Euler's Version of the Laplace Transform

1. Historical Introduction. The modern version of the Laplace Transform, now a standard part of most undergraduate mathematics and engineering courses, is, in its detailed working out and its widespread acceptance, a quite recent development. It may most conveniently be dated from the publication of Doetsch's Theorie und Anwendung der Laplace-Transformation [3] in 1937. The subject has, however, a much longer history than this. Its early beginnings have been traced back to Euler, by, inter alia, Laplace himself [6, p. 88]. It is not the purpose of this paper to trace this history (a detailed account is in preparation for publication elsewhere), but rather to show by specific examples how the earlier versions of the theory can be made to work in practice-made to work, in fact, in cases where the standard modem theory breaks down. 2. Eulerian Theory. Euler in several papers considers particular integrals of the form