Table Of ContentRamanujan's Notebooks
Part II
Springer
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Bust of Ramanujan by Paul Granlund
Bruce C. Berndt
Ramanujan's Notebooks
Part II
Springer
Bruce C. Berndt
Department of Mathematics
University of Illinois
Urbana, IL 61801
USA
The following journals have published earlier versions of
chapters in this book:
£Enseignement M athematique 26 (1980), 1-65.
Journal of the Indian Mathematical Society 46 (1982),31-76.
Bulletin London Mathematical Society 15 (1983),273-320.
Expositiones Mathematicae 2 (1984),289-347.
Journal fiir die reine und angewandte Mathematik 361 (1985), 118-134.
Rocky Mountain Journal of Mathematics 15 (1985),235-310.
Acta Arithmetica 47 (1986),123-142.
Mathematics Subject Classification (1980): 11-03, 11P99
Library of Congress Cataloging-in-Publication Data
(Revised for volume 2)
Ramanujan Aiyangar, Srinivasa, 1887-1920.
Ramanujan's notebooks.
Includes bibliographies and index.
1. Mathematics. I. Berndt, Bruce c., 1939-
II. Title.
QA3.R33 1985 510 84-20201
Printed on acid-free paper.
© 1989 by Springer-Verlag New York Inc.
Softcover reprint of the hardcover 1s t edition 1989
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Typeset by Asco Trade Typesetting Ltd., Hong Kong.
Printed and bound by Braun-Brumfield, Inc., Ann Arbor, MI.
9 8 7 6 5 4 3 2 (Corrected second printing, 1999)
ISBN-13: 978-1-4612-8865-7 e-ISBN-13: 978-1-4612-4530-8
001: 10.1007/978-1-4612-4530-8
Dedicated to my mother
Helen
and the memory of my father
Harvey
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.
Photograph reprinted with permission from Collected Papers of G. H. Hardy, Vol. 1,
Oxford University Press, Oxford, 1969.
Preface
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 offour volumes devoted to the editing of Ramanu
jan'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. 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 10-15 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 Bruce C. Berndt
November 1987
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
