Introduction to the Theory and Application of the Laplace Transformation

Table of Contents

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   Book Overview
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   Chapter 1 Introduction of the Laplace Integral from Physical and Mathematical Points of View
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   Chapter 2 Examples of Laplace Integrals. Precise Definition of Integration
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   Chapter 3 The Half-Plane of Convergence
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   Chapter 4 The Laplace Integral as a Transformation
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   Chapter 5 The Unique Inverse of the Laplace Transformation
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   Chapter 6 The Laplace Transform as an Analytic Function
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   Chapter 7 The Mapping of a Linear Substitution of the Variable
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   Chapter 8 The Mapping of Integration
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    Chapter 9 The Mapping of Differentiation
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    Chapter 10 The Mapping of the Convolution
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    Chapter 11 Applications of the Convolution Theorem: Integral Relations
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    Chapter 12 The Laplace Transformation of Distributions
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    Chapter 13 The Laplace Transforms of Several Special Distributions
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    Chapter 14 Rules of Mapping for the L -Transformation of Distributions
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    Chapter 15 The Initial Value Problem of Ordinary Differential Equations with Constant Coefficients
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    Chapter 16 The Ordinary Differential Equation, specifying Initial Values for Derivatives of Arbitrary Order, and Boundary Values
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    Chapter 17 The Solutions of the Differential Equation for Specific Excitations
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    Chapter 18 The Ordinary Linear Differential Equation in the Space of Distributions
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    Chapter 19 The Normal System of Simultaneous Differential Equations
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    Chapter 20 The Anomalous System of Simultaneous Differential Equations, with Initial Conditions which can be fulfilled
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    Chapter 21 The Normal System in the Space of Distributions
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    Chapter 22 The Anomalous System with Arbitrary Initial Values, in the Space of Distributions
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    Chapter 23 The Behaviour of the Laplace Transform near Infinity
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    Chapter 24 The Complex Inversion Formula for the Absolutely Converging Laplace Transformation. The Fourier Transformation
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    Chapter 25 Deformation of the Path of Integration of the Complex Inversion Integral
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    Chapter 26 The Evaluation of the Complex Inversion Integral by Means of the Calculus of Residues
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    Chapter 27 The Complex Inversion Formula for the Simply Converging Laplace Transformation
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    Chapter 28 Sufficient Conditions for the Representability as a Laplace Transform of a Function
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    Chapter 29 A Condition, Necessary and Sufficient, for the Representability as a Laplace Transform of a Distribution
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    Chapter 30 Determination of the Original Function by Means of Series Expansion of the Image Function
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    Chapter 31 The Parseval Formula of the Fourier Transformation and of the Laplace Transformation. The Image of the Product
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    Chapter 32 The Concepts: Asymptotic Representation, Asymptotic Expansion
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    Chapter 33 Asymptotic Behaviour of the Image Function near Infinity
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    Chapter 34 Asymptotic Behaviour of the Image Function near a Singular Point on the Line of Convergence
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    Chapter 35 The Asymptotic Behaviour of the Original Function near Infinity, when the Image Function has Singularities of Unique Character
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    Chapter 36 The Region of Convergence of the Complex Inversion Integral with Angular Path. The Holomorphy of the Represented Function
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    Chapter 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
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    Chapter 38 Ordinary Differential Equations with Polynomial Coefficients. Solution by Means of the Laplace Transformation and by Means of Integrals with Angular Path of Integration
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    Chapter 39 Partial Differential Equations
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    Chapter 40 Integral Equations