## Function Theory in Several Complex Variables``Kiyoshi Oka, at the beginning of his research, regarded the collection of problems which he encountered in the study of domains of holomorphy as large mountains which separate today and tomorrow. Thus, he believed that there could be no essential progress in analysis without climbing over these mountains ... this book is a worthwhile initial step for the reader in order to understand the mathematical world which was created by Kiyoshi Oka.'' --from the Preface This book explains results in the theory of functions of several complex variables which were mostly established from the late nineteenth century through the middle of the twentieth century. In the work, the author introduces the mathematical world created by his advisor, Kiyoshi Oka. In this volume, Oka's work is divided into two parts. The first is the study of analytic functions in univalent domains in ${\mathbf C}^n$. Here Oka proved that three concepts are equivalent: domains of holomorphy, holomorphically convex domains, and pseudoconvex domains; and moreover that the Poincare problem, the Cousin problems, and the Runge problem, when stated properly, can be solved in domains of holomorphy satisfying the appropriate conditions. The second part of Oka's work established a method for the study of analytic functions defined in a ramified domain over ${\mathbf C}^n$ in which the branch points are considered as interior points of the domain. Here analytic functions in an analytic space are treated, which is a slight generalization of a ramified domain over ${\mathbf C}^n$. In writing the book, the author's goal was to bring to readers a real understanding of Oka's original papers. This volume is an English translation of the original Japanese edition, published by the University of Tokyo Press (Japan). It would make a suitable course text for advanced graduate level introductions to several complex variables. |

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### Contents

Holomorphic Functions and Domains of Holomorphy | 3 |

Implicit Functions and Analytic Sets | 37 |

The Poincare Cousin and Runge Problems | 73 |

Pseudoconvex Domains and Pseudoconcave Sets | 105 |

Holomorphic Mappings | 147 |

Ramified Domains | 167 |

Analytic Sets and Holomorphic Functions | 209 |

Analytic Spaces | 267 |

Normal Pseudoconvex Spaces | 321 |

Bibliography | 359 |

### Common terms and phrases

ai(z analytic hypersurface analytic polyhedron analytic set analytic space analytically continued assume closed polydisk Cn+i compact set complex line complex variable continuous function converges coordinates Corollary Cousin I distribution Cousin I problem denote dimension disk domain in Cn domain of holomorphy domain over Cn equation exhaustion function exists a holomorphic exists a neighborhood Fi(p Fi(z finite number function of rank hence holomorphic function holomorphic function f(z holomorphic mapping holomorphic vector-valued function holomorphically complete integer irreducible component Jx(F Lemma Let f(z linear locally finite pseudobase meromorphic function n-dimensional notation number of holomorphic O-ideal O-module Oka's open set plurisubharmonic function polydisk centered polynomial projection Proof Proposition prove pseudoconcave set pseudoconvex domain pseudopolynomial ramified domain real number relative boundary satisfies condition satisfies the Weierstrass singular points solvable Stein space strictly pseudoconvex subset sufficiently small univalent unramified weakly holomorphic function Weierstrass condition zero set