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Asymptotic Behaviour of the Image Function near a Singular Point on the Line of Convergence

  • Gustav Doetsch

Abstract

In this Chapter we shall show that the functional behaviour of an original function f(t) as t → ∞, is reflected in the behaviour of the corresponding image function F(s) = L{f} near some finite point s0. At any point s0 in the interior of the half-plane of convergence of F(s), or on the line of convergence of F(s), where the image function is holomorphic, the behaviour of F(s) is trivial in the sense that F(s) is continuous at s0, and F(s) → F(s0) as s → s0. Moreover, the ℒ-integral cannot be called upon to provide information regarding the behaviour of F(s) near such points s0 outside the half-plane of convergence of F(s), where F(s) may happen to exist. Therefore, our interest concentrates upon singular points s0 on the line of convergence.

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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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