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

  • Gustav Doetsch

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 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 1974

Authors and Affiliations

  • Gustav Doetsch
    • 1
  1. 1.Emeritus of MathematicsUniversity of FreiburgFreiburg i. Br.Germany

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