Laplace Transform

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


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.


Laplace Transform (LT) Laplace Space Position Autocorrelation Function Escape Rate Friction Kernel 
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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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