# Sufficient Conditions for the Representability as a Laplace Transform of a Function

## Abstract

When asking what class of functions can be represented by a power series, \(\sum\limits_{v = 0}^\infty {{a_v}{z^v}}\), we obtain a simple answer: all those functions which are analytic on a circular disk centred at the origin. As an analogy to the above answer, one might expect that the class of functions which can be represented as L-transforms is composed of those functions which are analytic in right half-planes. However, merely a subset of all functions which are analytic in right half-planes constitutes, in fact, the class of functions which can be represented as L-transforms. Our aim is to delineate this subset. While investigating this problem, we shall have to distinguish between functions that can be represented as L-transforms of functions and those which can be represented as L-transforms of distributions. For the subset of those functions which can be represented as L-transforms of *functions* there exists no simple criterion in terms of the theory of analytic functions. Only *sufficient conditions* are known which describe merely a portion of the sought subset of representable functions. By contrast, the subset of functions which may be represented as L-transforms of *distributions* is characterized by a *necessary and sufficient condition* which is taken from the theory of functions. In this Chapter 28, we deal with the representability as L-transforms of functions.

## Keywords

Power Series Original Function Laplace Transform Inversion Formula Circular Disk## Preview

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