Arscott / Sneddon / Stark Periodic Differential Equations

An Introduction to Mathieu, Lamé, and Allied Functions
1. Auflage 2014
ISBN: 978-1-4831-6488-5
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 294 Seiten, Web PDF

Periodic Differential Equations: An Introduction to Mathieu, Lamé, and Allied Functions covers the fundamental problems and techniques of solution of periodic differential equations. This book is composed of 10 chapters that present important equations and the special functions they generate, ranging from Mathieu's equation to the intractable ellipsoidal wave equation. This book starts with a survey of the main problems related to the formation of periodic differential equations. The subsequent chapters deal with the general theory of Mathieu's equation, Mathieu functions of integral order, and the principles of asymptotic expansions. These topics are followed by discussions of the stable and unstable solutions of Mathieu's general equation; general properties and characteristic exponent of Hill's equation; and the general nature and solutions of the spheroidal wave equation. The concluding chapters explore the polynomials, orthogonality properties, and integral relations of Lamé's equation. These chapters also describe the wave functions and solutions of the ellipsoidal wave equation. This book will prove useful to pure and applied mathematicians and functional analysis.

Autoren/Hrsg.

Arscott, F. M.

Sneddon, I. N.

Stark, M.

Ulam, S.

Inhaltsverzeichnis

1;Front Cover;1
2;Periodic Differential Equations: An Introduction to Mathieu, Lamé, and Allied Functions;4
3;Copyright Page;5
4;Table of Contents;6
5;PREFACE;8
6;CHAPTER 1. FORMATION OF THE EQUATIONS: THE MAIN PROBLEMS;12
6.1;1.1. Problems leading to periodic differential equations;12
6.2;1.2. Separation of coordinates: examples;13
6.3;1.3. Elliptic coordinates: Mathieu's equation;19
6.4;1.4. Spheroidal coordinates: the spheroidal wave equation and associated Mathieu equation;23
6.5;1.5. Hill's equation;26
6.6;1.6. Ellipsoidal coordinates: Lamé's equation and the ellipsoidal wave equation;27
6.7;1.8. Two algebraic lemmas;31
6.8;1.9. The Sturm oscillation theorem;33
6.9;Miscellaneous examples on Chapter I;34
7;CHAPTER 2. MATHIEU'S EQUATION — GENERAL THEORY ;37
7.1;2.1. Introduction: some simple properties of solutions;37
7.2;2.2. Floquet's theorem;40
7.3;2.3. Periodicity factors and exponents: The periodicity equation: Definitions;42
7.4;2.4. Ince's theorem;45
7.5;2.5. The orthogonality property;50
7.6;2.6. An integral relationship;51
7.7;2.7. The basically-periodic solutions as functions of q;54
7.8;2.8. Mathieu's equation from the algebraic standpoint;58
7.9;Miscellaneous examples on Chapter II;60
8;CHAPTER 3. MATHIEU FUNCTIONS OF INTEGRAL ORDER;63
8.1;3.1. Recapitulation of main properties;63
8.2;3.2. Notation for Mathieu functions (of the first kind) of integral order;63
8.3;3.3. Expression of Mathieu functions as trigonometric series;65
8.4;3.4. Orthogonality;68
8.5;3.5. The modified Mathieu functions of integral order;69
8.6;3.6. Determination of the coefficients in the trigonometric series;70
8.7;3.7. Perturbation method of solution;78
8.8;3.8. The second solution of Mathieu's equation;81
8.9;3.9. The solution of Mathieu's algebraic equation;83
8.10;Miscellaneous examples on Chapter III;85
9;CHAPTER 4. MATHIEU FUNCTIONS OF INTEGRAL ORDER — FURTHER PROPERTIES ;90
9.1;4.1. Integral equations for periodic Mathieu functions;90
9.2;4.2. Extension to integral relations;96
9.3;4.3. Bessel function series;97
9.4;4.4. Another solution of Mathieu's equation;100
9.5;4.5. Integral relations with infinite limits;104
9.6;4.6. Bessel function nuclei;107
9.7;4.7. Bessel function product series;109
9.8;4.8. Tables of Mathieu functions;112
9.9;Miscellaneous examples on Chapter IV;113
10;CHAPTER 5. ASYMPTOTIC EXPANSIONS ;118
10.1;5.1. z-Asymptotic expansions for Mathieu functions of integral order;118
10.2;5.2. Asymptotic expansion of periodic solutions for large [q];122
10.3;5.3. The Horn–Jefferys asymptotic series;125
10.4;5.4. Further developments;130
10.5;Miscellaneous examples on Chapter V;130
11;CHAPTER 6. MATHIEU'S GENERAL EQUATION ;132
11.1;6.1. Introduction;132
11.2;6.2. Stable and unstable solutions;132
11.3;6.3. Hill's method;135
11.4;6.4. Calculation of µ: the constant-µ curves;139
11.5;6.5. Notation for solutions;141
11.6;6.6. The stable solutions;144
11.7;6.7. The unstable solutions;145
11.8;Miscellaneous examples on Chapter VI;148
12;CHAPTER 7. HILL'S EQUATION ;152
12.1;7.1. General properties;152
12.2;7.2. Determination of the characteristic exponent;153
12.3;7.3. The coexistence question for Hill's equation;155
12.4;7.4. Hill's equation with three terms;156
12.5;Miscellaneous examples on Chapter VH;160
13;CHAPTER 8. THE SPHEROIDAL WAVE EQUATION ;164
13.1;8.1. General nature of the equation;164
13.2;8.2. Solutions valid near z = 0;176
13.3;8.3. Solutions valid near z = 8
;183
13.4;8.4. Connections between the solutions;186
13.5;8.5. Spheroidal wave functions;187
13.6;8.6. Integral relations;191
13.7;8.7. Notation;192
13.8;8.8 The associated Mathieu equation;193
13.9;Miscellaneous examples on Chapter VIII;197
14;CHAPTER 9. LAMÉ'S EQUATION ;202
14.1;9.1. Introductory;202
14.2;9.2. Lamé polynomials of the first species;206
14.3;9.3. The Lamé polynomials—second, third and fourth species;209
14.4;9.4. Orthogonality properties;217
14.5;9.5. Integral relations;222
14.6;9.6. Lamé polynomial products: ellipsoidal harmonics;225
14.7;9.7. The second solution of Lamé's equation;234
14.8;9.8. Simply-periodic and other solutions;236
14.9;9.9. Tables of Lamé functions;241
14.10;Miscellaneous examples on Chapter IX;242
15;CHAPTER 10. THE ELLIPSOIDAL WAVE EQUATION ;248
15.1;10.1. Introduction;248
15.2;10.2. Ellipsoidal wave functions;249
15.3;10.3. Ellipsoidal wave functions—main properties;251
15.4;10.4. Perturbation solutions;254
15.5;10.5. Solutions in Lamé function and Bessel function series;257
15.6;10.6. Other solutions of the equation;259
15.7;Miscellaneous examples on Chapter X;261
16;APPENDIX A — BESSEL FUNCTIONS ;262
17;APPENDIX B — LEGENDRE, GEGENBAUER AND TCHEBYCHEFF FUNCTIONS ;267
18;APPENDIX C — ELLIPTIC FUNCTIONS ;272
19;REFERENCES;275
20;ADDITIONAL NOTES;280
20.1;A.1. Laplace's equation in paraboloidal coordinates;280
20.2;A.2. Calculation of the characteristic exponent of Mathieu's general equation;280
20.3;A.3. Asymptotic expansions for Mathieu functions and spheroidal wave functions;281
20.4;A.4. Hill's equation with three terms (Whittaker–Hill equation);281
20.5;A.5. Hill's equation—regions of stability;283
20.6;A.6. Integral equations and integral relations for Lamé functions;283
20.7;A.7. Two-parameter eigenvalue problems;284
20.8;A.8. Tables;284
20.9;A.9. Note on the proof of Lemma 2, p. 21;284
20.10;Supplementary References;285
21;INDEX;288
22;OTHER VOLUMES IN THE SERIES IN PURE AND APPLIED MATHEMATICS;294