Laplace Transforms

  • Fritz Oberhettinger
  • Larry Badii

Abstract

The function g(p) of the complex variable p defined by the integral
$$g(p) = \int\limits_0^\infty {f(t)e^{ - pt} dt}$$
is called the one sided Laplace transform of f(t) where f(t) is a function of the real variable t,(0 < t < ∞) which is integrable in every finite interval. If the integral converges at a point p = p0, then it converges for every p such that Re p > Re p0. The behavior of the integral (1) in the p-plane may be one of the following:
  1. (a)

    Divergent everywhere

     
  2. (b)

    Convergent everywhere

     
  3. (c)

    There exists a number 3 such that (1) converges, when Re p > β and diverges when Re p < β. The number β which is the greatest lower bound of Re p for the set of all p’s in the p-plane at which (1) converges is called the abscissa of convergence.

     

Keywords

Sine Larg 

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References

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Copyright information

© Springer-Verlag Berlin · Heidelberg 1973

Authors and Affiliations

  • Fritz Oberhettinger
    • 1
  • Larry Badii
    • 2
  1. 1.Oregon State UniversityCorvallisUSA
  2. 2.Eastern Michigan UniversityYpsilantiUSA

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