Ramanujan's Notebooks

Transcription

1 Ramanujan's Notebooks Part II Springer New York Berlin Heidelberg Barcelona Hong Kong London Milan Paris Singapore Tokyo

2 Bust of Ramanujan by Paul Granlund

3 Bruce C. Berndt Ramanujan's Notebooks Part II Springer

4 Bruce C. Berndt Department of Mathematics University of Illinois Urbana, IL USA The following journals have published earlier versions of chapters in this book: Enseignement M athematique 26 (1980), Journal of the Indian Mathematical Society 46 (1982), Bulletin London Mathematical Society 15 (1983), Expositiones Mathematicae 2 (1984), Journal fiir die reine und angewandte Mathematik 361 (1985), Rocky Mountain Journal of Mathematics 15 (1985), Acta Arithmetica 47 (1986), Mathematics Subject Classification (1980): 11-03, 11P99 Printed on acid-free paper. Library of Congress Cataloging-in-Publication Data (Revised for volume 2) Ramanujan Aiyangar, Srinivasa, Ramanujan's notebooks. Includes bibliographies and index. 1. Mathematics. I. Berndt, Bruce c., II. Title. QA3.R by Springer-Verlag New York Inc. Softcover reprint of the hardcover 1 st edition 1989 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer-Verlag, 175 Fifth Avenue, New York, NY 10010, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use of general descriptive names, trade names, trademarks, etc. in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone. Typeset by Asco Trade Typesetting Ltd., Hong Kong. Printed and bound by Braun-Brumfield, Inc., Ann Arbor, MI (Corrected second printing, 1999) ISBN-13: : / e-isbn-13:

5 Dedicated to my mother Helen and the memory of my father Harvey

6 The relation between Hardy and Ramanujan is unparalleled in scientific history. Each had enormous respect for the abilities of the other. Mrs. Ramanujan told the author in 1984 of her husband's deep admiration for Hardy. Although Ramanujan returned from England with a terminal illness, he never regretted accepting Hardy's invitation to visit Cambridge.

7 Photograph reprinted with permission from Collected Papers of G. H. Hardy, Vol. 1, Oxford University Press, Oxford, 1969.

8 Preface During the years , 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 This book is the second offour 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 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. Except in a few instances when Ramanujan's intent is not clear, we have been able to establish each result in these six chapters. Chapters are among the most interesting chapters in the notebooks. 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. Urbana, Illinois November 1987 Bruce C. Berndt

9 Contents Preface ix Introduction 1 CHAPTER 10 Hypergeometric Series, I 7 CHAPTER 11 Hypergeometric Series, II 48 CHAPTER 12 Continued Fractions 103 CHAPTER 13 Integrals and Asymptotic Expansions 185 CHAPTER 14 Infinite Series 240 CHAPTER 15 Asymptotic Expansions and Modular Forms 300 References 339 Index 355