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# Introduction to the Theory and Application of the Laplace Transformation by Gustav Doetsch (Paperback, 2011)

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## About this product

### Key Features

- Author(s)Gustav Doetsch
- PublisherSpringer-Verlag Berlin and Heidelberg GmbH & Co. KG
- Date of Publication12/11/2011
- Language(s)English
- FormatPaperback
- ISBN-103642656927
- ISBN-139783642656927
- GenreMathematics

### Publication Data

- Place of PublicationBerlin
- Country of PublicationGermany
- ImprintSpringer-Verlag Berlin and Heidelberg GmbH & Co. K
- Content Notebiography

### Dimensions

- Weight587 g
- Width170 mm
- Height244 mm
- Spine18 mm
- Pagination327

### Credits

- Edited byW. Nader
- Translated byW. Nader

### Editorial Details

- Format DetailsTrade paperback (US)
- Edition StatementSoftcover reprint of the original 1st ed. 1974

### Description

- Table Of Contents1. 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

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