Ramanujan’s Notebooks

1. Front Matter

   Pages i-xii

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2. Introduction

   • Bruce C. Berndt

   Pages 1-6

3. Elementary Results

   • Bruce C. Berndt

   Pages 7-50

4. Number Theory

   • Bruce C. Berndt

   Pages 51-110

5. Ramanujan’s Theory of Prime Numbers

   • Bruce C. Berndt

   Pages 111-137

6. Theta-Functions and Modular Equations

   • Bruce C. Berndt

   Pages 138-244

7. Inversion Formulas for the Lemniscate and Allied Functions

   • Bruce C. Berndt

   Pages 245-260

8. q-Series

   • Bruce C. Berndt

   Pages 261-286

9. Integrals

   • Bruce C. Berndt

   Pages 287-333

10. Special Functions

    • Bruce C. Berndt

    Pages 334-354

11. Partial Fraction Expansions

    • Bruce C. Berndt

    Pages 355-381

12. Elementary and Miscellaneous Analysis

    • Bruce C. Berndt

    Pages 382-400

13. Back Matter

    Pages 401-451

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During the years 1903-1914, Ramanujan worked in almost complete isolation in India. During this time, he recorded most of his mathematical discoveries without proofs in notebooks. Although many of his results were already found in the literature, most were not. Almost a decade after Ramanujan's death in 1920, G.N. Watson and B.M. Wilson began to edit Ramanujan's notebooks, but they 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 fourth of five volumes devoted to the editing of Ramanujan's notebooks. Parts I, II, and III, published in 1985, 1989, and 1991, contain accounts of Chapters 1-21 in Ramanujan's second notebook as well as a description of his quarterly reports. This is the first of two volumes devoted to proving the results found in the unorganized portions of the second notebook and in the third notebook. The author also proves those results in the first notebook that are not found in the second or third notebooks. For those results that are known, references in the literature are provided. Otherwise, complete proofs are given. Over 1/2 of the results in the notebooks are new. Many of them are so startling and different that there are no results akin to them in the literature.

• Ramanujan
• calculus
• equation
• function
• proof

• Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, USA

  Bruce C. Berndt

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