Introduction to transform theory

Front Matter......Page _0003.djvu
Preface......Page _0008.djvu
Symbols and Notation......Page _0012.djvu
1. Introduction......Page 0001.djvu
2. A brief Table of Transforms......Page 0003.djvu
3. Solution of Differential Equations......Page 0005.djvu
4. The Product Theorem......Page 0008.djvu
5. Integral Equations......Page 0011.djvu
1. Introduction......Page 0019.djvu
2. Convergence Tests......Page 0020.djvu
3. Convergence of Dirichlet Series......Page 0022.djvu
4. Analyticity......Page 0025.djvu
5. Uniform Convergence......Page 0027.djvu
6. Formulas for a sigma_c and sigma_a......Page 0029.djvu
7. Uniqueness......Page 0034.djvu
8. Behavior on Vertical Lines......Page 0036.djvu
9. Inversion......Page 0038.djvu
10. A Mean-Value Theorem......Page 0044.djvu
11. Analytic Behavior of the Sum of a Dirichlet Series......Page 0046.djvu
Exercises......Page 0048.djvu
2. Analytic Nature of zeta(s)......Page 0051.djvu
3. Euler Product for zeta(s)......Page 0053.djvu
4. The Zeros of zeta(s)......Page 0055.djvu
5. Order of zeta(s) and zeta'(s) on Vertical Lines......Page 0056.djvu
6. The Reciprocal of zeta(s)......Page 0058.djvu
7. The Functional Equation for zeta(s)......Page 0060.djvu
8. Summary......Page 0065.djvu
Exercises......Page 0066.djvu
1. Introduction......Page 0069.djvu
2. The Function pi(x)......Page
3. The Function phi(x)......Page 0074.djvu
4. The Function psi (x)......Page 0076.djvu
5. Five Lemmas......Page 0082.djvu
6. Background and Proof of the Prime Number Theorem......Page 0085.djvu
7. Further Developments......Page 0087.djvu
8. Summary......Page 0089.djvu
Exercises......Page 0090.djvu
1. Introduction......Page 0093.djvu
2. Definitions and Examples......Page 0094.djvu
3. Convergence......Page 0096.djvu
4. Uniform Convergence......Page 0098.djvu
5. Formulas for a sigma_c and sigma_a......Page 0099.djvu
6. Behavior on Vertical Lines......Page 0102.djvu
7. Inversion......Page 0104.djvu
8. Convolutions......Page 0110.djvu
9. Fractional Integrals......Page 0113.djvu
10. Analytic Behavior of Generating Functions......Page 0116.djvu
11. Representation......Page 0118.djvu
12. Generating Functions Analytic at Infinity......Page 0122.djvu
13. The Stieltjes Transform......Page 0125.djvu
15. Summary......Page 0126.djvu
Exercises......Page 0131.djvu
1. Introduction......Page 0133.djvu
2. Laplace's Asymptotic Method......Page 0134.djvu
3. Real Inversion of the Laplace Transform......Page 0140.djvu
4. The Stieltjes Transform......Page 0142.djvu
5. The Hausdorff Moment Problem; Uniqueness......Page 0145.djvu
6. Hausdorff's Moment Theorem......Page 0148.djvu
7. Bernstein's Theorem......Page 0154.djvu
8. Bounded Determining Function......Page 0157.djvu
9. An Application of Bernstein's Theorem......Page 0160.djvu
10. Completely Convex......Page 0161.djvu
Exercises......Page 0164.djvu
1. Introduction......Page 0169.djvu
2. Definitions and Examples......Page 0170.djvu
3. Operational Calculus......Page 0171.djvu
4. The Laguerre-Polya Class......Page 0173.djvu
5. Some Statistical Terms......Page 0174.djvu
6. Properties of the Laguerre-Polya Kernels......Page 0175.djvu
7. Inversion......Page 0179.djvu
8. The Laplace Transform as a Convolution......Page 0183.djvu
9. The Stieltjes Transform as a Convolution......Page 0186.djvu
Exercises......Page 0189.djvu
1. Introduction......Page 0193.djvu
2. Integral Analogs......Page 0196.djvu
3. A Basic Theorem......Page 0199.djvu
4. Hardy's and Littlewood's Integral Tauberian Theorems......Page 0203.djvu
5. One-Sided Tauberian Conditions......Page 0206.djvu
6. One-Sided Version of Littlewood's Integral Theorem......Page 0209.djvu
7. Classical Series Results......Page 0213.djvu
8. Summary......Page 0216.djvu
1. Introduction......Page 0219.djvu
2. The Potential Transfor......Page 0220.djvu
3. A Brief Table......Page 0221.djvu
4. The Inversion Algorithm......Page 0223.djvu
5. The Inversion Operator......Page 0225.djvu
6. Series Inversion......Page 0227.djvu
7. Relation to Potential Theory......Page 0230.djvu
8. Relation to the Sine Transform......Page 0231.djvu
9. The Laplace Transform......Page 0234.djvu
10. Series Inversion of the Laplace Transform......Page 0236.djvu
11. Summary......Page 0240.djvu
Exercises......Page 0241.djvu
Bibliography......Page 0243.djvu
Index......Page 0247.djvu