Definition and Elementary Properties

  • B. Davies
Part of the Applied Mathematical Sciences book series (AMS, volume 25)


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 convergence2 that F(p) is analytic in the half-plane Re(p) > α.


Asymptotic Expansion General Relationship LAPLACE Transform Elementary Property Converse Implication 
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  1. 2.
    AHLFORS (1966), Ch. 5.Google Scholar
  2. 3.
    Many more general relationships may be found in ERDELYI, et al. (1954), Ch. 4.Google Scholar
  3. 4.
    Extensive tables of Laplace transforms are available; for instance, ERDELYI, et. al. (1954).Google Scholar
  4. 5.
    Anticipating the result that the Laplace transform has a unique inverse.Google Scholar

Copyright information

© Springer Science+Business Media New York 1978

Authors and Affiliations

  • B. Davies
    • 1
  1. 1.The Australian National UniversityCanberraAustralia

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