EIGENFUNCTION EXPANSIONS ASSOCIATED WITH SECOND-ORDER DIFFERENTIAL EQUATIONS BY E. C. TITCHMARSH FJR. S. SAVILIAN PROFESSOR OF GEOMETRY IN THE UNIVERSITY OF OXFORD OXFORD AT THE CLARENDON PRESS 1946 OXFORD UNIVERSITY PRESS AMEN HOUSE, E. G. 4 LONDON EDINBURGH GLASGOW NEW YORK TORONTO MELBOURNE CAPE TOWN BOMBAY CALCUTTA MADRAS GEOFFREY CUMBERLEGE PUBLISHER TO THE UNIVERSITY PREFACE THE idea of expanding an arbitrary function in terms of the solutions of a second-order differential equation goes back to the time of Sturm and Liouville, more than a hundred years ago. The first satisfactory proofs were constructed by various authors early in the twentieth century. Later, a general theory of the singular cases was given by Weyl, who-based i on the theory of integral equations. An alternative method, proceeding via the general theory of linear operators in Hilbert space, is to be found in the treatise by Stone on this subject. Here I have adopted still another method. Proofs of these expansions by means of contour integration and the calculus of residues were given by Cauchy, and this method has been used by several authors in the ordinary Sturm-Liouville case. It is applied here to the general singular case. It is thus possible to avoid both the theory of integral equations and the general theory of linear operators, though of course we are sometimes doing no more than adapt the latter theory to the particular case considered. The ordinary Sturm-Liouville expansion is now well known. I therefore dismiss it as rapidly as possible, and concentrate on the singular cases, a class which seems to include all the most interesting examples. In order to present a clear-cut theory in a reasonablespace, I have had to reject firmly all generalizations. Many of the arguments used extend quite easily to other cases, such as that of two simultaneous first-order equations. It seems that physicists are interested in some aspects of these questions. If any physicist finds here anything that he wishes to know, I shall indeed be delighted but it is to mathematicians that the book is addressed. I believe in the future of mathematics for physicists, but it seems desirable that a writer on this subject should understand physics as well as mathematics. E. C. T. NEW COLLEGE, OXFOBD, 1946. CONTENTS I. THE STUEM-LIOUVILLE EXPANSION ... 1 II. THE SINGULAB CASE SERIES EXPANSIONS . . 19 III. THE GENERAL SINGULAR CASE . . . .39 IV. EXAMPLES 69 V. THE NATURE OF THE SPECTRUM . . .97 VI. A SPECIAL CONVERGENCE THEOREM . . .118 VII. THE DISTRIBUTION OF THE EIGENVALUES . . 124 VIII. FURTHER APPROXIMATIONS TO JV A . . .135 IX. CONVERGENCE OF THE SERIES EXPANSION UNDER FOUBIER CONDITIONS 148 X. SUMMABILITY OF THE SERIES EXPANSION . . 163 REFERENCES 172 THE STURM-LIOUVILLE EXPANSION 1.1. Introduction. Let L denote a linear operator operating on a function y y x. Consider the equation Ly - AT, 1.1.1 where A is a number. A function which satisfies this equation and also certain boundary conditions e. g. which vanishes at x a and x b is called an eigenfunction. The corresponding value of A is called an eigenvalue. Thus ifi t n x is an eigenfunction corresponding to an eigenvalue n, L x Mx. 1.1.2 The object of this book is to study the operator,72 where q x is a given function of x defined over some given interval a, b. In this case y satisfies the second-order differential equation and tff n x satisfies s A- W0- 1J. 5 If we take this and the corresponding equation with m instead of n, multiply by ift m x 9 n x respectively, and subtract, we obtain Hence b A M - AJ J lUaOiM dx 0 m a- a a if i m x and rl x both vanish at x a and x b or satisfy a more general condition of the same kind. If m A n, it follows that b t m x t n x dx Q. 1-1.6 a 4967 2 THE STURM-LIOUVILLE EXPANSION Chap. I By multiplying if necessary by a constant we can arrange that x dx l. 1.1.7 The functions n x then form a normal orthogonal set...

EIGENFUNCTION EXPANSIONS ASSOCIATED WITH SECOND-ORDER DIFFERENTIAL EQUATIONS BY E. C. TITCHMARSH FJR. S. SAVILIAN PROFESSOR OF GEOMETRY IN THE UNIVERSITY OF OXFORD OXFORD AT THE CLARENDON PRESS 1946 OXFORD UNIVERSITY PRESS AMEN HOUSE, E. G. 4 LONDON EDINBURGH GLASGOW NEW YORK TORONTO MELBOURNE CAPE TOWN BOMBAY CALCUTTA MADRAS GEOFFREY CUMBERLEGE PUBLISHER TO THE UNIVERSITY PREFACE THE idea of expanding an arbitrary function in terms of the solutions of a second-order differential equation goes back to the time of Sturm and Liouville, more than a hundred years ago. The first satisfactory proofs were constructed by various authors early in the twentieth century. Later, a general theory of the singular cases was given by Weyl, who-based i on the theory of integral equations. An alternative method, proceeding via the general theory of linear operators in Hilbert space, is to be found in the treatise by Stone on this subject. Here I have adopted still another method. Proofs of these expansions by means of contour integration and the calculus of residues were given by Cauchy, and this method has been used by several authors in the ordinary Sturm-Liouville case. It is applied here to the general singular case. It is thus possible to avoid both the theory of integral equations and the general theory of linear operators, though of course we are sometimes doing no more than adapt the latter theory to the particular case considered. The ordinary Sturm-Liouville expansion is now well known. I therefore dismiss it as rapidly as possible, and concentrate on the singular cases, a class which seems to include all the most interesting examples. In order to present a clear-cut theory in a reasonablespace, I have had to reject firmly all generalizations. Many of the arguments used extend quite easily to other cases, such as that of two simultaneous first-order equations. It seems that physicists are interested in some aspects of these questions. If any physicist finds here anything that he wishes to know, I shall indeed be delighted but it is to mathematicians that the book is addressed. I believe in the future of mathematics for physicists, but it seems desirable that a writer on this subject should understand physics as well as mathematics. E. C. T. NEW COLLEGE, OXFOBD, 1946. CONTENTS I. THE STUEM-LIOUVILLE EXPANSION ... 1 II. THE SINGULAB CASE SERIES EXPANSIONS . . 19 III. THE GENERAL SINGULAR CASE . . . .39 IV. EXAMPLES 69 V. THE NATURE OF THE SPECTRUM . . .97 VI. A SPECIAL CONVERGENCE THEOREM . . .118 VII. THE DISTRIBUTION OF THE EIGENVALUES . . 124 VIII. FURTHER APPROXIMATIONS TO JV A . . .135 IX. CONVERGENCE OF THE SERIES EXPANSION UNDER FOUBIER CONDITIONS 148 X. SUMMABILITY OF THE SERIES EXPANSION . . 163 REFERENCES 172 THE STURM-LIOUVILLE EXPANSION 1.1. Introduction. Let L denote a linear operator operating on a function y y x. Consider the equation Ly - AT, 1.1.1 where A is a number. A function which satisfies this equation and also certain boundary conditions e. g. which vanishes at x a and x b is called an eigenfunction. The corresponding value of A is called an eigenvalue. Thus ifi t n x is an eigenfunction corresponding to an eigenvalue n, L x Mx. 1.1.2 The object of this book is to study the operator,72 where q x is a given function of x defined over some given interval a, b. In this case y satisfies the second-order differential equation and tff n x satisfies s A- W0- 1J. 5 If we take this and the corresponding equation with m instead of n, multiply by ift m x 9 n x respectively, and subtract, we obtain Hence b A M - AJ J lUaOiM dx 0 m a- a a if i m x and rl x both vanish at x a and x b or satisfy a more general condition of the same kind. If m A n, it follows that b t m x t n x dx Q. 1-1.6 a 4967 2 THE STURM-LIOUVILLE EXPANSION Chap. I By multiplying if necessary by a constant we can arrange that x dx l. 1.1.7 The functions n x then form a normal orthogonal set...

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The Riemann zeta-function embodies both additive and multiplicative structures in a single function, making it our most important tool in the study of prime numbers. This volume studies all aspects of the theory, starting from first principles and probing the function's own challenging theory, with the famous and still unsolved "Riemann hypothesis" at its heart. The second edition has been revised to include descriptions of work done in the last forty years and is updated with many additional references; it will provide stimulating reading for postgraduates and workers in analytic number theory and classical analysis.