Titlebook: Integral Transforms and their Applications; B. Davies Book 19852nd edition Springer Science+Business Media New York 1985 Applications.Inte

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The Inversion Theoremm a powerful tool in applications, we must first take note of some results from classical analysis.. Suppose that f(x) is a function continuous on the closed interval a ≤ x ≤ b (and hence uniformly continuous).

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https://doi.org/10.1007/978-1-4899-2691-3Applications; Integral; Integraltransformation; Transforms; integral transform

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Integral Transforms and their Applications978-1-4899-2691-3Series ISSN 0066-5452 Series E-ISSN 2196-968X

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Ordinary Differential EquationsLinear differential equations with constant coefficients are an important area of application of the Laplace transform. As a prelude to the discussion of such problems we discuss first two particularly simple examples, since the connection with the classical methods of solution is readily apparent in these cases.

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Partial Differential EquationsAs an example to show how the Laplace transform may be applied to the solution of partial differential equations, we consider the diffusion of heat in an isotropic solid body.

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Definitions and Elementary PropertiesLet f(t) be an arbitrary function; then the (exponential) Fourier transform of f(t) is the function defined by the integral.for those values of ω for which the integral exists. We shall usually refer to (1) as the Fourier transform, omitting any reference to the term exponential.

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Generalized FunctionsThe subject of generalized functions is an enormous one, and we refer the reader to one of the excellent modern books. for a full account of the theory. We will sketch in this section some of the more elementary aspects of the theory, because the use of generalized functions adds considerably to the power of the Fourier transform as a tool.