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Complex Analysis

A Functional Analysis Approach

  • D. H. Luecking
  • L. A. Rubel

Part of the Universitext book series (UTX)

Table of contents

  1. Front Matter
    Pages i-vii
  2. D. H. Luecking, L. A. Rubel
    Pages 1-12
  3. D. H. Luecking, L. A. Rubel
    Pages 13-26
  4. D. H. Luecking, L. A. Rubel
    Pages 27-32
  5. D. H. Luecking, L. A. Rubel
    Pages 33-37
  6. D. H. Luecking, L. A. Rubel
    Pages 38-43
  7. D. H. Luecking, L. A. Rubel
    Pages 44-50
  8. D. H. Luecking, L. A. Rubel
    Pages 51-59
  9. D. H. Luecking, L. A. Rubel
    Pages 60-66
  10. D. H. Luecking, L. A. Rubel
    Pages 67-76
  11. D. H. Luecking, L. A. Rubel
    Pages 77-83
  12. D. H. Luecking, L. A. Rubel
    Pages 84-95
  13. D. H. Luecking, L. A. Rubel
    Pages 96-107
  14. D. H. Luecking, L. A. Rubel
    Pages 108-116
  15. D. H. Luecking, L. A. Rubel
    Pages 117-123
  16. D. H. Luecking, L. A. Rubel
    Pages 124-129
  17. D. H. Luecking, L. A. Rubel
    Pages 130-135
  18. D. H. Luecking, L. A. Rubel
    Pages 136-150
  19. D. H. Luecking, L. A. Rubel
    Pages 151-156
  20. D. H. Luecking, L. A. Rubel
    Pages 157-167
  21. D. H. Luecking, L. A. Rubel
    Pages 168-174
  22. Back Matter
    Pages 174-176

About this book

Introduction

The main idea of this book is to present a good portion of the standard material on functions of a complex variable, as well as some new material, from the point of view of functional analysis. The main object of study is the algebra H(G) of all holomorphic functions on the open set G, with the topology on H(G) of uniform convergence on compact subsets of G. From this point of vie~, the main theorem of the theory is Theorem 9.5, which concretely identifies the dual of H(G) with the space of germs of holomorphic functions on the complement of G. From this result, for example, Runge's approximation theorem and the global Cauchy integral theorem follow in a few short steps. Other consequences of this duality theorem are the Germay interpolation theorem and the Mittag-Leffler Theorem. The approach via duality is entirely consistent with Cauchy's approach to complex variables, since curvilinear integrals are typical examples of linear functionals. The prerequisite for the book is a one-semester course in com­ plex variables at the undergraduate-graduate level, so that the elements of the local theory are supposed known. In particular, the Cauchy Theorem for the square and the circle are assumed, but not the global Cauchy Theorem in any of its forms. The second author has three times taught a graduate course based on this material at the University of Illinois, with good results.

Keywords

Analysis Funktionalanalysis Funktionentheorie calculus functional analysis

Authors and affiliations

  • D. H. Luecking
    • 1
  • L. A. Rubel
    • 2
  1. 1.Department of MathematicsUniversity of ArkansasFayettevilleUSA
  2. 2.Department of MathematicsUniversity of IllinoisUrbana-ChampaignUSA

Bibliographic information

  • DOI https://doi.org/10.1007/978-1-4613-8295-9
  • Copyright Information Springer-Verlag New York 1984
  • Publisher Name Springer, New York, NY
  • eBook Packages Springer Book Archive
  • Print ISBN 978-0-387-90993-6
  • Online ISBN 978-1-4613-8295-9
  • Series Print ISSN 0172-5939
  • Series Online ISSN 2191-6675
  • Buy this book on publisher's site