(Almost) Impossible Integrals, Sums, and Series
This book contains a multitude of challenging problems and solutions that are not commonly found in classical textbooks. One goal of the book is to present these fascinating mathematical problems in a new and engaging way and illustrate the connections be
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Cornel Ioan Vălean
(Almost) Impossible Integrals, Sums, and Series With foreword by Paul J. Nahin
Problem Books in Mathematics Series Editor: Peter Winkler Department of Mathematics Dartmouth College Hanover, NH USA
More information about this series at http://www.springer.com/series/714
Cornel Ioan V˘alean
(Almost) Impossible Integrals, Sums, and Series With a Foreword by Paul J. Nahin
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Cornel Ioan V˘alean Teremia Mare, Timis, County Romania
ISSN 0941-3502 ISSN 2197-8506 (electronic) Problem Books in Mathematics ISBN 978-3-030-02461-1 ISBN 978-3-030-02462-8 (eBook) https://doi.org/10.1007/978-3-030-02462-8 Library of Congress Control Number: 2018966810 © Springer Nature Switzerland AG 2019 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. This Springer imprint is published by the registered company Springer Nature Switzerland AG. The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland
To my forever living parents, Ileana Ursachi and Ionel Valean
Foreword
Question: Answer:
“Can you tell me who can help me do some elliptic integrals?” “We’ve tried to get rid of anyone like that.” —Exchange between a physics graduate student and a professor of mathematics1
Shortly after my book Inside Interesting Integrals was published by Springer in August 2014, I began to receive e-mails from all over the world. They were from readers who were writing to show me how to do one or the other of the problems in my book in a way “easier,” or “more direct,” than was the solution I gave. Almost all were fascinating reading, and those communications confirmed my belief that people who buy math books are quite different from those who don’t. One of those correspondents was the author of the book you now hold in your hands. Cornel, in fact, wrote to me (from his home in Romania) numerous times over the following months. One of his first e-mails
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