Mathematical Models of Physics Problems PDF
Preview Mathematical Models of Physics Problems
PHYSICS RESEARCH AND TECHNOLOGY M M P ATHEMATICAL ODELS OF HYSICS P ROBLEMS No part of this digital document may be reproduced, stored in a retrieval system or transmitted in any form or by any means. The publisher has taken reasonable care in the preparation of this digital document, but makes no expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions. No liability is assumed for incidental or consequential damages in connection with or arising out of information contained herein. This digital document is sold with the clear understanding that the publisher is not engaged in rendering legal, medical or any other professional services. P R T HYSICS ESEARCH AND ECHNOLOGY Additional books in this series can be found on Nova’s website under the Series tab. Additional e-books in this series can be found on Nova’s website under the e-book tab. M R D ATHEMATICS ESEARCH EVELOPMENTS Additional books in this series can be found on Nova’s website under the Series tab. Additional e-books in this series can be found on Nova’s website under the e-book tab. PHYSICS RESEARCH AND TECHNOLOGY M M P ATHEMATICAL ODELS OF HYSICS P ROBLEMS LUIS ALFREDO ANCHORDOQUI AND THOMAS CANTZON PAUL New York Copyright © 2013 by Nova Science Publishers, Inc. All rights reserved. No part of this book may be reproduced, stored in a retrieval system or transmitted in any form or by any means: electronic, electrostatic, magnetic, tape, mechanical photocopying, recording or otherwise without the written permission of the Publisher. For permission to use material from this book please contact us: Telephone 631-231-7269; Fax 631-231-8175 Web Site: http://www.novapublishers.com NOTICE TO THE READER The Publisher has taken reasonable care in the preparation of this book, but makes no expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions. No liability is assumed for incidental or consequential damages in connection with or arising out of information contained in this book. The Publisher shall not be liable for any special, consequential, or exemplary damages resulting, in whole or in part, from the readers’ use of, or reliance upon, this material. Any parts of this book based on government reports are so indicated and copyright is claimed for those parts to the extent applicable to compilations of such works. Independent verification should be sought for any data, advice or recommendations contained in this book. In addition, no responsibility is assumed by the publisher for any injury and/or damage to persons or property arising from any methods, products, instructions, ideas or otherwise contained in this publication. This publication is designed to provide accurate and authoritative information with regard to the subject matter covered herein. It is sold with the clear understanding that the Publisher is not engaged in rendering legal or any other professional services. If legal or any other expert assistance is required, the services of a competent person should be sought. FROM A DECLARATION OF PARTICIPANTS JOINTLY ADOPTED BY A COMMITTEE OF THE AMERICAN BAR ASSOCIATION AND A COMMITTEE OF PUBLISHERS. Additional color graphics may be available in the e-book version of this book. Library of Congress Cataloging-in-Publication Data ISBN: (cid:28)(cid:26)(cid:27)(cid:16)(cid:20)(cid:16)(cid:25)(cid:21)(cid:25)(cid:20)(cid:27)(cid:16)(cid:26)(cid:23)(cid:21)(cid:16)(cid:28) (eBook) Published by Nova Science Publishers, Inc. † New York ToIsis Contents Preface ix 1 ComplexAnalysis 1 1.1. ComplexAlgebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2. FunctionsofaComplexVariable . . . . . . . . . . . . . . . . . . . . . . . 3 1.3. Cauchy’sTheoremandItsApplications . . . . . . . . . . . . . . . . . . . 10 1.4. IsolatedSingularitiesandResidues . . . . . . . . . . . . . . . . . . . . . . 22 Exercise.s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2 ElementsofLinearAlgebra 41 2.1. LinearSpaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 2.2. CauchySequencesandCompleteSpaces . . . . . . . . . . . . . . . . . . . 48 2.3. LinearOperatorsonEuclideanSpaces . . . . . . . . . . . . . . . . . . . . 52 Exercise.s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3 OrdinaryDifferentialEquations 59 3.1. SettingtheStage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 3.2. InitialValueProblem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 3.2.1. ExistenceandUniquenessofSolutions . . . . . . . . . . . . . . . 65 3.2.2. SystemofFirst-orderLinearDifferentialEquations . . . . . . . . . 69 3.2.3. GreenMatrixasaGeneralizedFunction . . . . . . . . . . . . . . . 75 3.3. BoundaryValueProblem . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 3.3.1. Self-adjointnessofSturm-LiouvilleOperator . . . . . . . . . . . . 82 3.3.2. GreenFunctionforSturm-LiouvilleOperator . . . . . . . . . . . . 87 3.3.3. SeriesSolutionstoHomogeneousLinearEquations . . . . . . . . . 91 3.4. FourierAnalysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 3.4.1. FourierSeries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 3.4.2. FourierTransform . . . . . . . . . . . . . . . . . . . . . . . . . . 119 Exercise.s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 4 PartialDifferentialEquations 127 4.1. Taxonomy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 4.2. WaveEquation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 4.2.1. VibratingString . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 4.2.2. d’AlembertSolution . . . . . . . . . . . . . . . . . . . . . . . . . 132 viii Contents 4.2.3. LinearAlgebraofSpacetime . . . . . . . . . . . . . . . . . . . . . 140 4.2.4. CausalGreenFunction . . . . . . . . . . . . . . . . . . . . . . . . 142 4.3. DiffusionEquation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 4.3.1. HeatFlow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 4.3.2. DiffusioninanInfinitel LongMetalBar . . . . . . . . . . . . . . 146 4.3.3. DiffusioninaFiniteMetalBbar . . . . . . . . . . . . . . . . . . . 149 4.4. LaplaceEquation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 4.4.1. HarmonicFunctions . . . . . . . . . . . . . . . . . . . . . . . . . 153 4.4.2. SphericalHarmonics . . . . . . . . . . . . . . . . . . . . . . . . . 155 4.4.3. GreenFunctionforLaplaceOperator . . . . . . . . . . . . . . . . 159 4.5. ApplicationstoQuantumMechanics . . . . . . . . . . . . . . . . . . . . . 163 4.5.1. Schro¨dingerandKlein-GordonEquations . . . . . . . . . . . . . . 163 4.5.2. FeynmanPropagator . . . . . . . . . . . . . . . . . . . . . . . . . 165 Exercise.s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 A ConformalTransformations 171 B GammaFunction 173 C GibbsPhenomenon 177 D LaplaceTransform 179 E Ausstrahlungsbedingung 181 AnswersandCommentsontheExercises 183 References 209 Index 211
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