Nahin - Inside interesting integrals: (with an introduction to contour integration): a collection of sneaky tricks, sly substitutions, and numerous other stupendously clever, awesomely wicked, and devilishly

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Undergraduate Lecture Notes in Physics

Series Editors Neil Ashby , William Brantley , Michael Fowler , Michael Inglis , Heinz Klose and Helmy Sherif

For further volumes: http://www.springer.com/series/8917

Undergraduate Lecture Notes in Physics (ULNP) publishes authoritative texts covering topics throughout pure and applied physics. Each title in the series is suitable as a basis for undergraduate instruction, typically containing practice problems, worked examples, chapter summaries, and suggestions for further reading.

ULNP titles must provide at least one of the following:

• An exceptionally clear and concise treatment of a standard undergraduate subject.
• A solid undergraduate-level introduction to a graduate, advanced, or non-standard subject.
• A novel perspective or an unusual approach to teaching a subject.

ULNP especially encourages new, original, and idiosyncratic approaches to physics teaching at the undergraduate level.

The purpose of ULNP is to provide intriguing, absorbing books that will continue to be the readers preferred reference throughout their academic career.

Paul J. Nahin

Inside Interesting Integrals A Collection of Sneaky Tricks, Sly Substitutions, and Numerous Other Stupendously Clever, Awesomely Wicked, and Devilishly Seductive Maneuvers for Computing Nearly 200 Perplexing Definite Integrals From Physics, Engineering, and Mathematics (Plus 60 Challenge Problems with Complete, Detailed Solutions)

Paul J. Nahin

Electrical and Computer Engineering, University of New Hampshire, Durham, NH, USA

ISSN 2192-4791 e-ISSN 2192-4805

ISBN 978-1-4939-1276-6 e-ISBN 978-1-4939-1277-3

DOI 10.1007/978-1-4939-1277-3

Springer New York Heidelberg Dordrecht London

Library of Congress Control Number: 2014945223

Springer Science+Business Media New York 2015

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. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publishers location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law.

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.

While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein.

Printed on acid-free paper

Springer is part of Springer Science+Business Media (www.springer.com)

Bernhard Riemann (18261866), the German mathematical genius whose integral is the subject of this book (AIP Emilio Segr Visual Archives, T. J. J. See Collection)

This book is dedicated to all who, when they read the following line from John le Carrs 1989 Cold War spy novel The Russia House , immediately know they have encountered a most interesting character:

as well as to all who understand how frustrating is the lament in Anthony Zee s book Quantum Field Theory in a Nutshell :

Also by Paul J. Nahin

Oliver Heaviside (1988, 2002)

Time Machines (1993, 1999)

The Science of Radio (1996, 2001)

An Imaginary Tale (1998, 2007, 2010)

Duelling Idiots (2000, 2002)

When Least Is Best (2004, 2007)

Dr. Eulers Fabulous Formula (2006, 2011)

Chases and Escapes (2007, 2012)

Digital Dice (2008, 2013)

Mrs. Perkinss Electric Quilt (2009)

Time Travel (1997, 2011)

Number-Crunching (2011)

The Logician and the Engineer (2012)

Will You Be Alive Ten Years From Now? (2013)

Holy Sci-Fi! (2014)

Preface

Engineering is like dancing; you dont learn it in a darkened lecture hall watching slides: you learn it by getting out on the dance floor and having your toes stepped on.

Professor Jack Alford (19202006), cofounder of the Engineering Clinic at Harvey Mudd College, who hired the author in 1971 as an assistant professor. The same can be said for doing definite integrals.

To really appreciate this book, one dedicated to the arcane art of calculating definite integrals, it is necessary (although perhaps it is not sufficient) that you be the sort of person who finds the following question fascinating, one right up there in a fierce battle with a hot cup of coffee and a sugar donut for first place on the list of sinful pleasures:

without actually calculating x, show that

if it then follows that

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