## Holomorphic Functions and Integral Representations in Several Complex VariablesThe subject of this book is Complex Analysis in Several Variables. This text begins at an elementary level with standard local results, followed by a thorough discussion of the various fundamental concepts of "complex convexity" related to the remarkable extension properties of holomorphic functions in more than one variable. It then continues with a comprehensive introduction to integral representations, and concludes with complete proofs of substantial global results on domains of holomorphy and on strictly pseudoconvex domains inC", including, for example, C. Fefferman's famous Mapping Theorem. The most important new feature of this book is the systematic inclusion of many of the developments of the last 20 years which centered around integral representations and estimates for the Cauchy-Riemann equations. In particu lar, integral representations are the principal tool used to develop the global theory, in contrast to many earlier books on the subject which involved methods from commutative algebra and sheaf theory, and/or partial differ ential equations. I believe that this approach offers several advantages: (1) it uses the several variable version of tools familiar to the analyst in one complex variable, and therefore helps to bridge the often perceived gap between com plex analysis in one and in several variables; (2) it leads quite directly to deep global results without introducing a lot of new machinery; and (3) concrete integral representations lend themselves to estimations, therefore opening the door to applications not accessible by the earlier methods. |

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### Contents

III | 1 |

IV | 18 |

V | 31 |

VI | 40 |

VII | 42 |

VIII | 43 |

IX | 48 |

X | 67 |

XXX | 216 |

XXXI | 217 |

XXXII | 227 |

XXXIII | 229 |

XXXIV | 232 |

XXXV | 239 |

XXXVI | 252 |

XXXVII | 272 |

### Other editions - View all

Holomorphic Functions and Integral Representations in Several Complex Variables R. Michael Range No preview available - 2010 |

Holomorphic Functions and Integral Representations in Several Complex Variables R. Michael Range No preview available - 1986 |

### Common terms and phrases

algebra analysis apply approximation arbitrary assume Banach space Bergman biholomorphic boundary bounded called Cauchy Chapter choose classical clearly closed compact complex condition connected consequence consider constant construction continuous converges coordinate Corollary Cousin defined defining function definition denote differential dimension discussed equations equivalent estimate example Exercise exhaustion extension fact finite fixed follows formula given gives global groups Hartogs hence holds holomorphic functions holomorphic map implies independent integral introduced kernel Lemma Levi linear locally manifold natural neighborhood Notice obtains open set operator particular principal problem proof proof of Theorem Proposition prove reader region Remark representation respect result satisfies sequence sheaf Show simple singular solution space Stein strictly pseudoconvex domains structure subset sufficiently Suppose theory topological unique valued variables vector zero

### Popular passages

Page 366 - Solutions with bounds for the equations of H. Lewy and PoincareLelong. Construction of functions of Nevanlinna class with given zeroes in a strongly pseudoconvex domain. Dokl. Akad. Nauk SSSR 224(1975), 771-774; Engl.

Page 367 - Soc. 73(1967), 373-377. 2. The Cauchy integral for differential forms. Bull. Amer. Math. Soc. 73(1967), 554-556. [Krai Krantz, SG: 1. Optimal Lipschitz and L' regularity for the equation flu = /on strongly pseudoconvex domains.

### References to this book

A Course in Robust Control Theory: A Convex Approach Geir E. Dullerud,Fernando Paganini No preview available - 2005 |

Invariant Distances and Metrics in Complex Analysis Marek Jarnicki,Peter Pflug No preview available - 1993 |