Laplace Transforms

We assume that the reader will have some degree of familiarity with the subject of classical Laplace transformation such as presented for example in [14, Graf]. After a discussion on loop integrals of Hankel type and their ramifications, some facts about the classical two-sided Laplace transformation are recalled. Then, the two subclasses of hyperfunctions, the right-sided and left-sided originals and their Laplace transforms, are defined by using loop integrals. The Laplace transform of a hyperfunction with an arbitrary support is handled by decomposing it into a sum of a left-sided and right-sided original (canonical splitting); it is then shown that its practical computation can be reduced to the evaluation of two right-sided Laplace transforms. Many concrete examples of Laplace transforms of hyperfunctions are presented. The operational rules of Laplace transforms of hyperfunctions are clearly stated. The subject of inverse Laplace transforms and convolutions follows. Fractional integrals and derivatives of right-sided hyperfunctions are briefly over-viewed. The application track with Volterra integral equations and convolution integral equations over an infinite range concludes the chapter. This chapter represents the core of the integral transformations part of the book. The following chapters on Fourier and Mellin transformations are heavily based on the results of this chapter. Another similar approach, due to Komatsu introducing the so-called Laplace hyperfunctions, is sketched in the Appendix.