Boehme / Sneddon Operational Calculus

2. Auflage 2014
ISBN: 978-1-4831-6145-7
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band Volume 110, 264 Seiten, Web PDF

Reihe: International Series in Pure and Applied Mathematics

Operational Calculus, Volume II is a methodical presentation of operational calculus. An outline of the general theory of linear differential equations with constant coefficients is presented. Integral operational calculus and advanced topics in operational calculus, including locally integrable functions and convergence in the space of operators, are also discussed. Formulas and tables are included. Comprised of four sections, this volume begins with a discussion on the general theory of linear differential equations with constant coefficients, focusing on such topics as homogeneous and non-homogeneous equations and applications of operational calculus to partial differential equations. The section section deals with the integral of an operational function and its applications, along with integral transformations. A definition of operators in terms of abstract algebra is then presented. Operators as generalized functions, power series of operators, and Laplace transform are also discussed. Formulas of the operational calculus and tables of functions round out the book. This monograph will be useful to engineers, who regard the operational calculus merely as a tool in their work, and readers who are interested in proofs of theorems and mathematical problems.

Autoren/Hrsg.

Boehme, Thomas K.

Sneddon, I. N.

Inhaltsverzeichnis

1;Front Cover;1
2;Operational Calculus;4
3;Copyright Page;5
4;Table of Contents;6
5;FOREWORD TO THE FIRST ENGLISH EDITION;11
6;FOREWORD TO THE SECOND ENGLISH EDITION;12
7;SUPPLEMENTS TO VOLUME I;13
7.1;(A) Supplement to Part I, Chapter VI;13
7.2;(B) Supplement to Part III, Chapter VIII;14
8;PART IV: AN OUTLINE OF THE GENERAL THEORY OF LINEAR DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS;17
8.1;CHAPTER I. Homogeneous equations;17
8.1.1;§ 1. Introductory remarks;17
8.1.2;§ 2. Characteristic equations;17
8.1.3;§ 3. On exponential functions;18
8.1.4;§ 4. Logarithms;19
8.1.5;§ 5. Multiple roots of the characteristic equation;20
8.1.6;§ 6. The general solution;21
8.1.7;§ 7. Theorem on uniqueness of solution;23
8.1.8;§ 8. The logarithmic equation;25
8.1.9;§ 9. Linear differential expressions;26
8.1.10;§ 10. Operations on linear differential expressions;27
8.1.11;§ 11. Characteristic polynomials of linear differential expressions;28
8.1.12;§ 12. Pure equations;28
8.1.13;§ 13. Mixed equations;28
8.1.14;§ 14. Adapting the solution to given initial, boundary and other conditions;29
8.2;CHAPTER II. Non-homogeneous equations;32
8.2.1;§ 15. The general solution of a non-homogeneous equation;32
8.2.2;§ 16. The case where the right side is a polynomial;33
8.2.3;§ 17. The case where the right side is an exponential function;34
8.2.4;§ 18. The case where the right side is a product of a polynomial and an exponential function;35
8.2.5;§ 19. The case where the right side is a linear combination of two functions;36
8.2.6;§ 20. The case where the right side is a trigonometric function;36
8.2.7;§ 21. Adapting the solution to additional conditions;37
8.3;CHAPTER III. Applications to partial differential equations;40
8.3.1;§ 22. Reducing partial operational equations to operational equations;40
8.3.2;§ 23. Remarks on additional conditions;45
8.3.3;§ 24. An incorrect solution;46
8.3.4;§ 25. Explaining the apparent contradiction;47
8.3.5;§ 26. The Cauchy conditions and the question of their being equivalent to the general conditions;49
8.3.6;§ 27. Solving restrictive equations;51
8.3.7;§ 28. The question of the equivalence of a partial equation and an operational equation;53
8.3.8;§ 29. Further examples of solving partial equations;54
8.3.9;§ 30. General remarks on solving partial equations by the operational method;57
8.3.10;§ 31. Mixed problems;60
9;PART V: INTEGRAL OPERATIONAL CALCULUS;67
9.1;CHAPTER I. The integral of an operational function and its applications;67
9.1.1;§ 1. Operational functions of class (K);67
9.1.2;§ 2. The definition of the integral;68
9.1.3;§ 3. Properties of the integral;69
9.1.4;§ 4. Operational functions of two variables;71
9.1.5;§ 5. Cutting down a function;73
9.1.6;§ 6. The integral form of a certain particular solution of the logarithmic differential equation;75
9.1.7;§ 7. Application to the equation of a vibrating string;77
9.1.8;§ 8. Application of infinite series and definite integrals;80
9.2;CHAPTER II. Integral transformations;83
9.2.1;§ 9. The Laplace transform;83
9.2.2;§ 10. The Laplace transform as a basis for the operational calculus;84
9.2.3;§ 11. A comparison of the direct method and the method of Laplace transform;85
9.2.4;§ 12. Other related methods;86
10;PART VI: ADVANCED TOPICS IN THE OPERATIONAL CALCULUS;87
10.1;CHAPTER I. Definition of operators in terms of abstract algebra;87
10.1.1;§ 1. Commutative ring;87
10.1.2;§ 2. Quotient field;88
10.1.3;§ 3. Operators;89
10.1.4;§ 4. Rings with divisors of zero;90
10.1.5;§ 5. Periodic operators;90
10.1.6;§ 6. The Fourier series of a periodic operator;91
10.2;CHAPTER II. Locally integrable functions;93
10.2.1;§ 1. The convolution of integrable functions;93
10.2.2;§ 2. Properties of this convolution;93
10.2.3;§ 3. Locally integrable functions as operators;94
10.2.4;§ 4. Functions of class K;94
10.2.5;§ 5. Absolutely continuous functions;96
10.2.6;§ 6. The ring of locally integrable functions;96
10.3;CHAPTER III. Operators as generalized functions;98
10.3.1;§ 1. Introduction;98
10.3.2;§ 2. The class of left-bounded functions on the whole real line;98
10.3.3;§ 3. The support number of an operator;101
10.3.4;§ 4. The support of continuous functions;102
10.3.5;§ 5. Approximate identities;103
10.3.6;§ 6. Convolution of functions defined on an arbitrary open set;104
10.3.7;§ 7. Almost uniform convergence on open sets;105
10.3.8;§ 8. Regular operators;106
10.3.9;§ 9. Examples of regular operators;107
10.3.10;§ 10. An example of an operator which is not regular;108
10.3.11;§ 11. The restriction of operators to an open set;109
10.3.12;§ 12. A characterization of support;111
10.3.13;§ 13. Derivatives of an operator;112
10.3.14;§ 14. The proof of Theorem A;112
10.3.15;§ 15. The proof of Theorem B;113
10.3.16;§ 16. Some examples of derivatives;114
10.3.17;§ 17. Regular convergence of operators;117
10.3.18;§ 18. A comparison with other types of generalized functions;119
10.3.19;§ 19. Defining sequences;120
10.3.20;§ 20. Generalized functions on the whole real line;121
10.4;CHAPTER IV. Convergence in the space of operators;123
10.4.1;§ 1. Introduction;123
10.4.2;§ 2. Approximation by convolution;123
10.4.3;§ 3. Infinite convolution products;124
10.4.4;§ 4. Continuous convolution multiples;126
10.4.5;§ 5. Common denominators for a sequence of operators;127
10.4.6;§ 6. The diagonal subsequence property;128
10.4.7;§ 7. The space of operators which have positive support numbers;128
10.4.8;§ 8. A convergent sequence;130
10.4.9;§ 9. Topological convergence;131
10.4.10;§ 10. The connection between sequential and topological convergence;132
10.4.11;§ 11. A topological convergence in the space of operators;135
10.4.12;§ 12. Functional on the space F0;136
10.4.13;§ 13. A characterization of operator convergence type I';138
10.4.14;§ 14. Metrizable topological vector spaces;139
10.4.15;§ 15. The space F0 as a metric space;142
10.4.16;§ 16. The space of operators as the union of metric spaces;143
10.4.17;§ 17. Bounded sets in the space of operators;144
10.4.18;§ 18. Precompact sets in the space of operators;145
10.4.19;§ 19. Precompact collections in &;145
10.4.20;§ 20. Convolution takes bounded sets into precompact sets;146
10.4.21;§ 21. Any bounded set of operators is precompact;147
10.4.22;§ 22. Periodic operators;150
10.4.23;§ 23. Continuous operator functions;151
10.4.24;§ 24. Differentiable operator functions;153
10.4.25;§ 25. Integrals of operator functions;156
10.5;CHAPTER V. Power series of operators;157
10.5.1;§ 1. Power series of integrable functions;157
10.5.2;§ 2. A more general case;158
10.5.3;§ 3. Some particular power series;160
10.5.4;§ 4. The classes C{Mn};164
10.5.5;§ 5. The logarithmically convex hull of a sequence;166
10.5.6;§ 6. Power series in the operator s;169
10.5.7;§ 7. The support of power series in s;169
10.6;CHAPTER VI. Laplace transform;173
10.6.1;§ 1. Fundamental properties of the Laplace transform;173
10.6.2;§ 2. Complex inversion formula;174
10.6.3;§ 3. Post's inversion formula;176
10.6.4;§ 4. Laplace transform of convolution;178
10.6.5;§ 5. Finite Laplace transform;178
10.6.6;§ 6. The Laplace transform of a function which vanishes on an initial interval;179
10.6.7;§ 7. The theorem of Titchmarsh for f = g;180
10.6.8;§ 8. The full theorem of Titchmarsh;181
10.6.9;§ 9. Convex sets in Rk;182
10.6.10;§ 10. Functions with compact support in Rk;184
10.6.11;§ 11. A theorem of Lions;184
10.6.12;§ 12. The proof of the theorem of Lions;186
10.6.13;§ 13. A Titchmarsh type theorem for Rk;187
10.6.14;§ 14. Operators in several variables;190
10.7;CHAPTER VII. A class of Dirichlet series;191
10.7.1;§ 1. Introduction;191
10.7.2;§ 2. Hirschman-Widder functions;192
10.7.3;§ 3. A sequence of translated Hirschman-Widder functions;192
10.7.4;§ 4. Entire function generated by a given sequence of exponents;193
10.7.5;§ 5. Generalized Euler's gamma function;196
10.7.6;§ 6. Generalized Phragmén's discontinuity factor;198
10.7.7;§ 7. A theorem on bounded moments;199
10.7.8;§ 8. Titchmarsh's theorem on convolution;201
10.8;CHAPTER VIII. The exponential function exp(–.sa);203
10.8.1;§ 1. Introduction;203
10.8.2;§ 2. Case a < 0;203
10.8.3;§ 3. Case a = 0;204
10.8.4;§ 4. Case 0 < a < 1;204
10.8.5;§ 5. Expansion into Taylor's series;208
10.8.6;§ 6. Function Ua(t);209
10.8.7;§ 7. Real formula of Ua(t);210
10.8.8;§ 8. Properties of Ua(t) on the real axis;212
10.8.9;§ 9. The case of a > 1 and real .;213
10.8.10;§ 10. Real and imaginary part of an operator;215
10.8.11;§ 11. The case of a > 1 and a complex, but not imaginary .;216
10.8.12;§ 12. The case of a = 1 and imaginary .;216
10.8.13;§ 13. The case of a = 1;218
10.8.14;§ 14. Table of existence;218
11;PART VII: FORMULAE AND TABLES;219
11.1;I. Special functions;219
11.1.1;1. Gamma function of Euler;219
11.1.2;2. Error function;219
11.1.3;3. Bessel functions;219
11.2;II. Formulae of the operational calculus;220
11.3;III. Electrotechnical applications;224
11.3.1;1. Equation of the circuit;224
11.3.2;2. Stationary current;224
11.3.3;3. Table of simple quadripoles and their matrices;225
11.4;IV. Tables of functions;226
11.4.1;1. Gamma function of Euler .(.);226
11.4.2;2. Error function erf .;226
11.4.3;3. Bessel function J0(.);227
11.4.4;4. Bessel function J1(.);227
11.4.5;5. Functions J0(i.) and–iJ1(i.);228
12;APPENDIX—SOME TOPICS FROM REAL ANALYSIS;229
12.1;§ 1. Introduction;229
12.2;§ 2. Some properties of Rk;229
12.3;§ 3. A theorem of Carathéodory;232
12.4;§ 4. The Riemann-Lebesgue lemma;236
12.5;§ 5. Metric and normed spaces;238
12.6;§ 6. The Hahn-Banach theorem;239
12.7;§ 7. A corollary to the Hahn-Banach theorem;242
12.8;§ 8. The Birkhoff-Kakutani theorem for topological vector spaces;243
12.9;§ 9. The Arzelä-Ascoli theorem;245
12.10;§ 10. Two lemmas;248
12.11;§ 11. The proof of the Denjoy-Carleman theorem;249
13;Answers to problems;253
14;Bibliography;255
15;Index;259
16;Other titles in the series;262