem as well as various extensions and generalizations to more complicated “crosses”. Additionally, several applications for other classes of “separately regular” functions are given.

A solid background in basic complex analysis is a prerequisite. In order to make the book self-contained, all the results needed for its understanding are collected in special introductory chapters and referred to at the beginning of each section.

The book is addressed to students and researchers in several complex variables as well as to mathematicians and theoretical physicists who are interested in this area of mathematics.

Keywords: Separately holomorphic/meromorphic functions, Riemann domains, N-fold crosses, generalized crosses, relative extremal functions, holomorphic extension with singularities

Further Information

Review in MR 2840986

Review in Zentralblatt MATH 05943530