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

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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.

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Integrals Involving a ParameterConsider the function g(γ) defined by..

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Dual Integral EquationsTo motivate this section, we first solve a classical problem of electrostatics. We wish to find the electrostatic potential φ created by an isolated thin conducting disc of radius a, whose potential is V.

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Integral Transforms Generated by Green’s FunctionsIn this section we will investigate (in a purely formal manner) some properties of the self-adjoint differential operator [see (10.15)].where p(x) and q(x) are given functions on the interval a ≤ x ≤ b, and the functions u(x) under consideration all satisfy homogeneous boundary conditions of the type [see (10.2)]