Ramanujan's theta functions / by Shaun Cooper.

Format:

Book

Author/Creator:

Cooper, Shaun, Author.

Language:

English

Subjects (All):

Number theory.

Geometry, Algebraic.

Number Theory.

Algebraic Geometry.

Local Subjects:

Number Theory.

Algebraic Geometry.

Physical Description:

1 online resource (XVIII, 687 p. 1 illus.)

Edition:

1st ed. 2017.

Place of Publication:

Cham : Springer International Publishing : Imprint: Springer, 2017.

Summary:

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.

Contents:

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.

Notes:

Includes bibliographical references and indexes.

ISBN:

3-319-56172-3