introduction to transform theory (eBook)

Front Cover 1
An Introduction to Transform Theory 4
Copyright Page 5
Contents 6
Preface 10
Symbols and Notation 14
Chapter 1. Introduction 16
1. Introduction 16
2. A brief Table of Transforms 18
3. Solution of Differential Equations 20
4. The Product Theorem 23
5. Integral Equations 26
Exercises 30
Chapter 2. Dirichlet Series 34
1. Introduction 34
2. Convergence Tests 35
3. Convergence of Dirichlet Series 37
4. Analyticity 40
5 . Uniform Convergence 42
6. Formulas for sc and sa 44
7. Uniqueness 49
8. Behavior on Vertical Lines 51
9. Inversion 53
10. A Mean-Value Theorem 59
11. Analytic Behavior of the Sum of a Dirichlet Series 61
12. Summary 63
Exercises 63
Chapter 3. The Zeta Function 66
1. Introduction 66
2. Analytic Nature of .(s) 66
3. Euler Product for .(s) 68
4. The Zeros of .(s) 70
5. Order of .(s) and .'(s) on Vertical Lines 71
6. The Reciprocal of .(s) 73
7. The Functional Equation for .(s) 75
8. Summary 80
Exercises 81
Chapter 4. The Prime Number Theorem 84
1. Introduction 84
2. The Function p(x) 84
3. The Function J(x) 89
4. The Function .(x) 91
5. Five Lemmas 97
6. Background and Proof of the Prime Number Theorem 100
7. Further Developments 102
8. Summary 104
Exercises 105
Chapter 5. The Laplace Transform 108
1. Introduction 108
2. Definitions and Examples 109
3. Convergence 111
4. Uniform Convergence 113
5. Formulas for sc and sa 114
6. Behavior on Vertical Lines 117
7. Inversion 119
8. Convolutions 125
9. Fractional Integrals 128
10. Analytic Behavior of Generating Functions 131
11. Representation 133
12. Generating Functions Analytic at Infinity 137
13. The Stieltjes Transform 140
14. Inversion of the Stieltjes Transform 141
15. Summary 144
Exercises 146
Chapter 6. Real Inversion Theory 148
1. Introduction 148
2. Laplace’s Asymptotic Method 149
3. Real Inversion of the Laplace Transform 155
4. The Stieltjes Transform 157
5. The Hausdorff Moment Problem Uniqueness
6. Hausdorff’s Moment Theorem 163
7. Bernstein’s Theorem 169
8. Bounded Determining Function 172
9. An Application of Bernstein’s Theorem 175
10. Completely Convex Functions 176
11. Summary 179
Exercises 180
Chapter 7. The Convolution Transform 184
1. Introduction 184
2. Definitions and Examples 185
3. Operational Calculus 186
4. The Laguerre–Pólya Class 188
5. Some Statistical Terms 189
6. Properties of the Laguerre–Pólya Kernels 190
7. Inversion 194
8. The Laplace Transform as a Convolution 198
9. The Stieltjes Transform as a Convolution 201
10. Summary 204
Exercises 204
Chapter 8. Tauberian Theorems 208
1. Introduction 208
2. Integral Analogs 211
3. A Basic Theorem 214
4. Hardy’s and Littlewood‘s Integral Tauberian Theorems 218
5. One-sided Tauberian Conditions 221
6. One-sided Version of Littlewood’s Integral Theorem 224
7. Classical Series Results 228
8. Summary 231
Exercises 231
Chapter 9. Inversion by Series 234
1. Introduction 234
2. The Potential Transform 235
3. A Brief Table 236
4. The Inversion Algorithm 238
5. The Inversion Operator 240
6. Series Inversion 242
7. Relation to Potential Theory 245
8. Relation to the Sine Transform 246
9. The Laplace Transform 249
10. Series Inversion of the Laplace Transform 251
11. Summary 255
Exercises 256
Bibliography 258
Index 262
Pure and Applied Mathematics 269