text:
Pagina principale Boundary Value Problems of Mathematical Physics 2 Volume Set: v. 1&2
You may be interested in
Most frequently terms
You can write a book review and share your experiences. Other readers will always be interested in your opinion of the books you've read. Whether you've loved the book or not, if you give your honest and detailed thoughts then people will find new books that are right for them.
BoundaryValue Problems of Mathematical Physics SIAM's Classics in Applied Mathematics series consists of books that were previously allowed to go out of print. These books are republished by SIAM as a professional service because they continue to be important resources for mathematical scientists. Editor-in-Chief Robert E. O'Malley,Jr., University of Washington Editorial Board Richard A. Brualdi, University of Wisconsin-Madison Herbert B. Keller,California Institute of Technology Andrzej Z. Manitius,George Mason University Ingram Olkin, Stanford University Ferdinand Verhulst,Mathematisch Instituut, University of Utrecht Classics in Applied Mathematics C. C. Lin and L.A. Segel,Mathematics Applied to Deterministic Problems in the Natural Sciences Johan G. F.Belinfante and Bernard Kolman,A Survey of Lie Groups and Lie Algebras with Applications and Computational Methods James M. Ortega, Numerical Analysis: A Second Course Anthony V. Fiacco and Garth P.McCormick, Nonlinear Programming:Sequential Unconstrained Minimization Techniques F. H. Clarke, Optimization and Nonsmooth Analysis George F.Carrier and Carl E. Pearson, Ordinary Differential Equations Leo Breiman, Probability R. Bellman and G. M. Wing, An Introduction to Invariant Imbedding Abraham Berman and RobertJ. Plemmons,Nonnegative Matrices in theMathematical Sciences Olvi L. Mangasarian, Nonlinear Programming *Carl Friedrich Gauss, Theory of the Combination of Observations Least Subject to Errors: Part One, Part Two, Supplement.Translated by G. W. Stewart Richard Bellman,Introduction to Matrix Analysis U. M. Ascher, R. M. M. Mattheij, and R. D. Russell,Numerical Solutionof Boundary Value Problems for Ordinary Differential Equations K. E. Brenan, S. L. Campbell, and L. R. Petzold,NumericalSolutionof Initial-Value Problems in Differential-Algebraic Equations Charles L. Lawson and Richard J. Hanson, Solving Least Squares Problems }. E. Dennis, Jr. and Robert B. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations Richard E. Ba; rlow and Frank Proschan, Mathematical Theory of Reliability Cornelius Lanczos, Linear Differential Operators Richard Bellman,Introductionto Matrix Analysis, Second Edition Beresford N. Parlett,The Symmetric Eigenvalue Problem Richard Haberman, Mathematical Models: Mechanical Vibrations, Population Dynamics, and Traffic Flow *First time in print. Classics in Applied Mathematics (continued) Peter W. M. John, Statistical Design and Analysis of Experiments Tamer Basar and Geert Jan Olsder, Dynamic Noncooperative Game Theory, Second Edition Emanuel Parzen, Stochastic Processes Petar Kokotovic, Hassan K. Khalil, and John O'Reilly, Singular Perturbation Methods in Control: Analysis and Design Jean Dickinson Gibbons, Ingram Olkin, and Milton Sobel, Selecting and Ordering Populations: A New Statistical Methodology James A. Murdock, Perturbations: Theory and Methods Ivar Ekeland and Roger Temam, Convex Analysis and Variational Problems Ivar Stakgold, Boundary Value Problems of Mathematical Physics, Volumes I and II J. M. Ortega and W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables David Kinderlehrer and Guido Stampacchia, An Introduction to Variational Inequalities and Their Applications F. Natterer, The Mathematics of Computerized Tomography Avinash C. Kak and Malcolm Slaney, Principles of Computerized TomographicImaging R. Wong, Asymptotic Approximations of Integrals O. Axelsson and V. A. Barker, Finite Element Soltttion of Boundary Value Problems: Theory and Computation David R. Brillinger, Time Series: Data Analysis and Theory Joel N. Franklin, Methods of Mathematical Economics: Linear and Nonlinear Programming, Fixed-Point Theorems Philip Hartman, Ordinary Differential Equations, Second Edition Michael D. Intriligator, Mathematical Optimization and EconomicTheory Philippe G. Ciarlet, The Finite Element Method for Elliptic Problems Jane K. Cullum and Ralph A. Willoughby, Lanczos Algorithmsfor LargeSymmetric Eigenvalue Computations, VoLI: Theory M. Vidyasagar, Nonlinear Systems Analysis, Second Edition Robert Mattheij and Jaap Molenaar, Ordinary Differential Equations in Theory and Practice Shanti S. Gupta and S. Panchapakesan, Multiple Decision Procedures: Theory and Methodology of Selecting and Ranking Populations Eugene L. Allgower and Kurt Georg, Introduction to Numerical Continuation Methods Leah Edelstein-Keshet, Mathematical Models in Biology Heinz-Otto Kreiss and Jens Lorenz,Initial-Boundary Value Problemsand the NavierStofces Equations J. L. Hodges, Jr. and E. L. Lehmann, Basic Concepts of Probabilityand Statistics, Second Edition George F.Carrier, Max Krook, and Carl E. Pearson, Functions of a Complex Variable: Theory and Technique This page intentionally left blank Boundary Value Problems of Mathematical Physics Volume I Ivar Stakgold University of Delaware Newark, Delaware Society for Industrial and Applied Mathematics Philadelphia siam Copyright � 2000 by the Society for Industrial and Applied Mathematics. This SIAM edition is an unabridged, corrected republication of the work first published byThe Macmillan Company, New York, 1968. 1098765432 All rights reserved. Printed in the United States of America. No part of this book may be reproduced, stored, or transmitted in any manner without the written permission of the publisher. For information, write to the Society for Industrial and Applied Mathematics, 3600 University City Science Center, Philadelphia, PA 19104-2688. Library of Congress Cataloging-in-Publication Data Stakgold, Ivar. Boundary value problems of mathematical physics / Ivar Stakgold. p. cm. �(Classics in applied mathematics ; 29) Originally published: NewYork : Macmillan, 1967-1968, in series: Macmillan series in advanced mathematics and theoretical physics. Includes bibliographical references and indexes. ISBN 0-89871-456-7 (set :pbk.) 1. Boundary value problems. 2. Mathematical physics. I. Title. II. Series. QC20.7.B6 .S73 2000 530.15'535--dc21 99-088661 is a registered trademark.siam Table of Contents PREFACE TO THE CLASSIC S EDITION ix PREFACE xi 1. THE GREEN'S FUNCTION 1.1 TheString Subject to Transverse Loading I 1.2 The Dirac Delta Function 18 1.3 The Theory of Distributions 28 1.4 Preliminary Results on Linear Equationsof the Second Order 58 1.5 Boundary Value Problems 64 1.6 Alternative Theorems and the Modified Green's Function 79 Suggested Readings for Chapter 1 91 2. INTRODUCTION TO LINEAR SPACES 2.1 Functions and Transformations 92 2.2 Linear Spaces 96 2.3 Metric Spaces, Normed Linear Spaces, and Inner Product Spaces 99 2.4 Properties of a Separable Hilbert Spa 2.6 Transformations 139 2.7 Linear Transformationson En(c) 146 2.8 The Inverse of a Linear Transformation in Hilbert Space 165 2.9 The Spectrumof an Operator 180 2.10 Completely Continuous Operators 184 2.11 Extremal Properties of BoundedOperators 187 Suggested Readings for Chapter 2 190 vii .5 \01\02\03 135 116 viii CONTENTS 3. LINEAR INTEGRAL EQUATIONS 3.1 Introduction 191 3.2 The Neumann Series (Method of SuccessiveApproximations) 206 3.3 The Spectrum of a Self-adjoint Hilbert-Schmidt Operator 212 3.4 The Solution of the Inhomogeneous Equation with a Symmetric Hilbert-Schmidt Kernel 220 3.5 Extremal Principles 223 3.6 Approximations Based on Extremal Principles 226 3.7 Questions Relating to Continuity and Uniform Convergence�The Bilinear Series for the Kernel and the Iterated Kernels 236 3.8 Approximate Methods for the Solution of Integral Equations 241 3.9 Nonsymmetric Hilbert-Schmidt Operators 250 Suggested Readings for Chapter 3 258 4. SPECTRAL THEORY OF SECOND-ORDER DIFFERENTIAL OPERATORS 4.1 introduction 259 4.2 The Regular Boundary Value Problem 268 4.3 Introductory Examples of Singular Problems 283 4.4 The General Singular Problem 295 Suggested Readings for Chapter 4 322 APPENDIX A A.1 Static and Dynamic Problems for Strings and Membranes 323 A.2 Static and Dynamic Problems for Beams and Plates 326 A.3 The Equation of Heat Conduction APPENDIX B B.1 Bessel Functions 329 B.2 Wronskian Relationships 330 B.3 The Modified Bessel Function 331 B.4 The Behavior of Cylinder Functions at Zero and at Infinity 332 INDEX 333 Preface to the Classics Edition I am pleased and honored that SIAM is reissuing the two volumes of Boundary Value Problems of Mathematical Physics, originally published (1967-68) in Macmillan's Series in Advanced Mathematics and Theoretical Physics under the editorship ofMark Kac,C.C. Lin, and GeorgeUhlenbeck. Soon afterwards, Macmillan was bought by Crowell-Collier, who discontinued the series and the marketing of graduate-level books. Thus, despite widespread university adoptions, the two volumes went out ofprint in 1974, with the publishing rights reverting tome. As my research interests were changing, I decided to write a new book, Green's Functions and Boundary Value Problems, which was published by Wiley-Interscience in 1979,with asecond edition appearing in late 1997. This book covers someofthe topics ofthe Macmillanvolumes,but somewhat more abstractly and with greater emphasis on nonlinear problems. Inlight ofthemanydevelopments incomputational applied mathematics since 1968, theprospective reader maywell askwhether thepresent reissue is nomorethananexerciseinnostalgia!Toallaysuchunderstandable doubts, let me point out that the Macmillan volumes are still used�through special arrangements�at anumberofuniversities.Theyreport that oneofthe attractive features of these volumes is the large number of concrete examplesof boundary value problems for partial differential equations. For instance, there is a substantial treatment of the Helmholtz equation and scattering theory, subjects which playa central role in contemporaryinverse problems in acoustics and electromagnetic theory. More generally, an examinationof the table ofcontents of the original volumes confirmsthat the topicscovered do belong in the arsenal of today's engineer, physical scientist, and applied mathematician.Itmustbeacknowledged, however, thatthepracticalneedsof themodernworkplacedemand supplementary credentials,particularly inthe area ofnumericaland computational methods. Inconclusion,letmethankSIAMand especially RobertE.O'Malley,Jr.for undertaking this reissue. Although I have received encouragement from manysourcesovertheyears,Iparticularlywantto thankmylong-time friend, W. Edward Olmstead, for his continuing interest in these volumes. I.S. ix This page intentionally left blank Preface The present book isan outgrowth of a series of courses whichI taught,first at Harvard and later at Northwestern,to classes consistingprimarilyof graduate students in engineeringand in the physical sciences. Addressing myself to a similar audience, I have attempted here to present a modern, comprehensive treatment of boundary value problems. By definition, a boundary value problem consists of an ordinary or partial differential equation with associated boundary or initial conditions. When E. P. Wigner, a Nobel Laureate in Physics, spoke of "the unreasonable effectiveness of mathematics in the physical sciences," he must have had boundary valueproblemsinmind, for nearly every branch of the physical sciences has been enlightened by the mathematical theory of boundary value problems. Attention will bedirected to both analytic and approximate methods for the solution of linear boundary value problems. The principal analytic methods employ either an eigenfunction expansion or a Green's function. Their full understanding will require the development of a number of mathematical topics such as the theory of distributions, generalized solutions of differential equations, extremal principles, and the spectral theory of differential and integral operators. The subject matter can be divided, in a natural and convenient way, into the parts whichcorrespond to the two volumesof this work. The first volume is devoted primarily to the mathematical topics just cited and to their application to ordinary differential equations and integral equations. The second volume deals with boundary value problems for partial differential equations. I have tried to make the material accessible to a wide circle of readers by adopting a frankly pedagogical approach and by postulating modest mathematical prerequisites�standard coursesinadvanced calculusand inelementary complex variables. In keepingwith this objective, I have often stated theorems in somewhat lessthan their most generalform;occasionally I have also omitted xi xii PREFACE or merely sketched a proof which appeared to be beyond the scope of the book. UsuallyI havestarted withsimpleconcrete examplesbeforeproceeding to the general theory and, sometimes, the same topicisdiscussed at increasing levels of sophistication as progress is made; I have therefore sacrificed compactness for the sake of greater ease in understanding. Considerable effort has been spent in devisingexercises that are more than routine drill and yet do not require the time and ingenuity of a doctoral dissertation. In fact, these exercises are important extensions of the text; often I have either sketched the method of solution or else offered clues as to how to proceed. The serious reader is encouraged to look over all exercises with care and to solve a substantial fraction of them. There remains the pleasant task of acknowledging the valuable help which I receivedthrough the years from a great many students and colleagues;their constant encouragement provided an essential stimulusfor this book. I. S. Chapter 1 TH E G R E E N ' S F U N C T I O N 1.1 THE STRING SUBJECT TO TRANSVERSE LOADING To acquire some insight and confidencein the concepts of this chapter, we introduce these concepts in relation to the simple problem of determining the static deflection of a taut string subject to transverse loading. The analysis of this example will guide us in the formulation of a general mathematical theory which will then be applied to problems of greater physical interest. A string under tension T is stretched on the x axis between the fixed end points x = 0 and x = l. A given arbitrary transverse load f(x), measured in units of force/length, is applied to the string; a load of this type will be called a pressure to distinguishit from a load consisting of concentrated forces. We wish to calculate the deflection u(x) resultingfrom the applied pressure f(x). According to Appendix A, the differential equation must be satisfied in the open interval 0 < x < l. Since the ends of the string are fixed on the x axis, u(x) must also obey the two boundary conditions If f(x) is given as a continuous function in 0 < x < l, we will show subsequently that there exists one and only one function u(x) which satisfies the differential equation and the boundary conditions. As a simple example The requirement that the string does not break implies that u(x) is continuous in 0 < x < I, so the boundary conditions can equally well be characterized as where 1 THE GREEN'S FUNCTION [Ch.1 consider the case wherethepressure isconstant along the string, say/(x) = q. The general solution of the differential equation �Tu" = q is the boundary conditions yield A = 0, B =ql/2T, so that the deflection is There are important reasons, both mathematical and physical, for considering problems in whichthe applied pressure/(x)is not continuous, butmerely piecewisecontinuous;for instance, onecan easilyenvisagea situation inwhich the applied pressure is the one of Figure 1.1;that is, Of course there are those who will insist that the actual pressuremust rise sharply but continuouslyfrom 0top inthe neighborhood of x0 �e/2and must exhibit a similardecrease near x0 + e/2(asin Figure 1.2). Wedo not choose to refute this argument on physical grounds; it suffices to point out that the pressures of Figures 1.1 and 1.2are essentiallyequivalent,for theygiverise to nearly the same deflections(a preliminaryresult in this direction is found in Exercise 1.7), and, needlessto say, the calculationsbased on thediscontinuous model are muchsimpler. FIGURE 1.1 2 �1.1] THE STRING SUBJECT TO TRANSVERSE LOADING FIGURE 1.2 The deflection M(JC) corresponding to the pressure (1.1) can be obtained by integrating the differential equation; we find after applying the boundary conditions, Now u'(x) is an integral of the piecewise continuous function �f(x)/T, hence u'(x) is continuous and of course u(x) is continuous; by usingthese properties at x = xQ -r e/2and x = x0 + e/2, wecan calculate ^4, B, C, D, to obtain The deflection is shown in Figure 1.3. We note that u" is discontinuous at jc = Jt0 �e/2 and x = x0 + e/2, the two points at which/(x) is discontinuous. At every other point, �Tu" =/; and, of course w(0) = u(l) =0. In the light of the above example, we now formulate the mathematical problem for the deflectionof a string, withfixedend points, subject to a piecewise continuous transverse pressure. 3 THE GREEN'S FUNCTION [Ch.1 FIGURE 1.3 Let f(x) be piecewise continuous,0 < x <I. We seek a function u(x) whichiscontinuous, hasacontinuousderivative, andapiecewis continuous secondderivative,such that at all points of continuity off(x) moreover, u(x) must satisfy the boundary conditionsw(0) = w(/) = 0. Such a combination of a differential equation and boundary conditions will be called a boundary valueproblem or system. Uniqueness of the Solution Consider two solutions u^(x), u2(x) of system (1.2). Their difference w(x) is continuous and hasa continuous derivativeand a piecewisecontinuous second derivative. Moreover, vt>(0) = w(l) = 0, and, at all points of continuity of f(x), w(x) satisfies �Tw" = 0. In each subintervalin which/is continuous we have w" = 0 and therefore wf =constant. We can therefore say that w' is piecewise constant in 0 < x < /. Since w' is continuous, the constant values in the various subintervals must be the same and w' = A, 0 < x </. Every continuous function w(x) whose derivativeis A must be of the form Ax + B, where B is another constant. Applying the boundary conditions, we find w(x) =0, 0 < x </. Therefore, system (1.2)has at most one solution. Wewill see later that such a solution actually exists. Superposition Principle Let u^x), u2(x) be the deflections corresponding to the pressures fi(x), f2(x), respectively, in system (1.2). Then, if q and c2 are arbitrary constants, 4 �1.1] THE STRING SUBJECT TO TRANSVERSE LOADING 5 ) + C2u2(x) is the deflection corresponding to the applied pressure The proof of the principle of superposition is simpleandis left tothe reader. It isadirect consequence ofthelinearityof the differential operator � T(d2/dx2) and of the fact that the boundary conditions are linear and homogeneous. A discussion of linear operators is found in Chapter 2. The Green's Function for the String with Fixed EndPoints We wish to solve system (1.2) in as economical a manner as possible for a variety of applied pressures. To accomplish this task, it is useful to introduce an accessory problem where a concentrated unit force acts on the string at the arbitrary point x = x0. This force is indicated by a spike in Figure 1.4, the corresponding deflection being shown in Figure 1.5. Although normally careful to distinguishbetween forceand pressure (force/ length), we now try to express the unit concentrated force in Figure 1.4 as an equivalent pressure p(x). Since the total force acting in any interval (a, b) is the integral of the pressure, we must have The second condition implies that p(x) = 0 for x ^ x0; on the other hand, p(x) must beinfinite near jc = x0 so asto satisfy thefirstof theabove conditions. It isclear that/?(jt) isnot an ordinary piecewisecontinuousfunction.We introduce the new symbol <5(x) to represent the pressure (as a function of x) corresponding to a unit force at x �0. Thus 6(x �x0) will represent the pressure FIGURE 1.4 c1f1(x)+c2f2(x) THE GREEN'S FUNCTION [Ch. 1 FIGURE 1.5 corresponding to a unit force at x = x0. The "function" 8(x �x0)is known as the Diracdeltafunction. One can visualize6(x �x0) as a limit of piecewise continuous functions; for instance, if we let e-�0 and p -� oo with ep = 1 in (1.1), we obtain d(x �XQ). We tentatively list below some of the properties of the delta function (subject to careful reexamination in subsequent sections.) a If <p(x) is continuous at x = x0, This last relation is made plausible by approximatingd(x �XQ)by (1.1),with s small and p =1/e. Indeed if x0 is in (a, b) we have since cp(x) is continuous at x0 and the range of integration is a smallinterval about x0, 6 �1.1] THE STRING SUBJECT TO TRANSVERSE LOADING 7 The Dirac delta function and related symbolic functions are presented in a more serious mathematical framework in Sections 1.2and 1.3. Our accessory problem is the boundary value problem The solution of this system is denoted by g(x \XQ) and is known as the Green's function for �T(d2/dx2) with vanishing boundary conditions at x = 0, x = /. Physically, g(x \XQ) is the deflection at x due to a unit force applied at x0. This physical interpretation justifiesthe name influence function, which is often given to g(x \x0). REMARK. It is extremely important to understand the distinction between system (1.4) and the completely homogeneous system System (1.5) has no load or pressure of any kind applied to it; therefore, the unique solution for the deflection is u(x) = 0, as was shown earlier. System (1.4) differs from (1.5) in only one respect, the unit force applied at x = x0. The corresponding deflection is certainly not identically 0 in 0 < x < /. Therefore, changing the right side of the differential equation at the single point x = x0 radically alters the nature of the solution. It must be realized that the change made at the point x = x0 is enormous, the pressure being changed from 0 for system (1.5) to infinity for system (1.4). It will be shown that this implies that the solution of (1.4) has a discontinuousfirstderivative at x = x0. We now solve for the Green's function [system (1.4)]. Since the pressure is 0 whenever x ^ x0, the differential equation reduces to Therefore, and Since the deflection vanishes at x = 0 and at x = / wefindthat 5 = 0 and D = - Cl, sothat T To determine A and C we require first that the solution be continuous at x = xQ; thatis, The shape of the string for various values of A is shown in Figure 1.6, the different curves corresponding to concentrated forces of different magnitudes applied at x0 . Thecurve with A �0 corresponds to a zero force at x = x0. The requirement that there be a unit upward (that is, positive) force at x = XQ enables us to determine A. The forces acting on an infinitesimal portion of the string, directly under the load, are shown in Figure 1.7. The equilibrium conditions are 1. S+ cos 6+ = �_ cos 0_ = T, where T is the horizontal component of sion. Combining these equations, we find 7\tan 6+ + tan 0_) = 1,or FIGURE 1.6 THEGREEN'S FUNCTION [Ch. 1 �1.1] THE STRING SUBJECT TO TRANSVERSE LOADING FIGURE 1.7 As expected, the slope of the string isdiscontinuous at x = XQ, the amount of the discontinuity being determined by the magnitude of the concentrated force. In our case The Green's function is therefore Let us introduce the notation We can then write(1.6) in the more compact form Sincedgjdx isdiscontinuousatx = x0,gisnot a solution oftheclassdescribed in connection with system(1.2); but wecan consider#asa limitof solutions of this class! Setting ps = 1 in (1.1) and letting e->0, we see that the applied pressure degenerates into a unit concentrated force at x =x0; by inspection of (1.1a) we observe that the corresponding deflection approaches g(x \x0). From (1.6) one verifies that g(x \x0) is symmetricin the variablesx and JCG: The deflection at x due to a unitforceat XQ is equal to the deflectionat JCQ due to a unitforce at x. In Section 1.5it will be shown that thissymmetryproperty holds for a wide class of problems. 9 10 THE GREEN'S FUNCTION [Ch. 1 The function g(x \x0)given by(1.6)wasconstructed to satisfy the equivalent systems The compactness of the delta function notation is apparent. But let us not be deceived!Although (1.8a) has a strong intuitive appeal, it means no more and no less than (1.8b); in the theory of distributions, which will be developed in the next two sections, it will be possible to assign a direct meaning to (1.8a) and then derive (1.8b) from (1.8a). We have shown that if (1.8b) has a solution it is given by (1.6), but wehave not yet proved that (1.6) is actually a solution of (1.8b) [or of its equivalent form (1.8a)]. Weconclude by inspection of the formula for gthatthe boundary conditions, the continuity requirement at x0, and the jump requirements on the derivative at XQ are all satisfied. By straightforward differentiation, we check that -T(d2gjdx2) = 0 in 0 < x< x0 and x0 < x < /. The function g(x \XQ)exhibited in (1.6) is the oneandonly solutionof system (1.8b) [or of its equivalent form (1.8a)]. Uses of the Green's Function for the String We return to the problem of finding the deflection of a string subject to an arbitrary piecewise continuous pressure /(*). This deflection M(X) satisfies system (1.2), repeated below: Divide the interval (0, /) in n equal parts. The center of the Ath subintervalis called �k, and the length of each subintervalis A� = //�. The deflection of the string subject to thepiecewise continuous pressure/(x)isclosely approximated by the deflection of a string subject to smallconcentrated forces,/(^)A^,..., located at &,..., ��, respectively(see Figure 1.8). According to the �1.1] THE STRING SUBJECT TO TRANSVERSE LOADING 11 FIGURE 1.8 principle of superposition extended to concentrated loads, the deflection at x due solely to the small load at x = �k isg(x \&)/(�k)A�, and the deflection at jc due to all the small concentrated loads is (and A� -> 0), we conclude Formula (1.10) was obtained by an intuitive (or heuristic) argument. Let us now show that (1.10) is the one and only solution of (1.9). Since uniqueness has been previously established, it remains only to verify that (1.10) actually satisfies system (1.9). The boundary conditions of (1.9) are satisfied because g(x \�) itself vanishes at x = 0 and x = /. By differentiating (1.10) with care, we will show that �Tu" =f(x) at any point x where /(x) is continuous. Because the derivativeof g is discontinuous at jc = <!;, it is prudent to split the interval of integration for � into the two parts (0,x�) and (x+, /); thus, In the separate intervals, wemay usethe formulafor differentiation under the integral sign to obtain Since g is a continuous function and/is continuous at x, the last two terms cancel and We see that the precaution of splitting the integral into two parts was unnecessary, but it will be essential in calculating the second derivative. We have 12 THE GREEN'S FUNCTION [Ch. 1 From thejump property of dg/dx at x = �, weconclude that Hence Sinced2g/dx2 = 0intheintervals � < x and <f > x, it follows that and our proof iscomplete. The oneand onlysolution of system(1.9)is As an illustration, consider the case of a uniform pressure f(x) = q, 0 < x < /. Then In the integral from 0 to x, � is smaller than x and we use the second line of (1.6), whereas in the integral from x to /, � is larger than x and the first line of (1.6) must be used (with x0 = � of course). Thus wefind a result previously obtained by elementary means. We now come to a valuable interpretation of the relation between g(x\ ^) and the differential expression �T(d2/dx2). Consider the operator L = �T(d2/dx2), which operates only on functions u(x) with the properties: u"(x) is piecewise continuous, w(0) = M(/) = 0. Define the integral operator G, which operates on functions/(*), which are piecewise continuous, by the formula Then wehave L(//=/and GLu= u, where/and u are any two functions in their respective classes. This leads to the terminology:Gand L are inversesof each other. This point of view is stressed in Chapter 2. We have established that the solution of system (1.9)is �1.1] THE STRING SUBJECT TO TRANSVERSE LOADING 13 where g(x \�) is givenby (1.6). System (1.9) could also have been solved by methods from elementary differential equations. Indeed, the solution of the differential equation in (1.9) can be written by inspection as We now require w(0) = u(l) = 0 to determine Ct and C2. The advantages of the Green's function method are 1. The boundaryconditions are incorporated in (1.10). We do not, for each different /(A:), have to recompute two constants to obtain the solution of the system. 2. The form (1.10) lendsitself to the valuableinterpretation that the deflection due to the distributed load isthe sum ofthe deflectionsdue to manysmall concentrated loads. 3. If the load on the string is a generalized pressuref(x), consisting of a piecewise continuous pressuref0(x) and concentrated forces/!, ...,/� located at jc1}...,xn, respectively,then Hence (1.10) is valid even if/(x) contains concentrated forces. 4. The Green's function also enables us to solve the completely inhomogeneous system: We recall that the Green's function g(x \�) satisfies Multiply the first ofthese differential equations byg, the second byu, subtract, and integrate from 0 to / to obtain which reduces to Since g vanishes at the end points, and 14 THE GREEN'S FUNCTION [Ch. 1 we have Interchanging the labels x and �, and using the symmetry of g, It iseasy to verify that the above actually satisfies system (1.11); uniqueness is shown in the same manner as for system (1.2). 5. The Green's function is also used in converting an eigenvalue problem consisting of a differential equation and boundary conditions into an eigenvalue integral equation. As an example consider the free vibrations of a string withfixedend points. In Appendix A it is shown that the deflectionu(x, t) must satisfy where p is the constant mass density of the string (per unit length). We look for solutions of the form u(x, t) = X(x)e~i<ot. This leads to an ordinary differential equation for the space part X(x) of the solution: where A= pea2. For most values of the parameter A (whichis at our disposal), the only solution of system (1.12) is X(x) = 0 in 0 < x <I. Any exceptional values of A, for which nontrivial solutions of the system exist, are called eigenvalues. The corresponding nontrivial solutions X are called the eigenfunctions. System (1.12) is of the form (1.9), with ATplayingthe role of the inhomogeneous term. Using (1.10) wehave This is an eigenvalue integral equation entirely equivalent to system (1.12). Integral equations of this type are studied in Chapter 3. EXERCISES 1.1 The solution of the system �1.1] THE STRING SUBJECT TO TRANSVERSE LOADING 15 can also be obtained by adding the solutions of thetwopartiallyinhomogeneous systems and The first system is solved using the Green's function,the second using elementary methods. Compare with the solution obtained under (4) above. 1.2 Another physical problem which leads to the same systemstreated in the previous sections isthe one-dimensional heatconductionproblem(steady state). Consider a rod 0 < x </withasmall constantcrosssection A.Therod ishomogeneous, theflowofheatisinthexdirectiononly,andthe temperature M is constant over anycross section. Byassumption, there is no heat flow through the lateral surface, but heat canflowthrough the plane boundaries, and there maybe heat sourcesintheinterior.Weassume that the heat generated by these sources per unit time per unit length is a piecewise continuous function f(x) [/(*) istherefore the density of heat input]. Let k be the thermal conductivity. Then according to Appendix A, an energy balance on an infinitesimal section of the rod yields There are a numberof boundary conditions at x = 0,x = I, which can be associated with this problem. In (b) a and bcannot be assignedarbitrarilywithoutviolatingthe steadystate condition! To find the relation between a, b and/(jc), we integrate the differential equation from 0 to / to obtain Thus the following relation must hold: 16 THE GREEN'S FUNCTION [Ch. 1 which guarantees that the heat flowingoutthrough theplane boundaries is equal to the net heat input in the rod from the interior heat sources. It can be shown that if this relation issatisfied,a solution actually exists but is determined only to an additive constant. (c) Heat transfer from the ends into a surrounding atmosphere at 0 temperature: where 6 is a given constant describingthe heat-transfer properties from the ends of the rod into the surroundings. (d) Acombination oftheaboveconditions, e.g., atoneendthe temperature is given,at the other the heat flow isspecified. 1.3 In the preceding heat-conduction problem, a more general differential equation is obtained if we assume: (a) thatthe crosssectionA is a slowly varyingfunctionofx (slowlyvaryingsoasnot tospoilthe one-dimensional feature of the problem), and/or that k isafunction of*; and (b)that heat transfer is allowed from the lateral surface into the surrounding atmosphere, the amount of heat transferred per unit timethrough the lateral surface between x and x + Ax being f*(x)u(x)Ax. Assuming that the temperature over a cross section isconstant, the appropriate differential equation for the temperature is Show that if f(x) = 8(x ��), that is, if a concentrated unit source is located at x = �, then (Take a heat balance for a small element of the rod at x = �.) 1.4 Construct the Green's function for the systems 1.5 Consider the motion of a particle alonga straight line. The force on the particle is along the line, and the particle starts from rest at the origin. �1.1] THE STRING SUBJECT TO TRANSVERSE LOADING 17 The displacement w(f) satisfiesNewton's law: and the initial conditions The Green's function g(t\t0) satisfies Show that Hence We observe that the solution at time t depends onlyon the forceacting before time t. 1.6 (a) For the string problem (1.6)showthat g(x\ JCG) is positive and that the maximumvalue ofg over the square 0 <x, x0 <I, is 1/4T. (b) Consider system (1.2) for two "nearly equal" pressures f�(x) and /2(x)satisfying Show that the corresponding deflections satisfy the inequality 1.7 Consider again the string problem (1.2). Let us define as the numericalmeasureofthe "separation" betweenthe two functions f^x) andf 2(x). Since d(fl,f2} is just the area between the graphs of/ and/2, the definition appears reasonable. In thissense itisquiteclearthat the separation betweenthe pressures in Figures 1.1 and 1.2is small. 18 THE GREEN'S FUNCTION [Ch. 1 Show that if d(f^,/2) < e, then 1.8 Consider a beam subject to the transverse pressure /(*). It is shown in Appendix A that the differential equation for the deflection u(x) is where E and / are certain constants. (a) Find the Green's function for a beam of length / whose ends are clamped. (b) Find the Green's function for a beam of length / whose ends are simply supported. (HINT: The deflection and its first two derivatives are continuous under the concentrated load, but d3g/dx3 is discontinuous under the load.) 1.2 THE DIRAC DELTA FUNCTION Introduction In various branches of physics one encounters sources which are nearly instantaneous (if time is the independent variable) or almost localized (if the independent variable is a space coordinate). To avoid the cumbersome study of the detailed functional dependence of these sources, one would like to replace them by idealized sources which are truly instantaneousor localized; such idealized sources are said to beimpulsive orconcentrated.Typical instances of such sources are the concentrated forces and moments in solid mechanics; the point masses in the theory of the gravitational potential; the impulsive forces and pressures in acoustics and in impact mechanics; and the point charges, dipoles, and multipoles in electrostatics. We wish to develop a mathematical theory of concentrated sources; we expect two thingsfrom this theory: (1)aclear and unambiguous wayto specify mathematically a concentrated source, and (2) a method for the calculation of the response to a concentrated source, that is, a method for interpreting and solvingdifferential equations whoseinhomogeneous termisa concentrated source. In the present section weconcern ourselves only with the first of these two aspects of the theory; the program will be fully developed in Section 1.3. Although it is natural to try to specify a concentrated source as the limitof continuous source distributions, such an approach quickly leads to mathematical difficulties. These difficulties are overcome by adopting an alternative point of view in which a concentrated source is characterized byits " action" on suitable accessory functions. �1.2] THE DIRAC DELTA FUNCTION 19 The Delta Function as the Limit of Sequences of Continuous Functions To fix ideas, we begin with a description of a point charge as the limiting case of continuous charge distributions on a line. Suppose that charge is distributed continuously on the real axis with the charge density where kisa positive integer. For large k, sk(x) isvery small, except for a peak of height k/n at the origin (see Figure 1.9). The total charge to the left of the point x is where rk(x) is known as the cumulativecharge distribution and is shown in Figure 1.10. We note that Hence the total charge on the lineisalways 1, independent of the indexk. It is clear that FIGURE 1.9 20 THE GREEN'S FUNCTION [Ch. 1 FIGURE 1.10 As k increases, the total charge (which is always 1) is pushed relentlessly toward the origin.Ifwetentativelywrites(x) = lim sk(x), thens(x) isthe charge fc-�00 density due to a positive unit charge located at x = 0. The limitingdensity s(x) isnot an ordinary function, since it isinfiniteat x = 0.In spite ofthis, itis convenient to introduce the symbolic function 6(x) to represent the limiting density for all x. It appears that 6(x) vanishes for x ^ 0 and is positively Too infinite at x = 0 in such a waythat d(x)dx = 1. (As Exercise 1.11 shows, J-oo these ideas are not precise and will have to be revised.) We note that the cumulative charge distribution rk(x) strictly approaches r(x), the cumulative distribution for apositiveunitchargelocated at x = O.f Thesymbolicfunction d(x) represents the charge density for a unit charge placed at jc = 0. The corresponding cumulativecharge distribution r(x) is knownas the Heaviside function and will henceforth be denoted by H(x); we observe that whereas <5(*) is a symbolicfunction, H(x) is a respectable piecewise continuousfunction. Wealsointroducethe symbolicfunction d(x �JCG)to representthecharge f There may be some question as to the correct value of the cumulative charge distribution at x = 0. One must define what is meant by the total charge to the left of x �0; do we include the charge at the origin or some part of it? There is no sensible physical criterion available, and infact,thevalueof r(x) at x =0is ofno consequence. Theimportant fact is that r(x) has a.jump [r(Q+) �r(Q�)] at the origin; it is this jumpwhich determines the charge located at x = 0. �1.2] THE DIRAC DELTA FUNCTION 21 density for a unit charge placed at x �XQ ; the corresponding cumulative distribution is clearly H(x �JCQ). EXERCISES 1.9 Consider the sequence of functions sk(x) = (k/^/n')e~k2x2. Sketch a few of these functions and the corresponding cumulativedistributions.Show that, in the light of all the properties so far mentioned, it is consistent to state 1.10 Same as Exercise 1.9 for the sequences 1.11 The sequence should serve to dispel some of the most cherished superstitions about the delta function. A glance at the graph of sk(x) shows that for k large the charge distribution is equivalentto a unit (positive) charge at jc = 0, yet lim sk(G) = �oo! Thus we must abandon the idea that <5(jc) is +oo at x =0. The reader should have no difficulty in constructinga sequence for which sk(0) = 0 yet lim sk(x) = d(x). 1.12 Consideran experimentwhoseoutcomecan be described by a singlereal number jc. The cumulativeprobabilitydistributionF(x) isdefined as the probability that the outcome be a number smaller than x. The function 22 THE GREEN'S FUNCTION [Ch. 1 /(*) = dF/dx isthe probability density. Notethat for anyexperiment we Too have F(� oo)= 0, F(+ oo)= 1, f(x)dx = 1. The situation isexactly J �oo the sameas whena unit of charge is spread on the x axis (except that the probability density must be nonnegative). The advantage of the delta function notation isthat it enables ustotreatbothdiscreteandcontinuous probability distributions in the sameframework. Take the particular example of the toss of a fair coin. The outcome "heads" will be assigned the value x = 0, the outcome "tails" the value x = 1. Show that The Sifting Property of the Delta Function Wehaveseenabove (Exercises 1.9, 1.10,and 1.11)that a varietyof sequenc have the "limit" <5(x), yet these sequences can behave quite differently at x = 0. There is a much more satisfactory way of characterizing the delta function and the sequences whichapproach the delta function. Let us return to the sequence sk(x) �k/n(l + k2x2) and let us define the action of sk(x) on the function q>(x) by the formula We regard Ak as a weightedaverage of sk(x) with respect to the weighting function <p(x). Here q>(x) is an arbitrary function, continuous at the origin, and behaving well enough at infinity to ensure the existence of the integral. For simplicity,let <p be bounded; that is, \<p\ < M for all x, �oo < x < oo. Theorem. If cp(x) is bounded, integrable, and continuous at x = 0, Proof. We first observe that where �1.2] THE DIRAC DELTA FUNCTION 23 To show that lim rjsk dx = 0, we must prove that for any e > 0 there k-�oo J-<*> exists an index N such that Let A be a positive number (to be chosen below) and divide the interval from � oo to oo into three parts so that We .examine first the integral 73; let the maximum of |?/| in �A <x <A be denoted byp(A). Then Since */(()) = 0 and y(x) is continuous at x = 0, wehave limp(A) = 0. Thus, for any e> 0, there exists A so small that and this holds independent of k. With A so chosen, it remains to show that |/x + 72| is small for sufficiently large k. Since \(p(x)\ < M and |?/| < |<p(x)| + I<P(0)|, M is bounded in � oo< x < oo, say |�j| < P. Then With ^4 fixed, lim (2/rc) arctan A:^4 = 1, so that we can find AT such that With this choice of N we have This shows that lim r\sk dx = 0 and completes the proof of (1.14). Thus, as /c approaches infinity, the action of kfn(l + k2x2) on cp(x) approaches the number ^(0). Since, as k increases, we regard k/n(l + k2x2) 24 THE GREEN'S FUNCTION [Ch.1 as an increasingly good approximation to S(x), it is natural to define d(x) as the " function "which hasthe action <p(Q). It willbe shown in the next section that no piecewise continuous function can have the action (p(0), so that 6(x) is a new mathematical object known as generalized or symbolic function. When d(x) acts on suitable accessory functions q>(x), it picks out the value of at x = 0; thus symbolically, Property (1.15) is known as the sifting property of the delta function. DEFINITION. A sequence sk(x) is a 6 sequence if for all (p(x) which are continuous and bounded �oo< x< oo.f If sk(x) is a 8 sequence, it is clear that Since, for large k, sk(x �x0) approximates 6(x �jc0), we define 6(x �Jt0) as the "function" which has the action (P(XQ). Thus 6(x �x0) has the sifting property We shall write EXERCISES 1.13 Show that the sequences in Exercises 1.9, 1.10, and 1.11 have the property t Occasionally, as in Exercise 1.18,one encounters sequences whichhave the property for a somewhat smaller class of accessory functions <p(x). Such sequences will still be known as 8 sequences. �1.2] THE DIRAC DELTA FUNCTION 25 for any <p(x) bounded, integrable, and continuous at x = 0. Therefore, all these sequences are 6 sequences. 1.14 Let a charge +k be located at x = s and a charge �A: be located at x = �e. The product 2ke is known as the dipole moment of the charge configuration. Now suppose that the charges are brought together in such a way as to keep the dipole moment equal to 1 (that is, we let � -> 0 and k -� oo so that 2ks = 1); the resulting charge configurationis known as a dipole source of unit moment. Such a dipole source can be approximated by a continuous charge distribution on the real line, as follows. Let tk(x) = �dsk/dx, where sk(x) = k/n(l + k2x2), so that tk(x) = 2k3xjn(l + k2x2)2; wehave sketched tk(x) in Figure 1.11 for a large value of k. We have FIGURE1.11 26 THE GREEN'S FUNCTION [Ch. 1 if neither a nor b is 0, each term approaches 0 as k ->� oo, and the total charge in any such interval goes to 0 as k -�oo. On the other hand, \tk(x)dx-+ +00 and j tk(x)dx-+ �oo; thus, for large k, we have a JO J ~~� large positive charge just to the right of the origin and a large negative charge just to the left of the origin. Further, the first moment of the Too charge distribution about the origin is xtk(x)dx = 1. Hence, for J � oo large k, tk(x) approximates a dipole ofunitmoment located at x = 0.We note with some surprise that for each x, limtk(x) = 0, so that the fc-�oo pointwise convergence to 0 does not tell the whole story. To characterize the behavior of tk(x), consider its action on suitable functions (p(x): Show that lim Ak[cp] =(d(p/dx)(0). This last property will be used to characterize a unit dipole at x = 0. 1.15 Lets(x) bea nonnegative function satisfying | �� s(x)dx = 1. Show tha J �00 ks(kx) is a 6 sequence. [The proof is practically identical to the one used for the special case s(x) = l/n(l + x2), which corresponds to (1.14).] Exercises 1.9 and 1.10 are illustrations of the above theorem. 1.16 (a) Let sk(x) satisfy the conditions sk(x) > 0, f �� sk(x)dx = 1, and J-OO if a and 6 are both positive or both negative; if a *s negative and b positive. Show that sk(x) is a 5 sequence. Note that the nonnegativitity of sk(x) is needed to exclude the possibility of dipole contributions to sk(x); the sequence satisfies all conditions except nonnegativity and is not a S sequence. [See Exercise 1.18(b)for an exampleofa sequenceof oscillatory functions which is a d sequence.] (b) The result in (a) implies Show that the convergence is uniform if <p(x) is uniformly continuous � 00 < JC <00. �1.2] THE DIRAC DELTA FUNCTION 27 1.17 The Weierstrassapproximation theorem. Consider the sequence using Stirling's formula, k\ ~ kke~k^/2nk, valid for large fc, show that (b) The sequence sk(x) =fk(x)/Ik satisfies the conditions bf Exercis 1.16, so Let (p(x) be a continuous function, �oo< x < oo, which vanishes identically for |x| > �. [Note that this implies <p(\) = q>(��) = 0, and that cp(x) is uniformly continuous �oo<x<oo.] Show that �� sk(x ��)(p(x)dx reduces to a polynomial in � when �\ < � < J �00 Therefore <p(�) has been uniformlyapproximated by a sequence of polynomials over the interval �\< �<$. (c) Let^(x)becontinuous �%< x< ^;show that\f/(x) canbeuniformly approximated by polynomials. [HINT: Consider <p(x) = i/^(x) �x[\j/($) >K�1)1 �i*Ki) �i>A(�i)-] Let i^(x) be continuous over the finite interval a<x <b; show that i^(x) can be uniformly approximated by polynomials over this interval. 1.18 (a) A sequence rk(x) is an H sequence if and �� rk(x)<p(x)dx -�� �� (p(x)dx. The latter property can be obtained J � oo JO from the pointwise convergence of rk(x) to H(x) by adding the requirement that |rk(x)| < r(x), wherer(x) isintegrable over every finite interval. Show that (l/7i) f* (sin ku/u)du is an // sequence. [HINT: Let ku = v, J- oo and show that f (sin v/v)dv < " (sin v/v)dv.] (b) Let sk(x) = sin kx/nx. Let (p(x) be differentiable with <p'(x) continuous and bounded, and <p(� oo)= 0. Show,by integration by parts, that Thus jfc(x) may be regarded as a <5 sequence. Observe that ^(x) changes signs an infinite number of times in (�00, oo). 1.19. Show that the sequence in Exercise 1.11 is a d sequence. 28 THE GREEN'S FUNCTION [Ch.1 1.3 THE THEORY OF DISTRIBUTIONS Functions Viewedas Linear Functional Consider a real-valued, continuous function y =/(*) defined on the real axis �oo<jc<oo.fltis customary to visualize such a function as a table of values which lists the values of the independent variable x in the first column and the corresponding values of the dependent variable y in the second column. Although this table cannot be written in its entirety (the variable x assuming an infinitenumber of valueseven in a finite interval), we have no difficulty in regarding this conceptual table as a characterization of the function/. Since a definite valueof y is associated with each point x on the real line,/is being viewed as a point function. We can equallywell characterize/by a completely different table of values. f �� Consider </, <p> = f(x)q>(x)dx for a variety of accessory functions </>(*) J �00 to be further specified below. The notation </, <p> is used to remind us of the interpretation of the integral as an innerproduct (also knownas a dot or scalar product; see Chapter 2). </, (p> is termed the actionof/on q> and is nothing more than the weighted average of/(x) with respect to the weightingfunction <p(x). We have thus constructed a new table of values which lists various accessory functions <p(x) in the first column and the corresponding actions </, <p> in the second column. Such a correspondence, which assigns to each function (p(x) an unambiguousreal number </, <p>, is known as a. functional [the independent variable being a function <p(x) and the dependent variable the number </, p>]. If the functional is to characterize the point function f(x) in an unambiguousmanner, it is essential to choose the class of accessory functions q>(x) so that the two different continuous functions/(jc) and/2(Ar) generate different functionals (that is, there must exist at least one <p for which the actions </1} <p> and </2, <p> are different). REMARK. The characterization of functions as functionals on a space of accessory functions isnot as unnatural as itfirstappears. In fact, the reader is probably familiar with the following examples, which illustrate the same idea. 1. If mass is continuouslydistributed on a finite portion of the real line, the mass density f(x) can be characterized by its moments t Strictly speaking, the function is the rule that establishes the correspondence between x andy; thus / isthefunction, whereas /(*) isthenumerical value assumed by/ at x. Here we shall adopt the classical notation of letting the symbol f(x) stand either for the function or for the value assumed by the function at x; the interpretation to be made should be clear from the context (seeSection 2.1 for more about functions). �1.3] THE THEORY OF DISTRIBUTIONS 29 The functions x" are the accessory functions. The specification of all the moments determines /(x). 2. If /(x) is a differentiable function in 0 < x <n, and /(O) = f(n) =0, then /(x) can be characterized by its actions on the accessory functions (2/7i) sinx, (2/n) sin2x,.... If we let bn = </, (2/n) sin�x> = (2/7t) | "/(x) sin�x dx, we can recover /(x) from {��} bythefam sine series: 3. Whenever a physical measurement is made on a function /(x) of time or position (such a pressure, velocity, voltage, etc.), it is never the instantaneous or local value of / which is measured but rather some average of / over a short time or space interval. Thus the observed value of / at x0 is really of the form I<p(x0 , x)f(x)dx, where <p(x0 , x) is nearly constant over a short interval about x0 , and vanishes outside this interval. We have seen that we can characterize/either as a point function or as a functional on a space of accessory functions. This second point of view will enable us to extend the notion of " function "in a very natural manner. The accessory functions which are useful in this connection are the so-called test functions defined below. Test Functions DEFINITION. A test function (p(x) is a function which is infinitely difTerentiable (that is, differentiable to every order) in �oo< x < oo, and which vanishes outside some finite interval (which may be different for different test functions). The space of all test functions will be denoted by D. REMARKS. 1. It is not entirely evident that there exist nontrivial test functions. Suppose that cp(x) is positive in a < x < b and vanishes outside this interval; according to the definition of a test function, every derivative of <p must vanish at x = a and x = b; it may at first be difficult to see how a function which is so "flat" at x = a and x = b can climb to positive values inside the interval. The following example shows that test functions do exist: This even function is sketched in Figure 1.12. It is easily seen that <p(x) is infinitely differentiable except possibly at x = + 1 . Because of the evenness of <p(x), it is enough to examine the point x = 1. Since 30 THE GREEN'S FUNCTION [Ch.1 FIGURE 1.12 lim exp [l/(x2 �1)] = 0, q>(x) is continuous at x = 1. A simple calculate-*! tion, based on the fact that, for any �, shows that all left-hand derivatives of <p vanish at x = 1; hence every derivative of <p vanishes at x = 1. 2. If cp(x) is the test function in the above example, then \J/(x) = (p{(i/e)(x �XQ)} is also a test function which is positive in x0e < x < x0 + B and vanishes outside this interval. 3. If <Pi(x) and q>2(x) are test functions, so is c^, + c2<P2 f�r any real numbers c, and c2 ',we can therefore say that the space D of test functions is a real linear space [see (2.2)]. 4. Derivatives of any order of a test function are again test functions. Convergence in the Space of Test Functions We introduce an extremely stringent form of convergence in the space of test functions. The usefulness of this definition will become apparent in what follows. Let <pn(x) be a sequence of test functions; we write q>n(x) -> 0 in D if 1. All the test functions tpn (x) vanish outside a common finite interval. 2. lim <pn(x) = 0 uniformly in �oo < x < oo, and, for every positive /2-;>-00 integer k, lim \dk<pn(x)/dxk} = 0, uniformly in �oo < jc < oo. �1.3] THE THEORY OF DISTRIBUTIONS 31 Distributions DEFINITION. A linear functional on the space D of testfunctions is a rule t which assigns to each test function q>(x) a real number <f, (p>_with the property for arbitrary real numbers ct and c2 and arbitrary test functions ^ and <p2. It follows that, for any linear functional, <f, 0> = 0 and DEFINITION. A linear functional on D is continuous if and only if O� <Pn> ~*0whenever</>� -� 0in D (here <f, <pn> -*0 refers of course to ordinary convergence of a sequence of numbers). A continuous linear functional on D will also be known as a distribution. The space of all distributions will be denoted by D'. By a slight abuse of terminologywe shall also refer to </, <p> as the distribution, when in fact <f, <p> is actually the value assumed by the distribution t at (p (the number <f, <p> will also be called more picturesquely the action of f on (p). Let fj and t2 be any two distributions and let cv and c2 be arbitrary real numbers; wethen define a newdistribution t = clti + c2t2 by the formula It is easily checked that t is in fact a linear functional on D and that t is continuous, so that a distribution has been defined. Thus if ^ and t2 are in D' so is cltl + c2t2; hence D' is itself a linear space. The simplest kindofdistributions (but not the onlykind)arethose generated by locally integrable functions. A function/(x) is said to be locallyintegrable if I \f(x)\dx exists for every finite interval 7.| The class of locally integrable functions includes allpiecewisecontinuous functionsand other functions such as I/A:1"*, where e > 0. An important property of locally integrable functions should be noted: Let 7l5 ...,/�,... be a sequence of nested intervals (thatis, each interval is included in the preceding one) whose lengths approach 0; then lim \f(x)\dx = 0. Consequently the class of locally integrable funcB-+00 J/� tions does not include the delta "function." t The integral used in this definition is the Lebesgue integral (some of whose properties are discussed in Chapter 2),but the reader who isunfamiliarwith this concept may interpret the integral in the Riemann sense for most of what follows. 32 THE GREEN'S FUNCTION [Ch.1 Theorem. Each locally integrablefunctionf(x) generatesa distributionvia the definition Proof. It is clear that a linear functional on the space oftest functions has been defined; to prove continuity, let q>n(x) be a sequence of test functions such that <pn(x) -> 0 in D. Then where Mn is the maximum of \<pn\ and / is the common interval outside of which all the (pn(x) vanish. The fact that (pn(x) -�0 in D implies the uniform convergence of (pn(x) to 0, so that Mn -> 0; hence </, <pny -� 0, and </, <p> is a distribution. The constant point function cislocally integrable and definesa distribution by the formula The samesymbolcwillbeused for the numberc,theconstant point function c, and for the distribution generated by the constant point function c. Do different locally integrable functions generate different distributions? The followingtheorem is a preliminary result in this direction. Theorem. The only continuousfunction f(x) for which </, <p> = 0/or every testfunction cp isf(x) = 0, �oo < ;c < oo. Proof. Supposef(x0) > 0;then there existsan intervalx0 �e< x< x0 +s in which/(jc) is positive. We have seen that there is a test function cp(x) which is positive in x0 �e < x < x0 + e and vanishes outside this interval. For this test function we have </, <p> > 0, which contradicts the hypothesis. A similar contradiction would have been obtained if we had assumed/(JCG) < 0. COROLLARY. Two different continuousfunctions generatedifferent distributions. Theabove theorem and itscorollary do not hold (without suitable modification) for locally integrable functions. Indeed the functions/^x) = 0 and generate the same distribution (the0 distribution). For the purposes of the present theory (and indeed for nearly all other important applications) such functions areregarded asindistinguishable.Thefollowing definition makesthe �1.3] THE THEORY OF DISTRIBUTIONS 33 idea precise: Two locally integrable functions/i(x) and/2(x) are equalalmost everywhere if \ft �f2\dx = 0 for every finite interval /. Theorem. Two locally integrable functions, equal almost everywhere, generate the same distribution; if they are not equal almost everywhere they generate different distributions. The proof of the theorem is omitted (only the second part presents any difficulty). From now on weshall take the point of view that functionswhichare equal almost everywhereare the same function. With this understanding, different locally integrable functions generate different distributions. So far, all wehave shown is that locally integrable functionscan be characterized as distributions by (1.17). We now make the important observation that there exist distributions which cannot be represented in the form (1.17) for any locally integrable f(x). Consider <?,(/>> = <p(0), which is clearly a linear functional; furthermore,if (pn(x) -> 0in D, thenthe sequenceof numbers <pn(0) certainly approaches 0 and hence <f, <p> is continuous and is a distribution. This distribution is known as the delta distributionand will be written <<5, (jt>>. We now show that <<5, <p> is not generated [via (1.17)] by a locally integrable function/^). If such a locallyintegrable functionexisted,wewould |*00 have f(x)<p(x)dx = G>(0) for every test function. Consider the test J-oo functions We have <pa(0) = Ife, \(pa(x)\< l(e; hence Since f(x) is locally integrable, we have lim I \f(x)\dx = 0 and lim J* f(x)(pa(x)dx = 0;ontheotherhand,wemust have P� f(x)<pa(x}dx = <pa(Q) = l/e. We have arrived at a contradiction, and consequently there cannot exist a locally integrable function /(jc) such that f(x)(p(x)dx � cp(0)J-co for every test function. DEFINITION. A distribution is regular if it can be generated by a locally integrable function through (1.17). All other distributionsaresingular. a 0 34 THE GREEN'S FUNCTION [Ch. 1 REMARKS. 1. Whether or not a distribution t is regular, we shall f 00 always feel free to write </, <p> = \_ t(x)<p(x)dx. If t is regular, the preceding formula is valid with t(x) a locally integrable function; if t is singular, the formula is purely symbolic and t(x) is known as the symbolic or generalized function corresponding to the distribution /. As we progress we shall pay less and less distinction to the difference between t(x) and <f, <p>. Any property ascribed to t(x) is, by definition, the direct consequence of the same property holding for <f, <p>. For instance, if two distributions are equal (that is, if <f1} <p> = </2, <p> for every <p), we say that t^x) = t 2. A distribution is a generalization of the notion of a real-valued function of a real variable. A similar generalization is easily made for complex-valued functionsof a real variable(this, of course, has nothing to do with functions of a complex variable). The test functions (p(x) are now infinitely differentiate complex-valued functions of the real variable x; this new space of test functions is denoted by D(c\ and the same definition is used for convergence -to 0 in Z)(c) as in D. A distribution t is now a complex-valued functional on D(c) satisfying the conditions for any complex numbers ct and c2. whenever (/>�-�0 in D(c). EXAMPLES Example 1. The Heaviside distribution <//, <^>> is generated by the locally integrable function Thus </f, q>y is a regular distribution, and The definitionused for //(*) at x = 0 isarbitrary; any otherfinitevalue could have been used for the definition of H(Q), and none of our results would be affected. Example 2. The rule which assigns to each test function cp(x) its value at x = 0 is the delta distribution (which has been shown to be singular). We write �1.3] THE THEORY OF DISTRIBUTION The generalized function <5(x) represents the charge density for a positive unit charge located at x =0. Example 3. The rule that assigns to each test function <p(x) its valueat the fixedpoint x =� is a singular distribution, The generalized function <55(x) represents the charge density for a positive unit charge located at x = <!;. We note that <50 and <5 are the samedistribution; furthermore, it will be shownbelow that ^(x) can be identifiedwith <5(x �<!;). Example 4. Let a positive charge of magnitude 1/e be located at x = e/2, and a like negative charge at x � �s/2. The limiting charge configuration, as e-�0, is said to be a dipole of positive unit moment located at jc = 0. The action of the limitingcharge configurationon a test function q>(x) is The singular distribution <p'(0) is known as the dipole distribution. We will show later that the corresponding generalized function can be interpreted as -*'(*)� Operations with Distributions 1. Translation. If /(x) is locally integrable, it generates the distribution </(x), (p(*)> = I �� f(x)<p(x)dx, whereas f(x � a) generates the distrib J - co The above relation suggests the definition to be used for t(x �a), where t(x) is an arbitrary generalized function. Thus, we define t(x �a) from This definitionmakes sense, since <p(x + a) isa test function and the action of t on <p(x + a) is thereforemeaningful. Usingthisdefinition wehave Therefore, <5(x �a) = da(x); henceforthwe use the notation <5(x �a) instead of<5a(x). 36 THE GREEN'S FUNCTION [Ch.1 2. Scale Expansion. If/(x) is locally integrable, so is/(ox) for any real number a. The distribution corresponding tof(ax) is Thus This suggests that, for any distribution t, we define f(ax) from We observe that <5(�x) is the generalized function corresponding to Therefore <5(�x) = (5(x), and ^(x) is an even generalized function. 3. Multiplication by a Locally Integrable Function. If t(x) is a generalized function and/(x) is locally integrable, it is natural to definef(x)t(x) from Unfortunately/<p is not necessarily a test function, so that the action of t on f<p may not be defined. If /(x) is infinitely differentiable, f(p will be a test function, and the definition (1.18) is used for//. If c is the constant point function, ct is always defined and <cf, <p> = <?, c</>> = c<f, <p>, the last equality being a consequence of the linearity of /. On the other hand, wehavepreviouslydefined aconstant multipleofadistribution by (1.16), namely <cf, <p> = c</, cp>; we note with satisfaction and relief that the two definitions coincide. As another example, consider /(x)<5(x), where /(x) is infinitely differentiable; then, </�, <p> = <(5,/<p> =/(0)<p(0). Thus/(x)<5(x) is the samegeneralized function as/(0)<5(.x). It is clear that this definition of/(x)<5(x) can be used even in the case where/(x) is onlycontinuous at x = 0. If/(x) is infinitely differentiable and t(x) is the dipole distribution [that is </, <p> = (p'(0)], we have from (1.18), </V, <p> =</, (</A/x)0�Uo =/'(0)<p(0) +/(0)<p'(0), or /(x)/(x) = /'(0>5( This definition can be used as long as /(x) is continuously differentiable at x = 0. We observe that the product of two distributions cannot alwaysbe defined; for instance, <52(x) = <5(x)<5(x) cannot be defined in a useful way as a distribution. �1.3] THE THEORY OF DISTRIBUTIONS 37 Values of a Distribution on an Open Interval The concept of the value of a generalized function at a point x is not particularly useful (and in some cases not even meaningful). On the other hand, we shall have occasion to speak of the valuesof a generalized functionover an open interval; in particular, wemay be able to identify a generalizedfunction with a locally integrable function over an open interval. DEFINITION. Two generalized functions ^(jc) and t2(x) are said to be equalovertheopenintervala < x < bif </l5 <p> = <72, (p> for alltestfunctions (p(x) which vanish outside the interval a < x < b. As an example we see that 6(x) = 0 in 0 < ;c < oo and �oo < x <0. Differentiation of Distributions If f(x) is a differentiable function whose derivative is locally integrable, (* defines the distribution Integrating by parts and recalling that (p(x) is 0 outside a finite interval, we find that Thus the action of/' on <p is equal to the action of/on the test function �9'. For any distribution t, the derivative t' is defined by the formula That (1.19) does in fact definea linear functional on D follows from To show that t' is a distribution, we must prove further that whenever <ptt -*0 in D then <f', cpn >-*� 0. Now if cpn -> 0 in D, then cp'n -� 0 in D and, since f is a distribution, <f, (p^>-�0. The definition (1.19) then shows that <?', (/>�> ->0. By repeated application of (1.19), we obtain Since the �th derivativeof a test function isalso a test function, one caneasily see that (1.19a) defines a distributionknownas the nth generalizedderivative of t. This distribution (or its corresponding generalized function) will be written t{n) or d"t/dx". 38 THE GREEN'S FUNCTION [Ch.1 EXAMPLES Example 1. Let H(x) be the Heaviside function; then, By (1.19) we have Therefore <#', <p> = <<5, <p> and we write An alternative way of defining dH/dx is by the formula Calculating the distribution on the right side, we obtain Thus the two natural definitions of dHjdx are in agreement. In ordinary analysis H(x) has derivative 0 except at the origin, where the derivative failsto exist.Onemight betempted to saythat H'(x) = 0and not to worry about what happens at x = 0; such a definitionwouldbe fruitless, since it would violate the desirable property that a function can be recovered by integrating its derivative. On the other hand, the more sophisticated equation (1.20) can be formally integrated to yield Example 2. Let/(A:) bedifferentiate except at the points a^,...,an,where f(x) jumps by the respective amounts A/l5 ..., A/n. It isfurther assumedthat the derivative of/ is locally integrable wherever the derivative exists. The generalized derivative of/is obtained from (1.19), �1.3] THE THEORY OF DISTRIBUTIONS 39 Integrating by parts and noting that/(afc+) �f(ak�) = A/fc, wefind We can therefore write or Thesymbol [/'] stands fortheusualderivativeof/(x) whereverthisderivative exists (at points wherethe usual derivative doesnot exist, [/'] can be assigned any finite value whatever).The symbol/' stands for the generalized derivative n of/. The term � A/fc<5(x �ak) in (1.21) ensures that the integral of/' yields k=l /rather than the continuous function whichwould be obtained by integrating [/'] alone. Formula (1.21) can also be derived by considering the function The presence of the term AfkH(x �ak) removes the jump in /(#) at jc = ak, so that g(x) is continuous; furthermore, the derivativeof g(x) coincides with [/']. It immediately follows that A jump in afunction results in a delta function appearing in its generalized derivative. Example3. The derivativeof the delta function is obtained from (1.19): Thus �d'(x) is the generalized function corresponding to the dipoledistribution <p'(0). We regard �d'(x) as the charge densitycorresponding to a dipole of positive unit moment located at x = 0. It is equallyreasonable to define 5' as the limit as e-*0 of the difference quotient [d(x + e) �6(x)]le, where S(x + e) is defined as a translation of 6(x). A simple calculationshows (see :-> 0, we again obtain �<p'(0), which agrees with the previous definition Exercise 1.22 for the general case). 40 THE GREEN'S FUNCTION [Ch. 1 Example 4. If/00 is infinitely differentiable, as follows: Example 5. Let Z, be the linear differential operator where all the coefficients are infinitely differentiable; then, for any distribution t, we have The action of Lt on <p is therefore equivalent to the action of / on the test function �^(x), where The wth-order linear differential operator L* is known as the formal adjoint ofL. Example 6. A distribution is said to be of order n if it can be expressed as the wth derivative of a regular distribution, but cannot be expressed as a derivative of lower order of a regular distribution. The higher the order of a distribution, the more singular it is. A distribution of order 0 is a regular distribution (that is, a distribution corresponding to a locally integrable function). A distribution is of order n over an open interval a < x < b if, over that interval, it coincides with the nth derivative of a regular distribution and does not coincide with a derivative of lower order of a regular distribution. It can be shown that every distribution is of finite order over any finite interval (but it ispossible for the order to increase withthe sizeofthe interval, so that the distribution may not be of finite order on the whole line). There is no difficulty in showing that 5(x) is of order 1, d'(x) of order 2, and d(n)(x) of order n + 1. EXERCISES 1.20 Construct a test function which has the value 1 in �1< x< 1. (HINT: Consider the integral of the test function in Exercise 1.21.) �1.3] THE THEORY OF DISTRIBUTIONS 41 1.21 Consider the test function where k has been chosen so that J_ <p(x)dx = 1. The function is then a test function which vanishes for \x ��|>e; furthermore, |�� (p�(x �Qdx � 1. Now let u(x) be a continuous function which J �00 vanishes outside a finite interval [u(x) is not a test function, since we have not assumed any differentiability properties]. Show that is a test function and that Thus we have the remarkable fact that, over a finite interval, any continuous function can be uniformlyapproximated by test functions. 1.22 We have defined t' from <f', <p> = �<f, <p'>. It is equally tempting to define t' by lim [t(x + e) �t(:c)]/e, wheret(x + s) is a translated distribute tion. From this point ofview For the two definitionsto be in agreement, we must have Since Ms a continuous functional on the space of test functions, the desired result followsif it can be shown that Prove thislast statement, therebyestablishingtheconcordance of thetwo definitions of t'. 42 THE GREEN'S FUNCTION [Ch. 1 Convergence of Distributions Consider a sequence /� of distributions. If, for each test function q>(x), the sequence of numbers <fn, <p> has a limitas n -* oo, we willsay that tnconverges as �-> oo. The set of limiting values defines a functional, easily seen to be linear, on the space of test functions. It can be shown (but not without difficulty) that this functional is continuous. Consequently, if the sequence of distributions /� converges as n -> oo, there exists a distribution / such that, for each test function q>, We shall use the notation lim tn = t, or tn -> t, and say that tn converges (distributionally) to t. The same definitions can be used if ta is a set of distributions which depend on the continuous parameter a; thus we write lim ta = t if, for each test a-��o function q>,we have lim <?a, <p> = <f, <p>. A series of distributions is said to converge to / provided the sequence of partial sums converges to t as k -� oo ; wethen write When dealing with a sequence of locally integrable functions/,(*), wemay wish to enquire about the relation between ordinary pointwise convergence and distributional convergence. Thefollowingtheorem willserveourpurposes: lffn(x) converges uniformly to/(jc) over each finite interval, then/,, -�/in the sense of distributions. Indeed it follows from uniform convergence that / is locally integrable, and The step n(x)(p(x)dx-�� fq>dxisaconsequenceoftheuniformconvergence offnq> to/</> and of the fact that q> vanishes outside a finite interval. Of course it is possible for/n(x) to approach/(jc) distributionally even if the sequence/.(A;) does not approach f(x) pointwise; for instance, the sequence /�(*) = sinnx does not converge pointwise to any function, yet fn(x) -�0 distributionally, as we shall see below. The following theorem on term-by-term differentiation plays a central role in the applications of the theory of distributions.If thesequenceof distributions tn converges to/, then thesequence t'n mustconverge tot'. Let us observe first that t' and t'n exist because all distributionsare differentiable. Furthermore, �1.3] THE THEORY OF DISTRIBUTIONS 43 which proves that t'n-*t'. This leads to the remarkable and useful result that every convergent sequenceor series of distributions can be differentiated term by term as often as required. In mathematical physics one frequently encounters sequences of functions which fail to converge uniformlyin �oo < x < oo. Such sequences fall into two categories: those which converge pointwise but not uniformly, and those which even fail to converge pointwise. Both categories should be interpreted distributionally, and the following procedure is usually effective. Let/rt(x) be the troublesome sequence; suppose that an integer k can be found such that /�(*) is the Ath derivative of a sequence gn(x) which converges uniformly to g(x). It follows that#n ->g distributionally and, by the differentiation theorem, that/n ->g(k\ whereg(k) is the /cth generalized derivative of g. EXAMPLES Example 1. It was shown in Section 1.2 that, for every test function cp, (In fact, this was shown for a much larger class of functions <p.) We have therefore a rigorous interpretation of the formula Example 2. Consider the locally integrable function t which depends on the parameter a. We have shown in Exercise 1.18 that, for every test function <p(x), Therefore, in the sense of distributional convergence, We note further that hence The foregoing equations have been popular among physicists (and used effectively by them) for a few decades; the theory of distributions now furnishes a rigorous interpretation of these equations. 44 THE GREEN'S FUNCTION [Ch. 1 Example 3. The sequencefn(x) = sinnx fails to converge in the classical sense(unlessx isamultiple of n). The sequence gn(x) = �cos nx/n approaches 0 uniformly in �oo < x < oo, therefore gn ->0 distributionally; by the differentiation theorem g'n(x) = sinnx -+0 distributionally. Example 4. Thesequence fn(x) - 2n3x/n(l +n2x2)2 (discussed previously in Section 1.2;see Figure 1.11) converges pointwise (butnot uniformly) to Oin� oo<x<oo. Thestatement thatfn(x) approaches 0 pointwise does not describe the situation satisfactorily, because we know that for large n, fn(x) has extremely large positive values for x = 0+ and extremely large negative values for x *= 0� .The distributional limit takes this into account. In fact, so that, from Example 1, Example 5. Consider the Fourier series If, for large \n\, \cn\ <M/n2, the series converges uniformly and hencedistributionally. If all we know is that, for large \n\, \cn\ <Mn", where a is a integer, the series may divergepointwise; the related series converges uniformly to some functiong(x). Ontaking the(a + 2)th derivative, we find that where g(a+2) is the (a + 2)thgeneralized derivative of g(x). Example 6. Consider the function (sketched in Figure 1.13) /(O) = 0;/(jt + k) =f(x). Thegeneralized derivative ofthis locally integrable, periodic function is given byf'(x) = 1� � d(x �k). Expandingf(x) in a Fourier series, we obtain �1.3] THE THEORY OF DISTRIBUTIONS 45 FIGURE 1.13 The above equality holds in the sense of pointwise convergence (from the elementary theory of Fourier series); it isalso validin the distributional sense, because the right side isthe term-by-term derivativeofa uniformly convergent series (see Example 5). By the differentiation theorem for series which converge distributionally, we have or A statement like (1.22) would create havoc in classical analysis because the right side is a divergent series, but it hardly causes a ripple of excitement in the theory of distributions. Let us be quite clear about (1.22); we are saying where tk(x) = Z d(x �i) and sn(x) = ]T cos 2jnx; here the distributions <fk, <p> and <5n, (py are given by 46 THE GREEN'S FUNCTION [Ch. 1 respectively. Since <p(x) is identically 0 outside a finite interval, the series � cp(i) converges. Equation (1.22) states that the series also converges and that these two series are equal for everytest function <p(x). Actually the equalitycan be extended to a muchlarger class of functions<p(x), but that problem is not on our agenda. With (x- �)/2 substituted forx, (1.22) becomes or In this last equation let us restrict x and � to lie between �1 and 1. Then x �� lies between �2 and 2, and all delta functions vanish except d(x ��)� Therefore, Formula (1.22a) contains the nucleus of the usual Fourier series expansion for the interval (�1, 1). Indeed, if we multiply both sides by an arbitrary function Z(x) and integrate from �1to 1,we obtain formally In Chapter 4 we shall develop methods for generating such formulas as (1.22a). The reader familiar with the theory of Fourier series knows that certain mild restrictions must be imposed on Z(x) for the expansion to bevalid. The method employed here givesno hintas to the nature of these restrictions. Example 7. The Poisson summation formula and transformation of series. From (1.22) it follows that or, setting Ax= y, �1.3] THE THEORY OF DISTRIBUTIONS 47 We multiply by an arbitrary test function q>(y) and integrate from �oo to oo; noting that S[OVA) �fc]�=W(y �kk), we obtain The above formulacan beextendedto functions<p whichare not test functions, but merely satisfy the milder conditions: q> continuous, piecewise differentiable, and \<p\dx < oo. We recall that the Fourier transform of <p(y) is a function <&(ct>) defined by so that (1.23a) can be written This transformation from one series to another often improves the convergence. Let uslook at the particular case where<p(y) = e~y2;an easy calculation shows that O(o>) =*Jn e~w2/4, so that (1.23b) becomes By examining the ratio of successive terms for these two series weseethat the series on the left convergesrapidly for |A| large, whereasthe series on the right converges rapidly for small values of |A|. EXERCISES 1.23 In the next few exercises we consider locally integrable functionswhose discontinuities are more severe than the simplejumps included in (1.21). These functions are still difFerentiable in the distributionalsense;investigation of the generalized derivatives will enable us to give meaning to certain divergent integrals. Consider the locally integrable function This function defines a distribution What is </', <p>? We have 48 THE GREEN'S FUNCTION [Ch.1 Integrating by parts, Although [<p(x)/x]dx = lim [(p(x)/x]dx fails to exist [unless <p(0) = 0], the limit of the quantity in brackets alwaysexists and isknown as the finite part of f �� [(p(x)/x]dx. The function . is not locally integrable and therefore does not generate a distribution. In the classical sense (l/x)+ is the derivative of (log x)+ except at x = 0. The distributional derivative of (logx)+ is a singular distribution g defined by We shall write where pf stands for pseudofunction. In this notation, we have Show that with appropriate definitions, 1.24 Find the generalized derivativeof With appropriate definitions show that 1.25 Consider the locally integrable function f(x} = log|x|, �oo < x < oo. For all x =� 0 we have (d/dx) log |x| = l/x in the ordinary sense of differentiation; since l/x is not a locally integrable function, the formula �1.3] THE THEORY OF DISTRIBUTIONS 49 cannot hold distributionally. We proceed to compute the distributional derivative of/(x). Wehave Hence Since <p is differentiable at x = 0, lim[<p(e) �</>(��)]loge = 0; therefore, Although the limiting values of the individual integrals fail to exist [unless<p(0) = 0],the sumconverges and isknownasthe Cauchy principal value of | �� [(p(x)/x]dx. We shall define the pseudofunction pf(l/x J �oo by so that Referring to Exercise 1.23, show that and, therefore, 1.26 Show that What you must prove is that for each test function cp(x), 50 THE GREEN'S FUNCTION [Ch. 1 1.27 Consider the complex-valued locally integrable function /a(x) = \j(x + /a) (where a is real, a =� 0). This function defines a distribution on Show that Therefore, we can write 1.28 Use the Poisson formula (1.23a) with A = 1,<p(x) = e~a|x| to show that By noting that the left side is essentially a geometric series, obtain Can you derive this result by any other method? 1.29 With 0 < � < 1,consider the even function Expand /(x) in a Fourier cosine series. By differentiating this series and comparing with the generalized derivative off obtained directly, show that If 0 < x, � < 1, this becomes Show that this is the nucleus for the usual sine-seriesexpansion for an arbitrary functionZ(x) in 0 < x < 1. �1.3] THE THEORY OF DISTRIBUTIONS 51 1.30 Combine (1.28) and (1.22a) to show that if 0 < x, � <1, Derive the usual cosine-series expansion for a functionZ(x) in0 < jc < 1. Differential Equationsin Distributions The equation may be considered as a differential equation for an unknown distribution t. We want to find every distribution whose derivative coincides with the 0 distribution, that is, every distribution t with the property <?', <p> = 0, or <f, �<p'> = 0. Thus the necessary and sufficient condition for / to be a solution of (1.30) is that for every test function $ which is the derivative of a test function. It iseasily seen that the necessary and sufficient condition for a test function \ii(x) to be the derivative of another test function is that f�� \b dx = 0. The J - Q O condition is necessary because ty =d(p/dx implies cp(co) �(p( �oo) = 0. To show that the condition is sufficient, let ^ be a test function with the property) \l/dx =Q; then cp(x) = \ \l/(ct)d<x, is inJ �oo J - oo finitely differentiable, <p'(x) =if/(x), and q> vanishes outside a finite interval; hence ^ is the derivative of a test function. We now return to the solution of (1.30). Let cp(x) be an arbitrary test function and cpQ(x) be a fixed test function withthe property We can write The function in brackets is a test function \l/ with the property ) �� ^ dx = 0. If ns a solution of (1.30), it follows from (1.31) that </, ^> = 0; hence The quantity <f, ^>0> is a constant independent of cp, therefore, so that Ms a constant distribution. 52 THE GREEN'S FUNCTION [Ch.1 We haveshown that the only distributional solutions of dt/dx = 0 are the constant distributions, in other words, regular distributions generated by constant functions.The distributionalsolutionsof '(1.30) therefore coincide with the classicalfunctional solutionsof (1.30). Consider now the inhomogeneous equation where / is an arbitrary given distribution. This equation is equivalent to <f', <p> = </, </>> or </, <p'> = </, �(/>>. Thus the action of t on any test function \l/ which is the derivative of a test function isknown. We decompose the arbitrary test function <p(x) by (1.32) to obtain O<P dxisthe derivative ofthe test function<pi= </, �<PJ�;it iseasily seen that it islegitimate to define a distribution f0 by the relation In terms of t0 wehave Since <^, ^>0> is a constant independent of <p, we have and (1.33) has solutions all of whichare of the form t0 + c, wheretQ is defined by (1.34). We proceed with a discussion of the general linear differential equation of the �th order. Let a0(x),..., an(x) beinfinitely differentiable functions,and let L be the differential operator The operation Z-/ is defined for any distribution /. Indeed wehave shown that = <?, L*q>y, whereL* is the formal adjoint of L: �1.3] THE THEORY OF DISTRIBUTIONS 53 If/is a given distribution, the equation may be regarded as a differential equation for an unknown distribution /. A distribution t is said to satisfy (1.37) if, for every test function cp, we have or,equivalently, The solutions of (1.37) may fall in any of the following categories: 1. If t corresponds to a function which is sufficiently differentiable so that the operations in (1.37) can be performed in the classical sense, and the resulting equation is an identity, we say that / is a classical solution [of course, / will also satisfy (1.38)]. 2. If t corresponds to a function whichis not sufficiently differentiable to be substituted in (1.37), but t satisfies (1.38), we say that t is a weak solution. 3. If fis a singulardistribution (that is, does not correspond to a function), but t satisfies (1.38), we say that Ms a distributionalsolution. The term generalized solutionis used for all three types of solutions listed above. There exists a large class of problems for whichall generalized solutions of (1.37) are classical solutions. We offer the following theorem without proof: Theorem. If a0(x) never vanishes andf(x) is a locally integrable function, every generalized solution of(\ .37) is a classicalsolution. In particular, for the homogeneous equation (/= 0),if a0(x) ^ 0 for all x, all solutions are classical solutions. The following examples show that nonclassical solutions can arise. EXAMPLES Example 1. x(dt/dx) =0 has the solution t = H(x)\ indeed in the sense of distributions dHjdx = 6(x) and xd(x) = 0. This solution is a weak solution, sinceH(x) isafunction but isnot differentiable in the classical sense. It can be shown that every generalized solution of the equation is of the form c�H(x) + c2 , whereas the only classical solutions are constants (see Exercise 1.31). Example 2. x2(dt/dx) = 0 has the distributional solution t = d(x) (see Exercise 1.32). 54 THE GREEN'S FUNCTION [Ch. 1 Example 3. d2tfdx2 = d"(x) has the solution t = d(x), which is a distributional solution. Every generalized solution must be of the form 8(x) + q 4c2x, since the difference between two solutions satisfies the homogeneous equation d2t/dx2 = 0, which has only the classical solutions ct + c2x. Fundamental Solutions and Green's Functions We assume throughout this section that a0(x) ^ 0 for every x. Let � be a fixed but arbitrary real number, and let L be the differential operator (1.35). Any solution of the equation is said to be a.fundamental solutionfor the operator L withpole at �. We will see below that fundamental solutions are actually functions which are weak solutions of (1.39).Weintroducethe notation t(x \�)to remindusofthe dependence of the fundamentalsolution on the point <!;. Since any two fundamental solutions (with the same pole) differ by a solution of the homogeneous equation, the general solution of (1.39) consists of the sum of a particular solution of (1.39) and the general solution of the homogeneous equation. A particular solution of (1.39) can be obtained by the following heuristic procedure. Let u(x) be a fixed but arbitrary solution of the homogeneous equation, and let v^(x) be a solution of the homogeneous equation still to be determined (as the notation implies, this solution will depend on �). Since Lt = 0 whenever x ^ �, where i>{(x) is to be computed from u(x) by appropriate matching conditions at x =f; these conditions will reflect the fact that Lt has a delta-function behavior at x - �. Integrating (1.39) from x = �-e to x = � +e, and assuming that t,..., d"~2t/dxn~2 arecontinuous at x = f, we obtain or or �1.3] THE THEORY OF DISTRIBUTIONS 55 We therefore conjecture that a solution of (1.39) is where u(x) is a fixed arbitrary solution of the homogeneous equation and v%(x) isanother solution ofthe homogeneousequation satisfying the conditions According to the uniqueness and existence theorem for ordinary differential equations, v^(x) existsand is unambiguouslydetermined [once u(x) and � are fixed]. It is clear from the jump condition on the (� �l)th derivative that u and u� are independent solutions of the homogeneous equation. We now prove that the function t(x\ �) defined by (1.40) is a solution of (1.39) [t is a weak solution since t(n~l) is discontinuous at x = �]. We must show that, for everytest function (p, By the definition of t, wehave where We integrate by parts enoughtimes so thatthe differentiations aretransferred from cp to u and v^; thus the terms involvingd\an-k(p)ldxk will be integrated by parts exactly k times. Since(pvanishesoutside a finite interval, there is no contribution from the integrated terms at x = +00. Because u(k)(�) = y(k)(0 for k = 0,..., n �2, the integrated contributions at �� and �+ cancel except for a term involvingw("~1) and one involvingt;^~1) (these terms arising from the nth integration by parts of the terms corresponding to k = n). Using the jump condition one obtains Since both Lu and Lu,. vanish, it follows that 56 THE GREEN'S FUNCTION [Ch1. We have shown that t(x\ �) as given by (1.40) is a solution of (1.39). Every fundamental solution is therefore the sum of this particular t(x\ �) and of a solution of the homogeneous equation. Since there exist n independent solutions of the homogeneous equation, n arbitrary constants will appear in the most general fundamental solution. Let us now introduce the concept of the Green's function; the Green's function is a fundamental solution which satisfies n additional, specified, homogeneous boundary conditions. Whereas the general fundamental solution contains n arbitrary constants, the Green's function is completely determined once the boundary conditions are given. (Certain limitations on the nature of the boundary conditions and the differential operator must be imposed to ensure the existence and uniquenessof the Green's function.)The theory of Green's functions for second-order linear differential operators is developed in the next section. EXAMPLES Example 1. It is easily seen that t = �\x ��|/2 is a solution of Formula (1.6) gives another fundamental solution satisfyingvanishing boundary conditions at x = 0, x = 1(with T = 1, / = 1).Thus is also a fundamental solution. We see that g(x \�) �t(x \�) = (x/2) + � �jt, which is a solution of the homogeneous equation. Example 2. The causalfundamental solution or Green's function for the initial-value problem. This fundamental solution is obtained by setting M = 0in(1.40). Thus where ^(f) = vffl = ��� = v({~2\^} = 0 and v�-l\?) = l/a0(�. The reason for calling t a Green's function is that, if � is restricted to positive values, t satisfies nhomogeneous conditions at x = 0; indeed t, t',..., t(n~l) allvanish at x = 0(sincet = 0for x < �}. IfL has constant coefficients, v$(x) = VQ(X �<!;), and where �1.3] THE THEORY OF DISTRIBUTIONS 57 Example 3. Fundamental solutions are used to solve the inhomogeneous equation where/is a given function. As was done in Section 1.1, we write/(x) as continuous superposition of delta functions /(x) = J_� d(x ��)f(�)d�\. The principle of superposition suggests the solution w(x) � t(x\ �)/(<!;)</<!;, where Lt = d(x �<!;). As we J-oo change / we obtain different solutions to (1.41). Often we are interested in solving (1.41) subject to n additional homogeneous conditions; in that case the Green's function satisfying these n conditions is the most convenient fundamental solution to employ. EXERCISES 1.31 The observant reader will have noted that the successful solution of (1.30) and (1.33) depended on the decomposition of an arbitrary test function (p(x) into two parts, one part proportional to the fixed test function (p0(x) with |�� <p0(x)dx = 1 and the other part the derivative J - 00 of a test function.Other decompositions are required to solve functional equations involving distributions. Consider, for instance, the equation We search for all solutions / of (1.42), that is, all distributions t for which (xt, <p> = 0, or by (1.18), <?, xq>y = 0. Thus the action of t is known on all test functions of the form x<p. The reader should prove that the necessary and sufficient condition for a test function ty to be of the form xcp is that i/f(0) = 0. This suggests the following decomposition for an arbitrarytest function cp(x). Let (p0(x) be a fixed test function such that (p0(0) = 1; then The test function in brackets vanishes at jc = 0. If Ms a solution of (1.42), it follows that The second term on the right side vanishes; hence Since </, q>0(x)y is a constant independent of <p(x), it follows that It is now trivialto showthat the general solution ofx(dt/dx) = 0isgiven by 58 THE GREEN'S FUNCTION [Ch.1 1.32 Find the general solution of where k is a positive integer. 1.33 Show that the general solution of is 1.34 Show that the solution of with the initial conditions w(G) = 0,..., H^""^ (0) = 0, is givenby where v^(x) is defined as in Example 2. 1.4 PRELIMINARY RESULTS ON LINEAR EQUATIONS OF THE SECOND ORDER The most general linear operator of the second order is In the last section we required that all coefficients beinfinitely differentiable; now we merely assume that #0(X)�ai(*)> o2(x) are continuous. Let f(x) be piecewise continuous and let / be an interval on the x axis. We say that u(x) is a solution (or classical solution) of if uhas a continuousderivative and a piecewisecontinuoussecond derivative, and ifthefunctionsLu and/are equal at all points of continuityof/ (Classically, the notion of solution can be extended to the case where / is only integrable instead of piecewise continuous.) Existence and Uniqueness Theorem Theorem. Let I be a closed interval in which a0(x) =� 0; let x0 be apoint in /, and let a and ft be given numbers. The problem has one and only onesolution. �1.4] PRELIMINARY RESULTS ON DIFFERENTIAL EQUATIONS 59 REMARKS. The problem (1.43) is known as an initial-value problem (even though x0 is not the left end point of the interval). The existence and uniqueness theorem will be proved in Exercise 3.10. We accept the theorem for the present but illustrate, by the followingexamples, the essential character of theassumption a0(x) ^ 0. Consider The general solution for x > 0 is A + Bx2. Imposing the initial conditions yields y = Bx2, which violates the uniqueness portion of the above theorem. The difficulty is caused by the fact that a0(x) = x vanishes at a point in the interval (more precisely the trouble is that aja0 is discontinuous at a point in the interval). A point at which either aja0 or a2ja0 is discontinuous is known as a singularpoint of the differential equation. Solutions often become infiniteat such points and the existence and uniqueness theorem maybeviolated. Consider, for instance, xy"+y' = 0 with XO) = a,y(0) - P ^ 0. For x > 0 the general solution of the differential equation is A + B log x, and no choice of A and B enables us to satisfy the conditions at x = 0. In all that follows, it will be assumed that a0(x) ^ 0 in /. COROLLARY 1. The only solutionof is u =0 in I. COROLL ARY 2. If u is the solutionof then v =yu is the solutionof Dependence and Independence; Wronskians Considernfunctions/!(x), ... ,/n(x).Thesefunctions aresaidtobe dependent over an interval /, if there exist constants c^, ... , cn, not all zero, such that A set of n functions is therefore dependent if any one of them can be expressed as a linear combination of the others. The set of n functions/^x),... ,/n(x) is independent over /if (1.44) can only be satisfied bysetting ct = ��� = cn = 0. Let/^x),... ,/n(x)befunctionswhich are n �1 times differentiable. The Wronskianof/l5 ...,/� is defined as the n by n determinant 60 THE GREEN'S FUNCTION [Ch. 1 The discussion in this section requires only the concept of the Wronskian of two functions/ and g, each of which is differentiable. From (1.45) wehave If/and g are dependent over /, it isclear that their Wronskian vanishesfor every x in /. Thefollowingexample shows that the converse is not true. Let / be the interval �1 < X < 1, /= x2, g =x\x\; then / and g are independent over / but W(/,g\ x) vanishes for everyx in �1< x< 1. Abel's Formula for the Wronskian Let M and v be any two solutions (independent or not) of Lu = 0. There exists a constant C such that The proof is simple. From vLu �uLv = 0 it follows that Therefore, Note that once m(x) is chosen, C is unambiguously determined from uand v; furthermore, if W vanishes at any point x, it vanishesfor all x. Theorem. Let u and v be solutions ofLu =0; the necessary and sufficient conditionfor u and v to be dependent is that their Wronskian vanish at asingle point XQ . (From the remark immediately preceding the theorem, W vanishes at onepoint if and only ifW=OorC =Q.) Proof. If Mand v are dependent, it is obvious that W = 0. If W = 0 at some point JCQ , then From the theory of linear algebraic equations, we know that this is precisely the necessary and sufficient conditionfor the existenceof a nontrivialsolution (that is, q and c2 not both 0) to the set of equations Let (q, c2) be a nontrivial solution of this set of equations and let U(x) = qw(x) + c2v(x). It follows that U(x0) = 0, and U'(x0) = 0; furthermore, �1.4] PRELIMINARY RESULTS ON DIFFERENTIAL EQUATIONS 61 U(x), being a linear combination of two solutions of Lu = 0, isitself a solution of Lu = 0. By Corollary 1, U = 0; hence u and v are dependent. EXERCISES 1.35 The functions x2 and x\x\ are independent over -1< jc < 1, yet their Wronskian is 0. Explain why this does not contradict the above theorem. 1.36 The existenceanduniqueness theoremfor theequationof order n. Let x0 be a point in the interval /; let a0,...,<*�_! be � given constants, and let L be the operator where a0(x),..., an(x) are continuous anda0(x) ^ 0 over /. There exists one and only one solution of where/is a given piecewise continuous function. Assuming the validity of this theorem, derive the equivalent forms of Corollaries 1 and 2. Show that if u^x),..., un(x) are n solutions of Lu = 0, then Show that the necessary and sufficient condition for the n solutions to be dependent is that W vanish at a single point x0 (or equivalently that W =0, or C =0). The Homogeneous Differential Equation Let a0(x) ^ 0 over / and consider the homogeneous differential equation Theorem. There exist twosolutionsu^x) and u2(x) of thisequationwhichare independent over I. Every solutionon the intervalI can be written in the form + Bu2(x) for some constantsA, B.~\ t The independent solutions MI(X), u2(x), can be chosen in many different ways. As an example, the equation (d2u/dx2) + u= 0 has independent solutions HI(X) = sinx, u2(x) = cosx, and every solution is of the form Auv + Bu2 . If we are in a particularly perverse mood we can choose instead the two independent solutions vt = sinx + cosx, v2 = 3 sinx �1cosx. Every solution of (d2u/dx2) + u= 0 can also be written Cv� + Dv2 . 11.4 Implicit and Iterative Differentiation 297 Mechanical AdjointRecurrence This similarity does occur, when we apply the reverse mode to the whole iteration in a mechanical fashion. Suppose we iterate over k steps and then evaluate The adjoint iteration is easily written down, namely, and Assuming that zk stays bounded, we seethat Pk converges to zero and can thus be neglected. In other words, we assume for simplicity that the Pk have been deactivated, i.e., are considered as real numbers that have fallen from the sky and are thus unrelated to x and the zk . Because ofour contractivity Assumption CP it is clear that zk\ ~ pk~k . This means the adjoints zk of early iterates, especially that of ZQ, should be small when k is sufficiently large. That effect makes sense, since the initial guess ZQshould not have any significant influence on the outcome of the overall computation. Hence the productive part is the incrementation ofthe parameter adjoint x. Assuming for a moment that zk = z* and Pk = P for all k < k includingnegative indices, we find using the Neumann expansion that and thus at the end, After bringing yfx onto the right-hand side, we obtain for x exactly the same expression as in (11.20). Without the unrealistic assumption zk = z* and Pk = P constant, we can only hope that the resulting x is reasonably close to that exact value. In contrast �1.4] PRELIMINARY RESULTS ON DIFFERENTIAL EQUATIONS 63 The usual superposition argument would lead one to believe that w(x) = | t(x|�)/(�W� furnishesasolution oftheinhomogeneous equation Lw =/. J � 00 This is indeed correct if/(<!;) and t(x \�) behave sufficiently well at � = + oo for the preceding integral to exist. In any event we observe that for any finite a and b, z(x) = (" t(x\ �)/(&% always exists and Lz - !"(Lt)f($d� = Ja Ja f *<5(x - Qf(t)dt. Thus, z(x) = f*f(x| �)/(��� is a solution of Ja Ja A proof of this statement is obtained by a procedure similar to that following (1.10). One can circumvent the limitation to finite intervals by introducing the two fundamental solutions where v$(x) is the solution of the homogeneous equation satisfyingthe initial conditions It follows that z Theorem. The one and only solution of the initial-valueproblem is giv