Cooper S. Ramanujan's Theta Functions

Springer International Publishing AG, 2017. — 696 p. — ISBN: 3319561715.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.Sum to Product Identities
Elliptic Functions
Transformations
Theta Functions
Levels 1, 2, 3, and 4: Jacobi’s Inversion Theorem and Ramanujan’s Alternative Theories
Level 5: The Rogers–Ramanujan Continued Fraction
Level 6: Ramanujan’s Cubic Continued Fraction
Level 7
Level 8: The Ramanujan–Göllnitz–Gordon Continued Fraction
Level 9
Level 10: Ramanujan’s Function k
Levels 11 and 23
Level 12
Hypergeometric Modular Transformations
Ramanujan’s Series for 1∕π