Laplace Transforms

  • Fritz Oberhettinger
  • Larry Badii


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.



Orthogonal Polynomial Laplace Transform Modify Bessel Function Absolute Convergence Whittaker Function 
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  1. Churchill, R. V., 1958. Operational Mathematics. McGraw-Hill.MATHGoogle Scholar
  2. Doetsch, G., 1950–1956. Handbuch der Laplace Transformation 3 vols. Verlag Birkhauser, Basel.Google Scholar
  3. Doetsch, G., 1961. Guide to the Application of Laplace Transforms. Von Nostrand.Google Scholar
  4. Doetsch, G., 1947. Tabellen zur Laplace Transformation. Springer Verlag.MATHGoogle Scholar
  5. Erdelyi, A. et al. 1952. Tables of Integral Transforms, Vol. 1. McGraw-Hill, 1954.Google Scholar
  6. McLachlan, N. W. and Humbert, P. 1950. Formulaires pour le calcul symbolique. Gauthier-Villars.Google Scholar
  7. Nixon, F. E. 1960. Handbook of Laplace Transformation. Prentice Hall.MATHGoogle Scholar
  8. Roberts, G. E. and H. Kaufmann, 1966. Tables of Laplace Transforms. W. B. Saunders Co.Google Scholar
  9. Sneddon, I. N. 1972. The Use of Integral Transforms. McGraw-Hill.MATHGoogle Scholar
  10. Widder, D. V., 1971. An Introduction to Transform Theory. Academic Press.MATHGoogle Scholar
  11. Widder, D. V., 1941. The Laplace Transform. Princeton University Press.Google Scholar

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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