1979 • 489 Pages • 4.46 MB • English

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Integral Transforms in Science and Engineering

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MATHEMATICAL CONCEPTS AND METHODS IN SCIENCE AND ENGINEERING Series Editor: Angelo Miele Mechanical Engineering and Mathematical Sciences Rice University Volume I INTRODUCTION TO VECTORS AND TENSORS Volume 1: Linear and Multilinear Algebra Ray M. Bowen and C.-C. Wang Volume 2 INTRODUCTION TO VECTORS AND TENSORS Volume 2: Vector and Tensor Analysis Ray M. Bowen and C.-C. Wang Volume 3 MULTICRITERIA DECISION MAKING AND DIFFERENTIAL GAMES Edited by George Leitmann Volume 4 ANALYTICAL DYNAMICS OF DISCRETE SYSTEMS Reinhardt M. Rosenberg Volume 5 TOPOLOGY AND MAPS Taqdir Husain Volume 6 REAL AND FUNCTIONAL ANALYSIS A. Mukherjea and K. Pot hoven Volume 7 PRINCIPLES OF OPTIMAL CONTROL THEORY R. V. Gamkrelidze Volume 8 INTRODUCTION TO THE LAPLACE TRANSFORM Peter K. F. Kuhfittig Volume 9 MATHEMATICAL LOGIC: An Introduction to Model Theory A. H. Lightstone Volume 10 SINGULAR OPTIMAL CONTROLS R. Gabasov and F. M. Kirillova Volume 11 INTEGRAL TRANSFORMS IN SCIENCE AND ENGINEERING Kurt Bernardo Wolf Volume 12 APPLIED MATHEMATICS: An Intellectual Orientation Francis J. Murray Volume 13 DIFFERENTIAL EQUATIONS WITH A SMALL PARAMETER AND RELAXATION OSCILLATIONS F. F. Mishchenko and N. Kh. Rozov Volume 14 PRINCIPLES AND PROCEDURES OF NUMERICAL ANALYSIS Ferenc Szidarovszky and Sidney Yakowitz A Con{inuation Order Plan is available for this series. A continuation order will bring delivery of each new volume immediately upon publication. Volumes are billed only upon actual shipment. For further information please contact the publisher.

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Integral Transforms in Science and Engineering Kurt Bernardo Wolf Jnstituto de Investigaciones en Matemdticas Aplicadas yen· Sistemas Universidad Nacional Aut6noma de Mexico Mexico D.F., Mexico SPRINGER SCIENCE+BUSINESS MEDIA, LLC

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Library of Congress Cataloging in Publication Data Wolf, Kurt Bernardo. Integral transforms in science and engineering. (Mathematical concepts and methods in science and engineering; v. 11) Bibliography: p. Includes index. I. Integral transforms. I. Title. QA432.W64 515 .723 78-12482 ISBN 978-1-4757-0874-5 ISBN 978-1-4757-0872-1 (eBook) DOI 10.1007/978-1-4757-0872-1 © 1979 Springer Science+ Business Media New York Originally published by Plenum Press, New York in 1979 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, mechanical, photocopying, microfilming, recording, or otherwise, without written permission from the Publisher

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To Ayala and little Gunnar

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Preface Integral transforms are among the main mathematical methods for the solution of equations describing physical systems, because, quite generally, the coupling between the elements which constitute such a system-these can be the mass points in a finite spring lattice or the continuum of a diffusive or elastic medium-prevents a straightforward "single-particle" solution. By describing the same system in an appropriate reference frame, one can often bring about a mathematical uncoupling of the equations in such a way that the solution becomes that of noninteracting constituents. The "tilt" in the reference frame is a finite or integral transform, according to whether the system has a finite or infinite number of elements. The types of coupling which yield to the integral transform method include diffusive and elastic interactions in "classical" systems as well as the more common quantum-mechanical potentials. The purpose of this volume is to present an orderly exposition of the theory and some of the applications of the finite and integral transforms associated with the names of Fourier, Bessel, Laplace, Hankel, Gauss, Bargmann, and several others in the same vein. The volume is divided into four parts dealing, respectively, with finite, series, integral, and canonical transforms. They are intended to serve as independent units. The reader is assumed to have greater mathematical sophistication in the later parts, though. Part I, which deals with finite transforms, covers the field of complex vector analysis with emphasis on particular linear operators, their eigen- vectors, and their eigenvalues. Finite transforms apply naturally to lattice structures such as (finite) crystals, electric networks, and finite signal sets. Fourier and Bessel series are treated in Part II. The basic theorems are proven here in the customary classical analysis framework, but when vii

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viii Preface introducing the Dirac S, we do not hesitate in translating the vector space con- cepts from their finite-dimensional counterparts, aiming for the rigor of most mathematical physics developments. The appropriate warning signs are placed where one is bound, nevertheless, to be led astray by finite-dimensional analogues. Applications include diffusive and elastic media of finite extent and infinite lattices. Fourier transforms occupy the major portion of Part III. After their introduction and the study of their main properties, we turn to the treatment of certain special functions which have close connection with Fourier trans- forms and which are, moreover, of considerable physical interest: the attrac- tive and repulsive quantum oscillator wave functions and coherent states. Other integral transforms (Laplace, Mellin, Hankel, etc.) related to the Fourier transform and applications occupy the rest of this part. "Canonical transforms" is the name of a parametrized continuum of transforms which include, as particular cases, most of the integral transforms of Part III. They also include Bargmann transforms, a rather modern tool used for the description of shell-model nuclear physics and second-quantized boson field theories. In the presentation given in Part IV, we are adapting recent research material such as canonical transformations in quantum mechanics, hyperdifferential operator realizations for the transforms, and similarity groups for a class of differential equations. We do not explicitly use Lie group theory, although the applications we present in the study of the diffusion and related Schrodinger equations should cater to the taste of the connoisseur. On the whole, the pace and tone of the text have been set by the balance of intuition and rigor as practiced in applied mathematics with the aim that the contents should be useful for senior undergraduate and graduate students in the scientific and technical fields. Each part contains a flux diagram showing the logical concatenation of the sections so as to facilitate their use in a variety of courses. The graduate student or research worker may be interested in some particular sections such as the fast Fourier transform computer algorithm, the Gibbs phenomenon, causality, or oscillator wave functions. These are subjects which have not been commonly included under the same cover. Part IV, moreover, may spur his or her interest in new directions. We have tried to give an adequate bibliography whenever our account of an area had to stop for reasons of specialization or space. References are cited by author's name and year of publication, and they are listed alphabetically at the end of the book. A generous number of figures and some tables should enable the reader to browse easily. New literals are defined by":="; thusf:= A means/is defined as the expression A. Vectors and unargumented functions are denoted by lowercase boldface type and matrices by uppercase boldface. Operators appear in "double" type, e.g., IQ, Ill, and sets in script, e.g., f!l, ~- A symbol list is included at the end.

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Preface ix Exercises are used mainly to suggest alternative proofs, extensions to the text material, or cross references, usually providing the answers as well as further comments. They are meant to be read at least cursorily as part of the text. Equations are numbered by chapter. I would like to express my gratitude to Professor Tomas Garza for his encouragement and support of this project; to my colleagues Drs. Charles P. Boyer, Jorge Ize, and Antonmaria Minzoni among many others at IIMAS, Instituto de Fisica, and Facultad de Ciencias, for their critical comments on the manuscript; and to my students for bearing with the first versions of the material in this volume. The graphics were programmed by the author on the facilities of the Centro de Servicios y Computo and plotted at the Instituto de Ingenieria, UNAM. Special acknowledgment is due to Miss Alicia Vazquez for her fine secretarial work despite many difficulties. Ciudad Universitaria, Mexico D.F. Kurt Bernardo Wolf

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Contents PART I. FINITE-DIMENSIONAL VECTOR SPACES AND THE FOURIER TRANSFORM 1. Concepts from Complex Vector Analysis and the Fourier Transform 3 1.1. N-Dimensional Complex Vector Spaces 3 1.2. Inner Product and Norm in "f/N 5 1.3. Passive and Active Transformations 9 1.4. Unitary Transformations: Permutations and the Fourier Transform . 13 1.5. Self-Adjoint Operators . 20 1.6. The Dihedral Group 26 1. 7. The Axes of a Transformation: Eigenvalues and Eigenvectors 33 2. The Application of Fourier Analysis to the Uncoupling of l,attices 43 2.1. Mechanical and Electric Analogies 43 2.2. The Equation of Motion of Coupled Systems and Solution 49 2.3. Fundamental Solutions, Normal Modes, and Traveling Waves 56 2.4. Farther-Neighbor Interaction, Molecular and Diatomic Lattices 71 2.5. Energy in a Lattice 85 2.6. Phase Space, Time Evolution, and Constants of Motion . ;88 3. Further Developments and Applications of the Finite Fourier Transform 101 3.1. Convolution: Filters and Windows 101 3.2. Correlation: Signal Detection and Noise. 114 3.3. The Fast Fourier Transform Algorithm . 125 3.4. The "Limit" N-+ oo: Fourier Series and Integral Transforms 130 xi

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