© 2017

Ramanujan's Theta Functions


Table of contents

  1. Front Matter
    Pages i-xviii
  2. Shaun Cooper
    Pages 1-57
  3. Shaun Cooper
    Pages 59-128
  4. Shaun Cooper
    Pages 129-169
  5. Shaun Cooper
    Pages 171-242
  6. Shaun Cooper
    Pages 423-466
  7. Shaun Cooper
    Pages 509-522
  8. Shaun Cooper
    Pages 523-551
  9. Shaun Cooper
    Pages 553-570
  10. Shaun Cooper
    Pages 571-593
  11. Shaun Cooper
    Pages 595-612
  12. Shaun Cooper
    Pages 613-666
  13. Back Matter
    Pages 667-687

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.


elliptic functions hypergeometric modular transformations partitions Weierstrass functions Jacobi's inversion theorem Rogers-Ramanujan continued fraction Euler's product

Authors and affiliations

  1. 1.Institute of Natural and Mathematical SciencesMassey UniversityAucklandNew Zealand

About the authors

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

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“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)