Ramanujan's Theta Functions

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.

Preface.- 0. Sum to Product Identities.- 1. Elliptic Functions.- 2. Transformations.- 3. Theta Functions.- 4. Levels 1, 2, 3, and 4: Jacobi's Inversion Theorem and Ramanujan's Alternative Theories.- 5. Level 5: The Rogers-Ramanujan Continued Fraction.- 6. Level 6: Ramanujan's Cubic Continued Fraction.- 7. Level 7.- 8. Level 8: The Ramanujan-Gollnitz-Gordon Continued Fraction.- 9. Level 9.- 10. Level 10: Ramanujan's Function k.- 11. Levels 11 and 23.- 12. Level 12.- 13. Hypergeometric Modular Transformations.- 14. Ramanujan's Series for 1/pi.- References.

"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)