Titlebook: Integral Transforms and Their Applications; Brian Davies Book 2002Latest edition Springer-Verlag New York 2002 Fourier series.Fourier tran

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Integral Equations,In this short chapter, we consider some integral equations that may be solved using the Laplace transform.

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The Fourier Transform,The Fourier transform has its origins in the concept of Fourier series, developed by Joseph Fourier early in the nineteenth century. It is often treated in the framework of real variable functions and/or Hilbert spaces; here we choose rather to emphasize the power of complex variable theory..

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Generalized Functions,The subject of generalized functions is an enormous one, and we refer to one of the excellent specialized books. for a full account of the theory. We will sketch in this chapter 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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The Mellin Transform,In this and the next chapter, we study the Mellin transform, which, while closely related to the Fourier transform, has its own peculiar uses. In particular, it turns out to be a most convenient tool for deriving asymptotic expansions, although it has other applications..

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Application to Sums and Integrals,This chapter explores some uses of the Mellin transform in obtaining analytic and asymptotic information about infinite sums and integrals involving a parameter. The Riemann zeta function, introduced in Section 1.8, plays a central role for the former.

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Methods Based on Cauchy Integrals,The major difficulty in using the Wiener-Hopf technique is the problem of constructing a suitable factorization. We consider in this chapter methods based on contour integration which leads, by natural extensions, to the use of Cauchy integrals in the solution of mixed boundary-value problems.

DEMUR

Numerical Inversion of Laplace Transforms,There are many problems whose solution may be found in terms of a Laplace or Fourier transform, which is then too complicated for inversion using the techniques of complex analysis. In this chapter, we discuss some of the methods that have been developed—and in some cases are still being developed—for the numerical evaluation of the inverse.

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