Introduction to the Theory and Application of the Laplace Transformation

1. Front Matter

   Pages I-VII

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2. Introduction of the Laplace Integral from Physical and Mathematical Points of View

   • Gustav Doetsch

   Pages 1-6

3. Examples of Laplace Integrals. Precise Definition of Integration

   • Gustav Doetsch

   Pages 7-13

4. The Half-Plane of Convergence

   • Gustav Doetsch

   Pages 13-18

5. The Laplace Integral as a Transformation

   • Gustav Doetsch

   Pages 19-20

6. The Unique Inverse of the Laplace Transformation

   • Gustav Doetsch

   Pages 20-25

7. The Laplace Transform as an Analytic Function

   • Gustav Doetsch

   Pages 26-30

8. The Mapping of a Linear Substitution of the Variable

   • Gustav Doetsch

   Pages 30-36

9. The Mapping of Integration

   • Gustav Doetsch

   Pages 36-39

10. The Mapping of Differentiation

    • Gustav Doetsch

    Pages 39-43

11. The Mapping of the Convolution

    • Gustav Doetsch

    Pages 44-55

12. Applications of the Convolution Theorem: Integral Relations

    • Gustav Doetsch

    Pages 55-57

13. The Laplace Transformation of Distributions

    • Gustav Doetsch

    Pages 58-60

14. The Laplace Transforms of Several Special Distributions

    • Gustav Doetsch

    Pages 61-63

15. Rules of Mapping for the L-Transformation of Distributions

    • Gustav Doetsch

    Pages 64-69

16. The Initial Value Problem of Ordinary Differential Equations with Constant Coefficients

    • Gustav Doetsch

    Pages 69-86

17. The Ordinary Differential Equation, specifying Initial Values for Derivatives of Arbitrary Order, and Boundary Values

    • Gustav Doetsch

    Pages 86-92

18. The Solutions of the Differential Equation for Specific Excitations

    • Gustav Doetsch

    Pages 92-102

19. The Ordinary Linear Differential Equation in the Space of Distributions

    • Gustav Doetsch

    Pages 103-108

20. The Normal System of Simultaneous Differential Equations

    • Gustav Doetsch

    Pages 109-115

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.

• Laplace-Transformation
• equation
• function
• proof
• theorem

• Emeritus of Mathematics, University of Freiburg, Freiburg i. Br., Germany

  Gustav Doetsch