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Ramanujan's Theta Functions

  • Book
  • © 2017

Overview

Authors:
  1. Shaun Cooper

    1. Institute of Natural and Mathematical Sciences, Massey University, Auckland, New Zealand

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  • Contains a detailed and systematic analysis of theta functions, by level and by weight

  • Serves as a useful, encyclopedic reference, designed to be a full and comprehensive study of select levels of Theta functions and modular forms

  • Features topics that have been the subject of much recent research, with the material organized into a systematic setting

  • Marks the first time many of the topics within will have appeared in book form

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Table of contents (15 chapters)

  1. Front Matter

    Pages i-xviii
    Download chapter PDF

  2. Sum to Product Identities

    • Shaun Cooper
    Pages 1-57

  3. Elliptic Functions

    • Shaun Cooper
    Pages 59-128

  4. Transformations

    • Shaun Cooper
    Pages 129-169

  5. Theta Functions

    • Shaun Cooper
    Pages 171-242

  6. Levels 1, 2, 3, and 4: Jacobi’s Inversion Theorem and Ramanujan’s Alternative Theories

    • Shaun Cooper
    Pages 243-293

  7. Level 5: The Rogers–Ramanujan Continued Fraction

    • Shaun Cooper
    Pages 295-360

  8. Level 6: Ramanujan’s Cubic Continued Fraction

    • Shaun Cooper
    Pages 361-422

  9. Level 7

    • Shaun Cooper
    Pages 423-466

  10. Level 8: The Ramanujan–Göllnitz–Gordon Continued Fraction

    • Shaun Cooper
    Pages 467-507

  11. Level 9

    • Shaun Cooper
    Pages 509-522

  12. Level 10: Ramanujan’s Function k

    • Shaun Cooper
    Pages 523-551

  13. Levels 11 and 23

    • Shaun Cooper
    Pages 553-570

  14. Level 12

    • Shaun Cooper
    Pages 571-593

  15. Hypergeometric Modular Transformations

    • Shaun Cooper
    Pages 595-612

  16. Ramanujan’s Series for 1∕π

    • Shaun Cooper
    Pages 613-666

  17. Back Matter

    Pages 667-687
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Keywords

About this book

Theta functions were studied extensively by Ramanujan. This book provides a systematic development of Ramanujan’s results and extends them to a general theory. The author’s treatment of the subject is comprehensive, providing a detailed study of theta functions and modular forms for levels up to 12.  Aimed at advanced undergraduates, graduate students, and researchers, the organization, user-friendly presentation, and rich source of examples, lends this book to serve as a useful reference, a pedagogical tool, and a stimulus for further research.

Topics, especially those discussed in the second half of the book, have been the subject of much recent research; many of which are appearing in book form for the first time. Further results are summarized in the numerous exercises at the end of each chapter.

Reviews

“Each chapter contains an extensive set of exercises, making the book suitable for students interested in an introduction to q-series, elliptic functions, and modular forms without necessarily requiring the theory of modular forms as a prerequisite. … it will be a valuable reference book on Ramanujan’s theta function identities together with their modern extensions and applications.” (Jeremy Lovejoy, Mathematical Reviews, April, 2018)

“This is a big and bountiful book, clearly written as a labor of love, and well worth the effort (both of writing and reading it). The book is pitched at advanced undergraduates, graduate students, and professionals or researchers, and this is entirely consonant with this kind of number theory … . It’s been a long time since I visited this material, but I am very happy to see it again.” (Michael Berg, MAA Reviews, November, 2017)

Authors and Affiliations

  • Institute of Natural and Mathematical Sciences, Massey University, Auckland, New Zealand

    Shaun Cooper

About the author

Shaun Cooper received a PhD in Mathematics from the University of Wisconsin at Madison in 1995 and has worked at Massey University in New Zealand ever since. He was a visiting Assistant Professor at the University of Minnesota for one semester in 2000, and has spent 12 months each at the National University of Singapore (2007/8) and the University of Newcastle, Australia (2015/16). He is the author of approximately 70 refereed journal articles and edited the book Development of Elliptic Functions According to Ramanujan.

Bibliographic Information

  • Book Title: Ramanujan's Theta Functions

  • Authors: Shaun Cooper

  • DOI: https://doi.org/10.1007/978-3-319-56172-1

  • Publisher: Springer Cham

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

  • Copyright Information: Springer International Publishing AG 2017

  • Hardcover ISBN: 978-3-319-56171-4Published: 26 June 2017

  • Softcover ISBN: 978-3-319-85843-2Published: 02 August 2018

  • eBook ISBN: 978-3-319-56172-1Published: 12 June 2017

  • Edition Number: 1

  • Number of Pages: XVIII, 687

  • Number of Illustrations: 1 b/w illustrations

  • Topics: Number Theory, Algebraic Geometry

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