This second extended edition of the classic reference on the extension problem of holomorphic functions in pluricomplex analysis contains a wealth of additional material, organized under the original chapter structure, and covers in a self-contained way all new and recent developments and theorems that appeared since the publication of the first edition about twenty years ago.

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.

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.

This book explains different types of subharmonic and harmonic functions. The book brings 12 chapters explaining general and specific types of subharmonic functions (eg. quasinearly subharmonic functions and other separate functions), related partial differential equations, mathematical proofs and extension results. The methods covered in the book also attempt to explain different mathematical analyses such as elliptical equations, domination conditions, weighted boundary behavior. The book serves as a reference work for scholars interested in potential theory and complex analysis.

This book provides an introduction to complex analysis in several variables. The viewpoint of integral representation theory together with Grauert's bumping method offers a natural extension of single variable techniques to several variables analysis and leads rapidly to important global results. Applications focus on global extension problems for CR functions, such as the Hartogs-Bochner phenomenon and removable singularities for CR functions. Three appendices on differential manifolds, sheaf theory and functional analysis make the book self-contained. Each chapter begins with a detailed abstract, clearly demonstrating the structure and relations of following chapters. New concepts are clearly defined and theorems and propositions are proved in detail. Historical notes are also provided at the end of each chapter. Clear and succinct, this book will appeal to post-graduate students, young researchers seeking an introduction to holomorphic function theory in several variables and lecturers seeking a concise book on the subject.

Introduction to Holomorphlc Functions of SeveralVariables, Volumes 1-111 provide an extensiveintroduction to the Oka-Cartan theory of holomorphicfunctions of several variables and holomorphicvarieties. Each volume covers a different aspect andcan be read independently.

This concise textbook gathers together key concepts and modern results on the theory of holomorphic foliations with singularities, offering a compelling vision on how the notion of foliation, usually linked to real functions and manifolds, can have an important role in the holomorphic world, as shown by modern results from mathematicians as H. Cartan, K. Oka, T. Nishino, and M. Suzuki. The text starts with a gentle presentation of the classical notion of foliations, advancing to holomorphic foliations and then holomorphic foliations with singularities. The theory behind reduction of singularities is described in detail, as well the cases for dynamics of a local diffeomorphism and foliations on complex projective spaces. A final chapter brings recent questions in the field, as holomorphic flows on Stein spaces and transversely homogeneous holomorphic foliations, along with a list of open questions for further study and research. Selected exercises at the end of each chapter help the reader to grasp the theory. Graduate students in Mathematics with a special interest in the theory of foliations will especially benefit from this book, which can be used as supplementary reading in Singularity Theory courses, and as a resource for independent study on this vibrant field of research.

This book introduces the reader to a wide range of topics in the theory of holomorphic functions of several variables, with fairly complete proofs. It reviews the relevant properties of subharmonic functions and discusses the basic properties of plurisubharmonic functions.