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.
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Notes
- 1.
In the following, references to the first volume of this course (L. Kantorovich, Mathematics for natural scientists: fundamentals and basics, Springer, 2015) will be made by appending the Roman number I in front of the reference, e.g. Sect. I.1.8 or Eq. (I.5.18) refer to Sect. 1.8 and Eq. (5.18) of the first volume, respectively.
- 2.
Recall that this means that one-sided limits, not equal to each other, exist on both sides of the discontinuity.
- 3.
Since f(t) is piecewise continuous, so is g a (t), and hence the other Dirichlet condition is also satisfied for the FT to exist.
- 4.
Strictly speaking, this condition is not necessary, but we shall assume it to simplify the proof.
- 5.
This follows from the fact that p λ → 0 on any part of C R in the R → ∞ limit.
- 6.
Note, however, that the two definitions of the convolution in the cases of LT and FT are not identical: the limits are 0 and t in the case of the LT, while when we considered the FT the limits were ±∞, Eq. (5.39).
- 7.
In the latter case the inverse LT of \(G(p) = 1/\left (p - p_{+}\right )^{2}\) can be obtained either directly or by taking the limit p − → p + in g(t) from the first formula in Eq. (6.58) obtained by assuming p + ≠ p −.
- 8.
For a more general discussion, including a derivation of the equations of motion for a multidimensional open classical system (the so-called Generalised Langevin Equation), as well as references to earlier literature, see L. Kantorovich, Phys. Rev. B 78, 094304 (2008).
- 9.
Here we loosely follow (and in a rather simplified form) a detailed discussion which a reader can find in L. Kantorovich, Phys. Rev. B 75, 064305 (2007).
- 10.
The solution to this problem was first obtained by J.E. Sader and S.P. Jarvis in their highly cited paper [Appl. Phys. Lett. 84, 1801 (2004)] using the methods of LT and the so-called fractional calculus. We adopted here some ideas of their method, but did not use the fractional calculus at all relying instead on conventional techniques presented in this chapter.
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Kantorovich, L. (2016). Laplace Transform. In: Mathematics for Natural Scientists II. Undergraduate Lecture Notes in Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-27861-2_6
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DOI: https://doi.org/10.1007/978-3-319-27861-2_6
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