Introduction to the Theory and Application of the Laplace Transformation

Front Matter....Pages I-VII
Introduction of the Laplace Integral from Physical and Mathematical Points of View....Pages 1-6
Examples of Laplace Integrals. Precise Definition of Integration....Pages 7-13
The Half-Plane of Convergence....Pages 13-18
The Laplace Integral as a Transformation....Pages 19-20
The Unique Inverse of the Laplace Transformation....Pages 20-25
The Laplace Transform as an Analytic Function....Pages 26-30
The Mapping of a Linear Substitution of the Variable....Pages 30-36
The Mapping of Integration....Pages 36-39
The Mapping of Differentiation....Pages 39-43
The Mapping of the Convolution....Pages 44-55
Applications of the Convolution Theorem: Integral Relations....Pages 55-57
The Laplace Transformation of Distributions....Pages 58-60
The Laplace Transforms of Several Special Distributions....Pages 61-63
Rules of Mapping for the L -Transformation of Distributions....Pages 64-69
The Initial Value Problem of Ordinary Differential Equations with Constant Coefficients....Pages 69-86
The Ordinary Differential Equation, specifying Initial Values for Derivatives of Arbitrary Order, and Boundary Values....Pages 86-92
The Solutions of the Differential Equation for Specific Excitations....Pages 92-102
The Ordinary Linear Differential Equation in the Space of Distributions....Pages 103-108
The Normal System of Simultaneous Differential Equations....Pages 109-115
The Anomalous System of Simultaneous Differential Equations, with Initial Conditions which can be fulfilled....Pages 115-125
The Normal System in the Space of Distributions....Pages 125-131
The Anomalous System with Arbitrary Initial Values, in the Space of Distributions....Pages 131-138
The Behaviour of the Laplace Transform near Infinity....Pages 139-147
The Complex Inversion Formula for the Absolutely Converging Laplace Transformation. The Fourier Transformation....Pages 148-161
Deformation of the Path of Integration of the Complex Inversion Integral....Pages 161-169
The Evaluation of the Complex Inversion Integral by Means of the Calculus of Residues....Pages 169-178
The Complex Inversion Formula for the Simply Converging Laplace Transformation....Pages 178-183
Sufficient Conditions for the Representability as a Laplace Transform of a Function....Pages 184-189
A Condition, Necessary and Sufficient, for the Representability as a Laplace Transform of a Distribution....Pages 189-191
Determination of the Original Function by Means of Series Expansion of the Image Function....Pages 192-200
The Parseval Formula of the Fourier Transformation and of the Laplace Transformation. The Image of the Product....Pages 201-217
The Concepts: Asymptotic Representation, Asymptotic Expansion....Pages 218-221
Asymptotic Behaviour of the Image Function near Infinity....Pages 221-230
Asymptotic Behaviour of the Image Function near a Singular Point on the Line of Convergence....Pages 231-233
The Asymptotic Behaviour of the Original Function near Infinity, when the Image Function has Singularities of Unique Character....Pages 234-238
The Region of Convergence of the Complex Inversion Integral with Angular Path. The Holomorphy of the Represented Function....Pages 239-249
The Asymptotic Behaviour of an Original Function near Infinity, when its Image Function is Many-Valued at the Singular Point with Largest Real Part....Pages 250-262
Ordinary Differential Equations with Polynomial Coefficients. Solution by Means of the Laplace Transformation and by Means of Integrals with Angular Path of Integration....Pages 262-278
Partial Differential Equations....Pages 278-303
Integral Equations....Pages 304-312
Back Matter....Pages 313-326