On a Generalisation of the Laplace Transformation

Abstract

On reading a recent paper by R. S. Varma (Varma 1949) I recalled that in May 1942 I investigated an integral transformation which is very similar to Varma's. Varma hasand points out that this reduces to a Laplace integral for k = ¼, m = ± ¼. Instead of (1), one could consider the integralwhich was introduced by C. S. Meijer (Meijer 1940 b); this integral reduces to a Laplace integral whenever k = m + ½. Now, apart from comparatively unimportant factors, the nucleus of (2) is a fractional derivative or integral, as the case may be, of e−st, and on carrying out a fractional integration by parts, it appears that (2) is essentially the Laplace transform of a fractional integral or derivative of f. Thus, the whole theory of the transformation (2), including inversion formulae, representation theorems, etc., can be deduced from the well-known theory of the Laplace transformation. It is not quite clear that a similar reduction is possible for (1), although it is certainly possible when k = 0.