# Definition and Elementary Properties

Chapter

## Abstract

Let f(t) be an arbitrary function defined on the interval 0 ≤ t < ∞; then is the Laplace transform of f(t), provided that the integral exists. We shall confine our attention to functions f(t) which are absolutely integrable on any interval 0 ≤ t ≤ a, and for which F(α) exists for some real α. It may readily be shown that for such a function F(p) is an analytic function of p for Re(p) > α, as follows. First note that the functions are analytic in p, and then that φ(p,T) converges uniformly to F(p) in any bounded region of the p plane satisfying Re(p) > α, as T → ∞. It follows from a standard theorem on uniform convergence

(1)

(2)

^{2}that F(p) is analytic in the half-plane Re(p) > α.## Keywords

Asymptotic Expansion General Relationship LAPLACE Transform Elementary Property Converse Implication
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## Footnotes

- 2.AHLFORS (1966), Ch. 5.Google Scholar
- 3.Many more general relationships may be found in ERDELYI, et al. (1954), Ch. 4.Google Scholar
- 4.Extensive tables of Laplace transforms are available; for instance, ERDELYI, et. al. (1954).Google Scholar
- 5.Anticipating the result that the Laplace transform has a unique inverse.Google Scholar

## Copyright information

© Springer Science+Business Media New York 1978