From Holomorphic Functions to Complex Manifolds

Author: Klaus Fritzsche,Hans Grauert

Publisher: Springer Science & Business Media

ISBN: 146849273X

Category: Mathematics

Page: 398

View: 8693

This introduction to the theory of complex manifolds covers the most important branches and methods in complex analysis of several variables while completely avoiding abstract concepts involving sheaves, coherence, and higher-dimensional cohomology. Only elementary methods such as power series, holomorphic vector bundles, and one-dimensional cocycles are used. Each chapter contains a variety of examples and exercises.

Time-Varying Vector Fields and Their Flows

Author: Saber Jafarpour,Andrew Lewis

Publisher: Springer

ISBN: 3319101390

Category: Science

Page: 119

View: 6252

This short book provides a comprehensive and unified treatment of time-varying vector fields under a variety of regularity hypotheses, namely finitely differentiable, Lipschitz, smooth, holomorphic, and real analytic. The presentation of this material in the real analytic setting is new, as is the manner in which the various hypotheses are unified using functional analysis. Indeed, a major contribution of the book is the coherent development of locally convex topologies for the space of real analytic sections of a vector bundle, and the development of this in a manner that relates easily to classically known topologies in, for example, the finitely differentiable and smooth cases. The tools used in this development will be of use to researchers in the area of geometric functional analysis.

Holomorphic Functions of Several Variables

An Introduction to the Fundamental Theory

Author: Ludger Kaup,Burchard Kaup

Publisher: Walter de Gruyter

ISBN: 3110838354

Category: Mathematics

Page: 363

View: 6967

The series is devoted to the publication of monographs and high-level textbooks in mathematics, mathematical methods and their applications. Apart from covering important areas of current interest, a major aim is to make topics of an interdisciplinary nature accessible to the non-specialist. The works in this series are addressed to advanced students and researchers in mathematics and theoretical physics. In addition, it can serve as a guide for lectures and seminars on a graduate level. The series de Gruyter Studies in Mathematics was founded ca. 30 years ago by the late Professor Heinz Bauer and Professor Peter Gabriel with the aim to establish a series of monographs and textbooks of high standard, written by scholars with an international reputation presenting current fields of research in pure and applied mathematics. While the editorial board of the Studies has changed with the years, the aspirations of the Studies are unchanged. In times of rapid growth of mathematical knowledge carefully written monographs and textbooks written by experts are needed more than ever, not least to pave the way for the next generation of mathematicians. In this sense the editorial board and the publisher of the Studies are devoted to continue the Studies as a service to the mathematical community. Please submit any book proposals to Niels Jacob.


Author: Eberhard Freitag,Rolf Busam

Publisher: Springer-Verlag

ISBN: 3662073498

Category: Mathematics

Page: 533

View: 1989

Die komplexen Zahlen haben ihre historischen Wurzeln im 16. Jahrhundert, sie entstanden bei dem Versuch, algebmische Gleichungen zu lösen. So führte schon G. CARDANO (1545) formale Ausdrücke wie zum Beispiel 5 ± v'-15 ein, um Lösungen quadratischer und kubischer Gleichungen angeben zu können. R. BOMBELLI rechnete um 1560 bereits systematisch mit diesen Ausdrücken 3 und fand 4 als Lösung der Gleichung x = 15x + 4 in der verschlüsselten Form 4 = ~2 + v'-121 + ~2 - v'-121. Auch bei G. W. LEIBNIZ (1675) findet man Gleichungen dieser Art, wie z. B. VI + v'=3 + Vl- v'=3 = v'6. Im Jahre 1777 führte L. EULER die Bezeichnung i = A für die imaginäre Einheit ein. Der Fachausdruck "komplexe Zahl" stammt von C. F. GAUSS (1831). Die strenge Einführung der komplexen Zahlen als Paare reeller Zahlen geht auf W. R. HAMILTON (1837) zurück. Schon in der reellen Analysis ist es gelegentlich vorteilhaft, komplexe Zahlen einzuführen. Man denke beispielsweise an die Integration rationaler Funktio nen, die auf der Partialbruchentwicklung und damit auf dem Fundamentalsatz der Algebra beruht: Über dem Körper der komplexen Zahlen zerfällt jedes Polynom in ein Produkt von Linearfaktoren.

Holomorphic Functions and Integral Representations in Several Complex Variables

Author: R. Michael Range

Publisher: Springer Science & Business Media

ISBN: 1475719183

Category: Mathematics

Page: 392

View: 6259

The 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.

Several Complex Variables and Complex Manifolds II

Author: Margaret. J. Field,Mike Field,Michael J. Field

Publisher: Cambridge University Press

ISBN: 9780521288880

Category: Mathematics

Page: 220

View: 1417

This self-contained and relatively elementary introduction to functions of several complex variables and complex (especially compact) manifolds is intended to be a synthesis of those topics and a broad introduction to the field. Part I is suitable for advanced undergraduates and beginning postgraduates whilst Part II is written more for the graduate student. The work as a whole will be useful to professional mathematicians or mathematical physicists who wish to acquire a working knowledge of this area of mathematics. Many exercises have been included and indeed they form an integral part of the text. The prerequisites for understanding Part I would be met by any mathematics student with a first degree and together the two parts provide an introduction to the more advanced works in the subject.

Lectures on Riemann Surfaces

Author: Otto Forster

Publisher: Springer Science & Business Media

ISBN: 1461259614

Category: Mathematics

Page: 256

View: 6446

This book grew out of lectures on Riemann surfaces given by Otto Forster at the universities of Munich, Regensburg, and Münster. It provides a concise modern introduction to this rewarding subject, as well as presenting methods used in the study of complex manifolds in the special case of complex dimension one. From the reviews: "This book deserves very serious consideration as a text for anyone contemplating giving a course on Riemann surfaces."—-MATHEMATICAL REVIEWS

Differentialgeometrie, Topologie und Physik

Author: Mikio Nakahara

Publisher: Springer-Verlag

ISBN: 3662453002

Category: Science

Page: 597

View: 7061

Differentialgeometrie und Topologie sind wichtige Werkzeuge für die Theoretische Physik. Insbesondere finden sie Anwendung in den Gebieten der Astrophysik, der Teilchen- und Festkörperphysik. Das vorliegende beliebte Buch, das nun erstmals ins Deutsche übersetzt wurde, ist eine ideale Einführung für Masterstudenten und Forscher im Bereich der theoretischen und mathematischen Physik. - Im ersten Kapitel bietet das Buch einen Überblick über die Pfadintegralmethode und Eichtheorien. - Kapitel 2 beschäftigt sich mit den mathematischen Grundlagen von Abbildungen, Vektorräumen und der Topologie. - Die folgenden Kapitel beschäftigen sich mit fortgeschritteneren Konzepten der Geometrie und Topologie und diskutieren auch deren Anwendungen im Bereich der Flüssigkristalle, bei suprafluidem Helium, in der ART und der bosonischen Stringtheorie. - Daran anschließend findet eine Zusammenführung von Geometrie und Topologie statt: es geht um Faserbündel, characteristische Klassen und Indextheoreme (u.a. in Anwendung auf die supersymmetrische Quantenmechanik). - Die letzten beiden Kapitel widmen sich der spannendsten Anwendung von Geometrie und Topologie in der modernen Physik, nämlich den Eichfeldtheorien und der Analyse der Polakov'schen bosonischen Stringtheorie aus einer gemetrischen Perspektive. Mikio Nakahara studierte an der Universität Kyoto und am King’s in London Physik sowie klassische und Quantengravitationstheorie. Heute ist er Physikprofessor an der Kinki-Universität in Osaka (Japan), wo er u. a. über topologische Quantencomputer forscht. Diese Buch entstand aus einer Vorlesung, die er während Forschungsaufenthalten an der University of Sussex und an der Helsinki University of Sussex gehalten hat.

Several Complex Variables

Author: H. Grauert,K. Fritzsche

Publisher: Springer Science & Business Media

ISBN: 1461298741

Category: Mathematics

Page: 208

View: 6728

The present book grew out of introductory lectures on the theory offunctions of several variables. Its intent is to make the reader familiar, by the discussion of examples and special cases, with the most important branches and methods of this theory, among them, e.g., the problems of holomorphic continuation, the algebraic treatment of power series, sheaf and cohomology theory, and the real methods which stem from elliptic partial differential equations. In the first chapter we begin with the definition of holomorphic functions of several variables, their representation by the Cauchy integral, and their power series expansion on Reinhardt domains. It turns out that, in l:ontrast ~ 2 there exist domains G, G c en to the theory of a single variable, for n with G c G and G "# G such that each function holomorphic in G has a continuation on G. Domains G for which such a G does not exist are called domains of holomorphy. In Chapter 2 we give several characterizations of these domains of holomorphy (theorem of Cartan-Thullen, Levi's problem). We finally construct the holomorphic hull H(G} for each domain G, that is the largest (not necessarily schlicht) domain over en into which each function holomorphic on G can be continued.


Author: New Zealand Mathematical Society

Publisher: N.A


Category: Mathematics

Page: N.A

View: 5208


Complex Analysis

In the Spirit of Lipman Bers

Author: Rubí E. Rodríguez,Irwin Kra,Jane P. Gilman

Publisher: Springer Science & Business Media

ISBN: 1441973230

Category: Mathematics

Page: 306

View: 7667

The authors’ aim here is to present a precise and concise treatment of those parts of complex analysis that should be familiar to every research mathematician. They follow a path in the tradition of Ahlfors and Bers by dedicating the book to a very precise goal: the statement and proof of the Fundamental Theorem for functions of one complex variable. They discuss the many equivalent ways of understanding the concept of analyticity, and offer a leisure exploration of interesting consequences and applications. Readers should have had undergraduate courses in advanced calculus, linear algebra, and some abstract algebra. No background in complex analysis is required.

Maß und Kategorie

Author: J.C. Oxtoby

Publisher: Springer-Verlag

ISBN: 364296074X

Category: Mathematics

Page: 112

View: 8861

Dieses Buch behandelt hauptsächlich zwei Themenkreise: Der Bairesche Kategorie-Satz als Hilfsmittel für Existenzbeweise sowie Die "Dualität" zwischen Maß und Kategorie. Die Kategorie-Methode wird durch viele typische Anwendungen erläutert; die Analogie, die zwischen Maß und Kategorie besteht, wird nach den verschiedensten Richtungen hin genauer untersucht. Hierzu findet der Leser eine kurze Einführung in die Grundlagen der metrischen Topologie; außerdem werden grundlegende Eigenschaften des Lebesgue schen Maßes hergeleitet. Es zeigt sich, daß die Lebesguesche Integrationstheorie für unsere Zwecke nicht erforderlich ist, sondern daß das Riemannsche Integral ausreicht. Weiter werden einige Begriffe aus der allgemeinen Maßtheorie und Topologie eingeführt; dies geschieht jedoch nicht nur der größeren Allgemeinheit wegen. Es erübrigt sich fast zu erwähnen, daß sich die Bezeichnung "Kategorie" stets auf "Bairesche Kategorie" be zieht; sie hat nichts zu tun mit dem in der homologischen Algebra verwendeten Begriff der Kategorie. Beim Leser werden lediglich grundlegende Kenntnisse aus der Analysis und eine gewisse Vertrautheit mit der Mengenlehre vorausgesetzt. Für die hier untersuchten Probleme bietet sich in natürlicher Weise die mengentheoretische Formulierung an. Das vorlie gende Buch ist als Einführung in dieses Gebiet der Analysis gedacht. Man könnte es als Ergänzung zur üblichen Grundvorlesung über reelle Analysis, als Grundlage für ein Se minar oder auch zum selbständigen Studium verwenden. Bei diesem Buch handelt es sich vorwiegend um eine zusammenfassende Darstellung; jedoch finden sich in ihm auch einige Verfeinerungen bekannter Resultate, namentlich Satz 15.6 und Aussage 20.4. Das Literaturverzeichnis erhebt keinen Anspruch auf Vollständigkeit. Häufig werden Werke zitiert, die weitere Literaturangaben enthalten.

Complex Analysis 2

Riemann Surfaces, Several Complex Variables, Abelian Functions, Higher Modular Functions

Author: Eberhard Freitag

Publisher: Springer Science & Business Media

ISBN: 9783642205545

Category: Mathematics

Page: 506

View: 1908

The book contains a complete self-contained introduction to highlights of classical complex analysis. New proofs and some new results are included. All needed notions are developed within the book: with the exception of some basic facts which can be found in the ̄rst volume. There is no comparable treatment in the literature.

Hyperbolic Manifolds and Holomorphic Mappings

An Introduction

Author: Shoshichi Kobayashi

Publisher: World Scientific

ISBN: 9812564969

Category: Mathematics

Page: 148

View: 5567

The first edition of this influential book, published in 1970, opened up a completely new field of invariant metrics and hyperbolic manifolds. The large number of papers on the topics covered by the book written since its appearance led Mathematical Reviews to create two new subsections ?invariant metrics and pseudo-distances? and ?hyperbolic complex manifolds? within the section ?holomorphic mappings?. The invariant distance introduced in the first edition is now called the ?Kobayashi distance?, and the hyperbolicity in the sense of this book is called the ?Kobayashi hyperbolicity? to distinguish it from other hyperbolicities. This book continues to serve as the best introduction to hyperbolic complex analysis and geometry and is easily accessible to students since very little is assumed. The new edition adds comments on the most recent developments in the field.