Introduction to the Theory and Application of the Laplace Transformation

Cover......Page 1
Title......Page 2
Copyright......Page 3
Preface......Page 4
Table of Contents......Page 6
1. Introduction of the Laplace Integral from Physical and Mathematical Points of View......Page 10
2. Examples of Laplace Integrals. Precise Definition of Integration......Page 16
3. The Half-Plane of Convergence......Page 22
4. The Laplace Integral as a Transformation......Page 28
5. The Unique Inverse of the Laplace Transformation......Page 29
6. The Laplace Transform as an Analytic Function......Page 35
7. The Mapping of a Linear Substitution of the V mabIe......Page 39
8. The Mapping of Integration......Page 45
9. The Mapping of Differentiation......Page 48
10. The Mapping of the Convolution......Page 53
11. Applications of the Convolution Theorem: Integral Relations......Page 64
12. The Laplace Transformation of Distributions......Page 67
13. The Laplace Transforms of Several Special Distributions......Page 70
14. Rules of Mapping for the ~-Transformation of Distributions......Page 73
15. The Initial Value Problem of Ordinary Differential Equations with Constant Coefficients......Page 78
16. The Ordinary Differential Equation, specifying Initial Values for Derivatives of Arbitrary Order, and Boundary Values......Page 95
17. The Solutions of the Differential Equation for Specific Excitations......Page 101
18. The Ordinary Linear Differential Equation in the Space of Distributions......Page 112
19. The Normal System of Simultaneous Differential Equations......Page 118
20. The Anomalous System of Simultaneous Differential Equations, with Initial Conditions which can be fulfilled......Page 124
21. The Normal System in the Space of Distributions......Page 134
22. The Anomalous System with Arbitrary Initial Values, in the Space of Distributions......Page 140
23. The Behaviour of the Laplace Transform near Infinity......Page 148
24. The Complex Inversion Formula for the Absolutely Converging Laplace Transformation. The Fourier Transformation......Page 157
25. Deformation of the Path of Integration of the Complex Inversion Integral......Page 170
26. The Evaluation of the Complex Inversion Integral by Means of the Calculus of Residues......Page 178
27. The Complex Inversion Formula for the Simply Converging Laplace Transformation......Page 187
28. Sufficient Conditions for the Representability as a Laplace Transform of a Function......Page 193
29. A Condition, Necessary and Sufficient, for the Representability as a Laplace Transform of a Distribution......Page 198
30. Determination of the Original Function by Means of Series Expansion of the Image Function......Page 201
31. The Parseval Formula of the Fourier Transformation and of the Laplace Transformation. The Image of the Product......Page 210
32. The Concepts: Asymptotic Representation, Asymptotic Expansion......Page 227
33. Asymptotic Behaviour of the Image Function near Infinity......Page 230
34. Asymptotic Behaviour of the Image Function near a Singular Point on the Line of Convergence......Page 240
35. The Asymptotic Behaviour of the Original Function near Infinity, when the Image Function has Singularities of Unique Character......Page 243
36. The Region of Convergence of the Complex Inversion Integral with Angular Path. The Holomorphy of the Represented Function......Page 248
37. The Asymptotic Behaviour of an Original Function near Infinity, when its Image Function is Many-Valued at the Singular Point with Largest Real Part......Page 259
38. Ordinary Differential Equations with Polynomial Coefficients. Solution by Means of the Laplace Transformation and by Means of Integrals with Angular Path of Integration......Page 271
39. Partial Di1ferential Equations......Page 287
40. Integral Equations......Page 313
APPENDIX......Page 322
Table of Laplace Transforms......Page 326
INDEX......Page 331