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Hermitian Analysis

From Fourier Series to Cauchy-Riemann Geometry

  • Textbook
  • © 2019

Overview

Authors:
  1. John P. D'Angelo

    1. Department of Mathematics, University of Illinois, Urbana-Champaign, Urbana, USA

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  • Includes a new chapter that provides an in-depth exploration of Cauchy-Riemann geometry
  • Uses the concept of orthogonality to unify various mathematical topics
  • Includes accessible content designed to lead students from undergraduate to research-level mathematics

Part of the book series: Cornerstones (COR)

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Table of contents (6 chapters)

  1. Front Matter

    Pages i-x
    Download chapter PDF

  2. Introduction to Fourier series

    • John P. D’Angelo
    Pages 1-42

  3. Hilbert spaces

    • John P. D’Angelo
    Pages 43-92

  4. Fourier transform on \(\mathbf{R}\)

    • John P. D’Angelo
    Pages 93-117

  5. Geometric considerations

    • John P. D’Angelo
    Pages 119-176

  6. The unit sphere and CR geometry

    • John P. D’Angelo
    Pages 177-204

  7. Appendix

    • John P. D’Angelo
    Pages 205-218

  8. Back Matter

    Pages 219-229
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Keywords

About this book

This textbook provides a coherent, integrated look at various topics from undergraduate analysis. It begins with Fourier series, continues with Hilbert spaces, discusses the Fourier transform on the real line, and then turns to the heart of the book, geometric considerations. This chapter includes complex differential forms, geometric inequalities from one and several complex variables, and includes some of the author's original results. The concept of orthogonality weaves the material into a coherent whole. This textbook will be a useful resource for upper-undergraduate students who intend to continue with mathematics, graduate students interested in analysis, and researchers interested in some basic aspects of Cauchy-Riemann (CR) geometry. The inclusion of several hundred exercises makes this book suitable for a capstone undergraduate Honors class.​

This second edition contains a significant amount of new material, including a new chapter dedicated to the CR geometry of the unit sphere. This chapter builds upon the first edition by presenting recent results about groups associated with CR sphere maps. 


From reviews of the first edition:


The present book developed from the teaching experiences of the author in several honors courses. …. All the topics are motivated very nicely, and there are many exercises, which make the book ideal for a first-year graduate course on the subject. …. The style is concise, always very neat, and proofs are given with full details. Hence, I certainly suggest this nice textbook to anyone interested in the subject, even for self-study. Fabio Nicola, Politecnico di Torino, Mathematical Reviews


D’Angelo has written an eminently readable book, including excellent explanations of pretty nasty stuff for even the more gifted upper division players .... It certainly succeeds in hooking the present browser: I like this booka great deal. Michael Berg, Loyola Marymount University, Mathematical Association of America

Authors and Affiliations

  • Department of Mathematics, University of Illinois, Urbana-Champaign, Urbana, USA

    John P. D'Angelo

About the author

John P. D'Angelo, PhD, is a Professor in the Department of Mathematics at the University of Illiniois at Urbana-Champaign, USA

Bibliographic Information

  • Book Title: Hermitian Analysis

  • Book Subtitle: From Fourier Series to Cauchy-Riemann Geometry

  • Authors: John P. D'Angelo

  • Series Title: Cornerstones

  • DOI: https://doi.org/10.1007/978-3-030-16514-7

  • Publisher: Birkhäuser Cham

  • eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)

  • Copyright Information: Springer Nature Switzerland AG 2019

  • Hardcover ISBN: 978-3-030-16513-0Published: 04 June 2019

  • Softcover ISBN: 978-3-030-16516-1Published: 14 August 2020

  • eBook ISBN: 978-3-030-16514-7Published: 24 May 2019

  • Series ISSN: 2197-182X

  • Series E-ISSN: 2197-1838

  • Edition Number: 2

  • Number of Pages: X, 229

  • Number of Illustrations: 8 b/w illustrations, 20 illustrations in colour

  • Topics: Functions of a Complex Variable, Differential Geometry, Fourier Analysis

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