Introduction to the Theory and Application of the Laplace Transformation

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.

1. Introduction of the Laplace Integral from Physical and Mathematical Points of View.- 2. Examples of Laplace Integrals. Precise Definition of Integration.- 3. The Half-Plane of Convergence.- 4. The Laplace Integral as a Transformation.- 5. The Unique Inverse of the Laplace Transformation.- 6. The Laplace Transforrp. as an Analytic Function.- 7. The Mapping of a Linear Substitution of the Variable.- 8. The Mapping of Integration.- 9. The Mapping of Differentiation.- 10. The Mapping of the Convolution.- 11. Applications of the Convolution Theorem: Integral Relations.- 12. The Laplace Transformation of Distributions.- 13. The Laplace Transforms of Several Special Distributions.- 14. Rules of Mapping for the Q-Transformation of Distributions.- 15. The Initial Value Problem of Ordinary Differential Equations with Constant Coefficients.- The Differential Equation of the First Order.- Partial Fraction Expansion of a Rational Function.- The Differential Equation of Order n.- 1. The homogeneous differential equation with arbitrary initial values.- 2. The inhomogeneous differential equation with vanishing initial values 82 The Transfer Function.- 16. The Ordinary Differential Equation, specifying Initial Values for Derivatives of Arbitrary Order, and Boundary Values.- 17. The Solutions of the Differential Equation for Specific Excitations.- 1. The Step Response.- 2. Sinusoidal Excitations. The Frequency Response.- 18. The Ordinary Linear Differential Equation in the Space of Distributions.- The Impulse Response.- Response to the Excitation ?(m).- The Response to Excitation by a Pseudofunction.- A New Interpretation of the Concept Initial Value.- 19. The Normal System of Simultaneous Differential Equations.- 1. The Normal Homogeneous System, for Arbitrary Initial Values.- 2. The Normal Inhomogeneous System with Vanishing Initial Values.- 20. The Anomalous System of Simultaneous Differential Equations, with Initial Conditions which can be fulfilled.- 21. The Normal System in the Space of Distributions.- 22. The Anomalous System with Arbitrary Initial Values, in the Space of Distributions.- 23. The Behaviour of the Laplace Transform near Infinity.- 24. The Complex Inversion Formula for the Absolutely Converging Laplace Transformation. The Fourier Transformation.- 25. Deformation of the Path of Integration of the Complex Inversion Integral.- 26. The Evaluation of the Complex Inversion Integral by Means of the Calculus of Residues.- 27. The Complex Inversion Formule for the Simply Converging Laplace Transformation.- 28. Sufficient Conditions for the Representability as a Laplace Transform of a Function.- 29. A Condition, Necessary and Sufficient, for the Representability as a Laplace Transform of a Distribution.- 30. Determination of the Original Function by Means of Series Expansion of the Image Function.- 31. The Parseval Formula of the Fourier Transformation and of the Laplace Transformation. The Image of the Product.- 32. The Concepts: Asymptotic Representation, Asymptotic Expansion.- 33. Asymptotic Behaviour of the Image Function near Infinity.- Asymptotic Expansion of Image Functions.- 34. Asymptotic Behaviour of the Image Function near a Singular Point on the Line of Convergence.- 35. The Asymptotic Behaviour of the Original Function near Infinity, when the Image Function has Singularities of Unique Character.- 36. The Region of Convergence of the Complex Inversion Integral with Angular Path. The Holomorphy of the Represented Function.- 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.- 38. Ordinary Differential Equations with Polynomial Coefficients. Solution by Means of the Laplace Transformation and by Means of Integrals with Angular Path of Integration.- The Differential Equation of the Bessel Functions.- The General Linear Homogeneous Differential Equation with Linear Coefficients.- 39. Partial Differential Equations.- 1. The Equation of Diffusion or Heat Conduction.- The Case of Infinite Length.- The Case of Finite Length.- 1. Thermal Conductor with Vanishing Initial Temperature.- 2. Thermal Conductor with Vanishing End Temperatures.- Asymptotic Expansion of the Solution.- 2. The Telegraph Equation.- Asymptotic Expansion of the Solution.- 40. Integral Equations.- 1. The Linear Integral Equation of the Second Kind, of the Convolution Type.- 2. The Linear Integral Equation of the First Kind, of the Convolution Type.- The Abel Integral Equation.- APPENDIX: Some Concepts and Theorems from the Theory of Distributions.- Operations.- Functions and Distributions.