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Laplace Transform

  • Lev Kantorovich
Chapter
  • 2.5k Downloads
Part of the Undergraduate Lecture Notes in Physics book series (ULNP)

Abstract

In Chap.  5 we considered the Fourier transform (FT) of a function f(x) defined on the whole real axis − < x < . Recall that for the FT to exist, the function f(x) must tend to zero at x → ± to ensure the convergence of the integral \(\int _{-\infty }^{\infty }\left \vert f(x)\right \vert dx\). However, sometimes it is useful to study functions f(x) defined only for x ≥ 0 which may also increase indefinitely as x → + (but not faster than an exponential function, see below). For instance, this kind of problems is encountered in physics when dealing with processes in time t when we are interested in the behaviour of some function f(t) at times after some “initial” time (considered as t = 0), at which the system was “prepared” (the initial value problem), e.g. prior to some perturbation acting on it. In such cases another integral transform , named after Laplace, the so-called Laplace transform (LT), has been found very useful in many applications.

Keywords

Laplace Transform (LT) Laplace Space Position Autocorrelation Function Escape Rate Friction Kernel 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Lev Kantorovich
    • 1
  1. 1.Physics Department School of Natural and Mathematical SciencesKing’s College London, The StrandLondonUK

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