## Analysis of Several Complex VariablesOne of the approaches to the study of functions of several complex variables is to use methods originating in real analysis. In this concise book, the author gives a lucid presentation of how these methods produce a variety of global existence theorems in the theory of functions (based on the characterization of holomorphic functions as weak solutions of the Cauchy-Riemann equations). Emphasis is on recent results, including an $L^2$ extension theorem for holomorphic functions, that have brought a deeper understanding of pseudoconvexity and plurisubharmonic functions. Based on Oka's theorems and his schema for the grouping of problems, the book covers topics at the intersection of the theory of analytic functions of several variables and mathematical analysis. It is assumed that the reader has a basic knowledge of complex analysis at the undergraduate level. The book would make a fine supplementary text for a graduate-level course on complex analysis. |

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### Contents

Holomorphic Functions | 1 |

Rings of Holomorphic Functions and Cohomology | 23 |

Pseudoconvexity and Plurisubharmonic Functions | 35 |

Lē Estimates and Existence Theorems | 55 |

Solutions of the Extension and Division Problems | 83 |

Bergman Kernels | 105 |

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### Common terms and phrases

algebraic analytic functions analytic subset application approximation theorem argument Bergman kernel biholomorphic boundary of class Cauchy-Riemann equations Cauchy's estimate Chapter class C2 compact sets compact subsets complex converges COROLLARY defining functions definition denoted differential forms Division Problems domain of holomorphy element F existence theorem exists an element follows formula functions on Q fundamental geometry Hartogs pseudoconvex Hence holo holomorphic functions holomorphic mapping implies inequality Kerd L2 estimates L2 norm lemma Let Q Levi form Levi pseudoconvex linear Mittag-Leffler theorem morphic functions obtain open ball open set Q plurisubharmonic function positive number power series PROPOSITION pseudoconvex boundary point pseudoconvex domain pseudoconvex open set PSH(J Reinhardt domain restriction map right hand side satisfies sequence Serre's condition solutions solved strictly plurisubharmonic subharmonic function surjection tion topology uniformly on compact variables Weierstrass Weierstrass product theorem weight function