Ramanujan's Lost Notebook Part V
In the spring of 1976, George Andrews of Pennsylvania State University visited the library at Trinity College, Cambridge, to examine the papers of the late G.N. Watson. Among these papers, Andrews discovered a sheaf of 138 pages in the handwriting of Srin
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Ramanujan’s Lost Notebook Part V
Ramanujan’s Lost Notebook
George E. Andrews • Bruce C. Berndt
Ramanujan’s Lost Notebook Part V
123
George E. Andrews Department of Mathematics The Pennsylvania State University University Park, PA, USA
Bruce C. Berndt Department of Mathematics University of Illinois Urbana, IL, USA
ISBN 978-3-319-77832-7 ISBN 978-3-319-77834-1 (eBook) https://doi.org/10.1007/978-3-319-77834-1 Library of Congress Control Number: 2018935289 © Springer International Publishing AG, part of Springer Nature 2018 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Printed on acid-free paper This Springer imprint is published by the registered company Springer International Publishing AG part of Springer Nature. The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland
Some CEOs of Mock Theta Functions
We are especially grateful to these mathematicians whose work and guidance on mock theta functions made it possible to complete this volume.
I have shown you today the highest secret of my own realization. It is supreme and most mysterious indeed. Verse 575 of Vivekachudamani, by Adi Shankaracharya Sixth Century, A.D.
Preface
This is the fifth and final volume that the authors have written in their examination of all the claims made by S. Ramanujan in The Lost Notebook and Other Unpublished Papers. Published by Narosa in 1988, the treatise contains the “Lost Notebook,” which was discovered by the first author in the spring of 1976 at the library of Trinity College, Cambridge. Also included in this publication are partial manuscripts, fragments, and letters from Ramanujan to G.H. Hardy. In his last letter, Ramanujan introduced mock theta functions to the mathematical world for the first time. Most of this volume is devoted to Ramanujan’s beautiful identities involving mock theta functions, which populate his “Lost Notebook.” A
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