METHODS OF MATHEMATICAL PHYSICS By COURANT and D. HILBERT R. First English edition Translated and Revised from the German Original VOLUME I Wiley-VCH Verlag GmbH & Co. KGaA This Page Intentionally Left Blank METHODS OF MATHEMATICAL PHYSICS VOLUME I METHODS OF MATHEMATICAL PHYSICS By COURANT and D. HILBERT R. First English edition Translated and Revised from the German Original VOLUME I Wiley-VCH Verlag GmbH & Co. KGaA This Page Intentionally Left Blank All books published by Wiley-VCH are carehlly produced. Nevertheless, authors, editors, and publisher do not warrant the information contained in these books, including this book, to be free of errors. Readers are advised to keep in mind that statements, data, illustrations, procedural details or other items may inadvertently bhe inaccurate. Library of Congress Card Number 53-7164 British Library Cataloging-in-Publication Data: A catalogue record for this book is available from the British Library Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at <http://dnb.ddb.de>. Copyright 1937 by Julius Springer, Berlin vested in the Attorney General of the United States, pursuant to law. Manufactured and sold under License A-1 556 Copyright 1953 by Interscience Publishers, inc. 0 1989 by John Wiley & Sons, Inc. 0 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim All rights reserved (including those of translation into other languages). No part of this book may be reproduced in any form - nor transmitted or translated into machine language without written permission from the publishers. Registered names, trademarks, etc. used in this book, even when not specifically marked as such, are not to be considered unprotected by law. Printed in Singapore Printed on acid-free paper Printing and Bookbinding Markono Print Media Pte Ltd, Singapore ISBN-13: 978-0-471-50447-4 ISBN-10: 0-47 1-50447-5 PREFACE The first German edition of this volume was published by Julius Springer, Berlin, in 1924. A second edition, revised and improved with the help of K. 0. Friedrichs, R. Luneburg, F. Rellich, and other unselfish friends, followed in 1930. The second volume appeared in 1938. In the meantime I had been forced to leave Germany and was fortunate and grateful to be given the opportunities open in the United States. During the Second World War the German book be- came unavailable and later was even suppressed by the National Socialist rulers of Germany. Thus the survival of the book was secuied when the United States Government seized the copyright and licensed a reprint issued by Interscience Publishers, New York. Such a license also had to be obtained from the Alien Property Cus- todian for the present English edition. This edition follows the German original fairly closely but contains a large number of additions and modifications. I have had to post- pone a plan to completely rewrite and modernize the book in collabo- ration with K. 0. Friedrichs, because the pressure for publication of an English “Courant-Hilbert” has become irresistible. Even so, it is hoped that the work in its present form will be useful to mathe- maticians and physicists alike, as the numerous demands from all sides seem to indicate. The objective of the book can still today be expressed almost as in the preface to the first German edition. “Since the seventeenth century, physical intuition has served as a vital source for mathe- matical problems and methods. Recent trends and fashions have, however, weakened the connection between mathematics and physics; mathematicians, turning away from the roots of mathematics in intuition, have concentrated on refinement and emphasized the postu- lational side of mathematics, and at times have overlooked the unity of their science with physics and other fields. In many cases, physi- cists have ceased to appreciate the attitudes of mathematicians. This rift is unquestionably a serious threat to science as a whole; the broad stream of scientific development may split into smaller and V Vi PREFACE smaller rivulets and dry out. It seems therefore important to direct our efforts toward reuniting divergent trends by clarifying the com- mon features and interconnections of many distinct and diverse scientific facts. Only thus can the student attain some mastery of the material and the basis be prepared for further organic develop- ment of research. “The present work is designed to serve this purpose for the field of mathematical physics. Mathematical methods originating in prob- lems of physics are developed and the attempt is made to shape re- sults into unified mathematical theories. Completeness is not at- tempted, but it is hoped that access to a rich and important field will be facilitated by the book. “The responsibility for the present book rests with me. Yet the name of my teacher, colleague, and friend, D. Hilbert, on the title page seems justified by the fact that much material from Hilbert’s papers and lectures has been used, well by the hope that the as as book expresses some of Hilbert’s spirit, which has had such a decisive influence on mathematical research and education.” I greatly indebted to many helpers in all phases of the task of am preparing this edition: to Peter Ceike, Ernest Courant, and Anneli Lax, who provided most of the first draft of the translation; to Hanan Rubin and Herbert Kranzer, who have given constructive criticism; to Wilhelm Magnus, who is responsible for the appendix to Chapter VII; and to Natascha Artin and Lucile Gardner, who carried the burden of the editorial work. Most cordial thanks also are due to Interscience Publishers for their patient and helpful attitude and to my old friend and publisher, Dr. Ferdinand Springer in Heidelberg, the great pioneer of modern scientific publishing, for his sympathetic understanding of the situation, which has so greatly changed since the old days of our close cooperation. R. COURANT New Rochelle, New York June 1965 CONTENTS I . The Algebra of Linear Transformations and Quadratic Forms .... 1 . Linear equations and linear transformations ................... $1 1 . Vectors ................................................... 1 1 2 . Orthogonal systems of vectors . Completeness .............. . Linear transformations . Matrices ....... 3 3 5 . Bilinear, quadratic, and Hermitian forms .... 4 11 5 . Orthogonal and unitary transformations ................... 14 . Linear transformations with a linesf parameter ............... $2 17 . Transformation to principal axes of quadratic and Hermitian $3 forms .......... ...... ................ 23 . Transformatio axes on the basis of a maximum 1 . . principle ......................................... 23 2 . Eigenvalues ............................................... 26 3 . Generalization to Hermitian forms ........................ 28 . Inertial theorem for quadratic farms ...................... 4 28 5 . Representation of the resolvent of a form ................. 29 6 . Solution of systems of linear equations associated with forms ................................................... 30 . Minimum-maximum property of eigenvslues ................... 54 31 . Characterization of eigenvalues by a minimum-maximum 1 problem ................................................ 31 2 . Applications . Constraints ................................. 33 55 . Supplement and problems ..................................... 34 1 . Linear independence and the Gram determinant .......... 34 2 . Hadamard’s inequality for determinants .................. 36 3 . Generalized t.reatment of canonical transformations ........ 37 4. Bilinear and quadratic forms of infinitely many variables .. 41 5 . Infinitesimal linear transformations ....................... 41 6 . Perturbations ............................................ 42 . Constraints ............................................... 7 44 8. Elementary divisors of a matrix or a bilinear form ......... 45 . Spectrum of a uni .................. 9 46 ............................. 47 . I1 Series Expansions of Arbitrary Functions ............. 48 $1 . Orthogonal systems of functions. ........................... 49 . Definitions ........................................... 1 49 2 . Orthogonalization of functions. ......................... 50 . Bessel’s inequality . Completeness relation . Approximation 3 in the mean ............................................. 51 vii ... Vlll CONTENTS 4 . Orthogonal and unitary transformations with infinitely many variables ............ ....................... 55 5 . Validity of the results for se independent variables. ............................... More general assumptions 56 . 6 Construction of complete systems of functions of several variables ............................. .......... 56 . $2 The accumulation principle for functions. ... 57 1. Convergence in function space .......... 57 . $3 Measure of independence and dimension number .............. 61 1 . Measure of independence ................................. 61 . 2 Asymptotic dimension of a sequence of functions .......... 63 $4. Weierstrass’s approximation theorem . Completeness of powers and of trigonometric functions ..................... .... 65 1. Weierstrass’s approximation theorem ...................... 65 2 . ‘Extension to functions of several variables ................ 68 3 . Simultaneous approximation of derivatives ................ 68 4 . Completeness of the trigonometric functions .............. 68 . $5 Fourier series ............. ..................... 69 1. Proof of the fundamental theorem ........................ 69 2 . Multiple Fourier series ...................... 73 3 . Order of magnitude of 74 4 . Change in length of ba 74 5 . Examples ............. 74 . $6 The Fourier integral ......................................... 77 . 1 The fundamental theorem ................................. 77 . 2 Extension.of the result to several variables ............. 79 3. Reciprocity formulas .................. ...... 80 . Examples of Fourier integrals ............................... 81 $7 $8. Legendre polynomials ........................................ 82 1. Construction of the Legendre polynomials by orthogonaliza- of the powers 1, 5, 21, .................................. 82 2 . The generating function .................................. R5 3. Other properties of the Legendre polynomialR ............. A6 (a) Recursion formula (b) Differential equation (c) Minimum property . 69 Examples of other orthogonal systems . . R7 . Generalizat.i on of the problem leadi 1 nomials .......................... .......... 87 . 2 Tchebycheff polynomials ............ 88 3 . Jacobi polynomials .................. 90 4 . Hermite polynomials ...................................... 91 5 . Laguerre polynomials ..................................... 93 . 6 Completeness of the Laguerre and Hermite functions . . . . . 95 $10. Supplement and problems ................................... 97 1 . Hurwitz’s solution of the isoperimebric problem ........... 97 . 2 Reciprocity formulas ...................................... 98 3 . The Fourier integral and convergence in the mean ......... 98