Introduction to the theory and application of the Laplace transformation with a table of Laplace transforms

In anglo-american literature there exist numerous books, devoted to the application of the Laplace transformation in technical domains such as electrotechnics, mechanics etc. Chiefly, they treat problems which, in mathematical language, are governed by ordi­ nary and partial differential equations, in various physically dressed forms. The theoretical foundations of the Laplace transformation are presented usually only in a simplified manner, presuming special properties with respect to the transformed func­ tions, which allow easy proofs. By contrast, the present book intends principally to develop those parts of the theory of the Laplace transformation, which are needed by mathematicians, physicists a,nd engineers in their daily routine work, but in complete generality and with detailed, exact proofs. The applications to other mathematical domains and to technical prob­ lems are inserted, when the theory is adequately· developed to present the tools necessary for their treatment. Since the book proceeds, not in a rigorously systematic manner, but rather from easier to more difficult topics, it is suited to be read from the beginning as a textbook, when one wishes to familiarize oneself for the first time with the Laplace transforma­ tion. For those who are interested only in particular details, all results are specified in'Theorems'with explicitly formulated assumptions and assertions. Chapters 1-14 treat the question of convergence and the mapping properties of the Laplace transformation. The interpretation of the transformation as the mapping of one function space to another (original and image functions) constitutes the dom­ inating idea of all subsequent considerations. 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 In anglo-american literature there exist numerous books, devoted to the application of the Laplace transformation in technical domains such as electrotechnics, mechanics etc. Chiefly, they treat problems which, in mathematical language, are governed by ordi nary and partial differential equations, in various physically dressed forms. The theoretical foundations of the Laplace transformation are presented usually only in a simplified manner, presuming special properties with respect to the transformed func tions, which allow easy proofs. By contrast, the present book intends principally to develop those parts of the theory of the Laplace transformation, which are needed by mathematicians, physicists a, nd engineers in their daily routine work, but in complete generality and with detailed, exact proofs. The applications to other mathematical domains and to technical prob lems are inserted, when the theory is adequately. developed to present the tools necessary for their treatment. Since the book proceeds, not in a rigorously systematic manner, but rather from easier to more difficult topics, it is suited to be read from the beginning as a textbook, when one wishes to familiarize oneself for the first time with the Laplace transforma tion. For those who are interested only in particular details, all results are specified in "Theorems" with explicitly formulated assumptions and assertions. Chapters 1-14 treat the question of convergence and the mapping properties of the Laplace transformation. The interpretation of the transformation as the mapping of one function space to another (original and image functions) constitutes the dom inating idea of all subsequent considerations."