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Handbook of Mathematical Formulas and Integrals
FOURTH EDITION

Handbook of Mathematical Formulas and Integrals
FOURTH EDITION

Alan Jeffrey
Professor of Engineering Mathematics University of Newcastle upon Tyne Newcastle upon Tyne United Kingdom

Hui-Hui Dai
Associate Professor of Mathematics City University of Hong Kong Kowloon, China

AMSTERDAM • BOSTON • HEIDELBERG • LONDON NEWYORK • OXFORD • PARIS • SAN DIEGO SAN FRANCISCO • SINGAPORE • SYDNEY • TOKYO Academic Press is an imprint of Elsevier

Acquisitions Editor: Lauren Schultz Yuhasz Developmental Editor: Mara Vos-Sarmiento Marketing Manager: Leah Ackerson Cover Design: Alisa Andreola Cover Illustration: Dick Hannus Production Project Manager: Sarah M. Hajduk Compositor: diacriTech Cover Printer: Phoenix ColorPrinter: Sheridan Books Academic Press is an imprint of Elsevier 30 Corporate Drive, Suite 400, Burlington, MA 01803, USA 525 B Street, Suite 1900, San Diego, California 92101-4495, USA 84 Theobald’s Road, London WC1X 8RR, UK This book is printed on acid-free paper. Copyright c 2008, Elsevier Inc. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by anymeans, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher. Permissions may be sought directly from Elsevier’s Science & Technology Rights Department in Oxford, UK: phone: (+44) 1865 843830, fax: (+44) 1865 853333, E-mail: permissions@elsevier.com. You may also complete your request online viathe Elsevier homepage (http://elsevier.com), by selecting “Support & Contact” then “Copyright and Permission” and then “Obtaining Permissions.” Library of Congress Cataloging-in-Publication Data Application Submitted British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. ISBN: 978-0-12-374288-9 For information on all Academic Presspublications visit our Web site at www.books.elsevier.com Printed in the United States of America 08 09 10 9 8 7 6 5 4 3 2 1

Contents

Preface Preface to the Fourth Edition Notes for Handbook Users Index of Special Functions and Notations 0 Quick Reference List of Frequently Used Data 0.1. Useful Identities 0.1.1. Trigonometric Identities 0.1.2. Hyperbolic Identities 0.2. Complex Relationships0.3. Constants, Binomial Coefficients and the Pochhammer Symbol 0.4. Derivatives of Elementary Functions 0.5. Rules of Differentiation and Integration 0.6. Standard Integrals 0.7. Standard Series 0.8. Geometry Numerical, Algebraic, and Analytical Results for Series and Calculus 1.1. Algebraic Results Involving Real and Complex Numbers 1.1.1. Complex Numbers 1.1.2. Algebraic Inequalities InvolvingReal and Complex Numbers 1.2. Finite Sums 1.2.1. The Binomial Theorem for Positive Integral Exponents 1.2.2. Arithmetic, Geometric, and Arithmetic–Geometric Series 1.2.3. Sums of Powers of Integers 1.2.4. Proof by Mathematical Induction 1.3. Bernoulli and Euler Numbers and Polynomials 1.3.1. Bernoulli and Euler Numbers 1.3.2. Bernoulli and Euler Polynomials 1.3.3. The Euler–Maclaurin SummationFormula 1.3.4. Accelerating the Convergence of Alternating Series 1.4. Determinants 1.4.1. Expansion of Second- and Third-Order Determinants 1.4.2. Minors, Cofactors, and the Laplace Expansion 1.4.3. Basic Properties of Determinants v

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Contents

1.5.

1.6.

1.7.

1.8.

1.9.1.10. 1.11. 1.12.

1.13. 1.14.

1.15.

1.4.4. Jacobi’s Theorem 1.4.5. Hadamard’s Theorem 1.4.6. Hadamard’s Inequality 1.4.7. Cramer’s Rule 1.4.8. Some Special Determinants 1.4.9. Routh–Hurwitz Theorem Matrices 1.5.1. Special Matrices 1.5.2. Quadratic Forms 1.5.3. Differentiation and Integration of Matrices 1.5.4. The Matrix Exponential 1.5.5. The Gerschgorin Circle Theorem Permutations and...