Laplace Transform Conditions of Existence

Topics: Laplace Transform

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Concerning Laplace transforms, we can guarantee their existence by means of the consequences of the following theorems.

(theorem, 8)

Note that, as $t$ increases, the growth of the module of the function $f(t)$ is not greater than that of an exponential function. That is, there exists $M>0$ and $s_0\ge 0$ such that, for every value of $t$:

$$\left|f(t)\right|<M{e}^{s_{0}t}$$

We call $s_0$ the growth exponent of the function $f(t)$. This condition guarantees the existence of the Laplace transform.

(theorem)

If $f(t)$ is piecewise continuous in every finite set $0<t<M$ of exponential order $s_0$ for $t>M$, then the Laplace transform $F(s)$ exists as long as $s_0>0$.