Ramanujan’s Notebooks

1. Front Matter

   Pages i-xi

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2. Introduction

   • Bruce C. Berndt

   Pages 1-6

3. Hypergeometric Series, I

   • Bruce C. Berndt

   Pages 7-47

4. Hypergeometric Series, II

   • Bruce C. Berndt

   Pages 48-102

5. Continued Fractions

   • Bruce C. Berndt

   Pages 103-184

6. Integrals and Asymptotic Expansions

   • Bruce C. Berndt

   Pages 185-239

7. Infinite Series

   • Bruce C. Berndt

   Pages 240-299

8. Asymptotic Expansions and Modular Forms

   • Bruce C. Berndt

   Pages 300-338

9. Back Matter

   Pages 339-359

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During the years 1903-1914, Ramanujan recorded many of his mathematical discoveries in notebooks without providing proofs. Although many of his results were already in the literature, more were not. Almost a decade after Ramanujan's death in 1920, G.N. Watson and B.M. Wilson began to edit his notebooks but never completed the task. A photostat edition, with no editing, was published by the Tata Institute of Fundamental Research in Bombay in 1957. This book is the second of four volumes devoted to the editing of Ramanujan's Notebooks. Part I, published in 1985, contains an account of Chapters 1-9 in the second notebook as well as a description of Ramanujan's quarterly reports. In this volume, we examine Chapters 10-15 in Ramanujan's second notebook. If a result is known, we provide references in the literature where proofs may be found; if a result is not known, we attempt to prove it. Not only are the results fascinating, but, for the most part, Ramanujan's methods remain a mystery. Much work still needs to be done. We hope readers will strive to discover Ramanujan's thoughts and further develop his beautiful ideas.

• Finite
• Mathematica
• Modular form
• integral
• proof

• Department of Mathematics, University of Illinois, Urbana, USA

  Bruce C. Berndt

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