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Where ζ denotes the famous Euler–Riemann zeta function, well-known to physicists and mathematicians alike. And then, turning to the final page of Cornel’s five-page derivation that he had mercifully included in his e-mail (else I would have gone mad with frustration), his answer was just too wonderful to simply be made-up: using his general result that is a function of n, he gave the explicit solutions for the first five values of n, the first two of which I’ll repeat here: ġ 3 dx dy dz = 1 − ζ (2) ζ (3) ζ (2) 4 6ġ 7 1 dx dy dz = 1 − ζ (2) − ζ (3) ζ (6) 2 2 48 1 1 2 ζ (3) ζ (2) ζ (3) 18 18 1 ζ (3) ζ (4), 12 #Almost impossible integrals sums and series how to#But how to do the triple integration completely baffled me. It is, of course, immediately clear that the integral exists, as the integrand is always in the interval 0 to 1 over the entire finite region of integration (the volume of the unit cube). I had never seen anything like it before. Π ecos(x) sin denotes the fractional part of x and the integer n ≥ 1. One of his first e-mails was to take exception to my claim that a result attributed to the great Cauchy, himself, 0 Cornel, in fact, wrote to me (from his home in Romania) numerous times over the following months. One of those correspondents was the author of the book you now hold in your hands. 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. 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. 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. “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 To my forever living parents, Ileana Ursachi and Ionel Valean The registered company address is: GewerbestraCham, Switzerland This Springer imprint is published by the registered company Springer Nature Switzerland AG. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. 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, 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. #Almost impossible integrals sums and series free#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 use of general descriptive names, registered names, trademarks, service marks, etc. 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. ISSN 0941-3502 ISSN 2197-8506 (electronic) Problem Books in Mathematics ISBN 978-1-1 ISBN 978-2-8 (eBook) Library of Congress Control Number: 2018966810 © Springer Nature Switzerland AG 2019 This work is subject to copyright. NahinĬornel Ioan V˘alean Teremia Mare, Timis, County Romania #Almost impossible integrals sums and series series#(Almost) Impossible Integrals, Sums, and Series With a Foreword by Paul J. Problem Books in Mathematics Series Editor: Peter Winkler Department of Mathematics Dartmouth College Hanover, NH USA (Almost) Impossible Integrals, Sums, and Series With foreword by Paul J. ![]()
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