The Laplace Transform

Abstract

The chapter presents a study on the Laplace transform. Treatment of mixed problems is easier when the coefficients of the differential operator A(x, t, ∂/∂x) do not depend on the time variable t. In this case, two fairly general methods, in addition to the abstract method are available: the Laplace transform method and the theory of semigroups. These methods lead to rather precise results, and have a wide range of application, beyond the particular situation in which they are going to be used. The chapter begins with the Laplace transform. The chapter briefly recalls and includes its definition, its main properties—with a twist in the direction of functions, and distributions valued in a Banach space E (where the norm will be denoted by ‖ ‖E. The chapter discusses functions and distributions defined in the real line R. The chapter includes the traditional definition of the Laplace transform of a function f (t). © 1975, Academic Press Inc.