Definition of Laplace

Question

If a function is not of exponential order, then is it possible that function's Laplace transform exist? If yes, then how can we determine for a function that its Laplace transform exists or not?

Answer 1

consider for example $f(x)=2xe^{x^2}\cos(e^{x^2})u(x)$. for this function we must define if there exist $M , \alpha$ such that $|2xe^{x^2}\cos(e^{x^2})|< Me^{\alpha x}$ and we see that the limit of the form $lim_{x \rightarrow +\infty}\frac{2xe^{x^2}\cos(e^{x^2})}{Me^{\alpha x}}$ for any finite choice of $M , \alpha$ is unbounded so $f(x)$ is not of exponential order but we observe that: $$\int_0^{+\infty}{2xe^{x^2}\cos(e^{x^2})}{e^{-sx}}=e^{-sx}\sin(e^{x^2})|^{+\infty}_{0}+s\int_0^{+\infty}{\sin(e^{x^2})}{e^{-sx}}=-\sin(1) +s\times \int_0^{+\infty}{\sin(e^{x^2})}{e^{-sx}}$$ which was calculated using by-part technique. in the last integral since $|\sin(e^{x^2})|<|{e^{-sx}}|$ the last integral does exist and is valid.

so we se that if the function is of exponential order then Laplace Transform exist but if function is not of exponential order then we must see if the integral $\int_{0}^{+\infty}f(x)e^{-sx}$ exist or not.

Answer 2

I don't think there's a single test for that, as little as there's a single test for convergence of integrals.

However there's some conditions that will guarantee or prohibit the existence of a Laplace transform.

One such condition is that if a function is locally integrable (integrable on every finite interval) vanishes (ie the value is zero) on an infinite interval (this means that we only have to bother about the single sided Laplace transform) and it's $O(e^{\sigma t})$ for some $\sigma$ as $t\to\pm\infty$. Then the Laplace transform exists for some $s$.

Conditions of the other kind is that if the absolute value of a function is bounded below by a constant (or constant multiplied with an exponential), then there's no Laplace transform (in the function sense).

With these kind of condition you can normally find function that fulfill the neither condition that guarantees the existence nor the condition that prohibits it and you cannot use those conditions to determine whether the transform exists or not.

Note that your condition about not being of exponential order is not such a condition that rules out the existence of the Laplace transform. While being of exponential order guarantees the existence that condition doesn't say anything about what happens if the function is not of exponential order.