Mathematics for Natural Scientists II pp 445-498 | Cite as

# Laplace Transform

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## 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.