UID:
almafu_9958085679702883
Umfang:
1 online resource (271 p.)
ISBN:
1-281-76771-9
,
9786611767716
Serie:
Pure and applied mathematics ; v. 42
Inhalt:
An introduction to transform theory
Anmerkung:
Description based upon print version of record.
,
Front Cover; An Introduction to Transform Theory; Copyright Page; Contents; Preface; Symbols and Notation; Chapter 1. Introduction; 1. Introduction; 2. A brief Table of Transforms; 3. Solution of Differential Equations; 4. The Product Theorem; 5. Integral Equations; Exercises; Chapter 2. Dirichlet Series; 1. Introduction; 2. Convergence Tests; 3. Convergence of Dirichlet Series; 4. Analyticity; 5 . Uniform Convergence; 6. Formulas for σc and σa; 7. Uniqueness; 8. Behavior on Vertical Lines; 9. Inversion; 10. A Mean-Value Theorem; 11. Analytic Behavior of the Sum of a Dirichlet Series
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12. SummaryExercises; Chapter 3. The Zeta Function; 1. Introduction; 2. Analytic Nature of ζ(s); 3. Euler Product for ζ(s); 4. The Zeros of ζ(s); 5. Order of ζ(s) and ζ'(s) on Vertical Lines; 6. The Reciprocal of ζ(s); 7. The Functional Equation for ζ(s); 8. Summary; Exercises; Chapter 4. The Prime Number Theorem; 1. Introduction; 2. The Function π(x); 3. The Function J(x); 4. The Function ψ(x); 5. Five Lemmas; 6. Background and Proof of the Prime Number Theorem; 7. Further Developments; 8. Summary; Exercises; Chapter 5. The Laplace Transform; 1. Introduction; 2. Definitions and Examples
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3. Convergence4. Uniform Convergence; 5. Formulas for σc and σa; 6. Behavior on Vertical Lines; 7. Inversion; 8. Convolutions; 9. Fractional Integrals; 10. Analytic Behavior of Generating Functions; 11. Representation; 12. Generating Functions Analytic at Infinity; 13. The Stieltjes Transform; 14. Inversion of the Stieltjes Transform; 15. Summary; Exercises; Chapter 6. Real Inversion Theory; 1. Introduction; 2. Laplace's Asymptotic Method; 3. Real Inversion of the Laplace Transform; 4. The Stieltjes Transform; 5. The Hausdorff Moment Problem; Uniqueness; 6. Hausdorff's Moment Theorem
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7. Bernstein's Theorem8. Bounded Determining Function; 9. An Application of Bernstein's Theorem; 10. Completely Convex Functions; 11. Summary; Exercises; Chapter 7. The Convolution Transform; 1. Introduction; 2. Definitions and Examples; 3. Operational Calculus; 4. The Laguerre-Pólya Class; 5. Some Statistical Terms; 6. Properties of the Laguerre-Pólya Kernels; 7. Inversion; 8. The Laplace Transform as a Convolution; 9. The Stieltjes Transform as a Convolution; 10. Summary; Exercises; Chapter 8. Tauberian Theorems; 1. Introduction; 2. Integral Analogs; 3. A Basic Theorem
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4. Hardy's and Littlewood's Integral Tauberian Theorems5. One-sided Tauberian Conditions; 6. One-sided Version of Littlewood's Integral Theorem; 7. Classical Series Results; 8. Summary; Exercises; Chapter 9. Inversion by Series; 1. Introduction; 2. The Potential Transform; 3. A Brief Table; 4. The Inversion Algorithm; 5. The Inversion Operator; 6. Series Inversion; 7. Relation to Potential Theory; 8. Relation to the Sine Transform; 9. The Laplace Transform; 10. Series Inversion of the Laplace Transform; 11. Summary; Exercises; Bibliography; Index; Pure and Applied Mathematics
,
English
Weitere Ausg.:
ISBN 0-12-748550-3
Weitere Ausg.:
ISBN 0-08-087355-3
Sprache:
Englisch
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