# Separately Analytic Functions

Share this page
*Marek Jarnicki; Peter Pflug*

A publication of the European Mathematical Society

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, from its beginnings to
current research. Most of the results presented have never been published
before in book form.

The text is divided into two parts. The first part deals with
separately holomorphic functions, “without singularities”.
The second part 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. To
make the book self contained, all the results are collected in special
introductory chapters and referred to at the beginning of each
section.

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

A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society.

#### Readership

Graduate students, research mathematicians, and theoretical physicists interested in complex variables and analysis.