Titlebook: Introduction to the Theory and Application of the Laplace Transformation; Gustav Doetsch Book 1974 Springer-Verlag Berlin Heidelberg 1974

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The Mapping of the Convolution,vestigate operations which involve more than one function, like sum or product of functions. The formula for the sum of functions . is immediately obvious. However, the image of the product of two original functions, . · . say, is quite complicated, thus necessitating that its study be deferred to Chapter 31.

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Introduction of the Laplace Integral from Physical and Mathematical Points of View,The integral . is known as the Laplace integral; ., the dummy variable of integration, scans the real numbers between 0 and ∞, and the parameter . may be real-valued or complex-valued. Should this integral converge for some values of ., then it defines a function .(.):..

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The Half-Plane of Convergence,Reviewing the examples of Chapter 2, we observe that for each of these functions the Laplace integral converges in a right half-plane. We shall show in_this Chapter that this is generally true for Laplace integrals. Prior to that, we shall determine the domain of absolute convergence of a Laplace integral.

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The Laplace Transform as an Analytic Function,On p. 5 we developed the Laplace integral as a continuous analogue of the power series. In this Chapter, we shall demonstrate that a Laplace integral, like a power series, always represents an analytic function.

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The Mapping of Differentiation,Using Theorem 8.1, we shall derive, in this Chapter, Theorem 9.1, which provides the image of differentiation. The latter will prove extremely useful in practical applications of the .-transformation. A few introductory remarks will aid the subsequent development.

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Applications of the Convolution Theorem: Integral Relations,The .-transformation permits the transformation of the convolution, a complicated integral representation, into a simple algebraic product. This facility can be utilized to produce simple proofs of integral relations which are otherwise difficult to verify.

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