Necessary and sufficient conditions for the representation of a function as a Laplace integral

considering such an integral as a generalization of a Taylor series. All developments of that paper were on the assumption that f(x) permitted of the integral representation (1.1). We wish to study here conditions on f(x), both necessary and sufficient, for the validity of such representation. Following the analogy of Taylor's series we might at first be tempted to suppose that the analyticity of f(x) in a half-plane, the region of convergence of an integral (1.1), would be the condition required. That this is not the case we see at once by recalling that such a function as sin x, analytic in the entire plane, admits of no representationt of the form (1.1). We are led, however, to a correct conjecture by considering our problem as the analogue of the moment problem of F. Hausdorff . ? This is the problem of determining a function X(x) bounded and non-decreasing in the interval O?? _ and such that