Is there a deeper explanations for why Laplace transforms work (like Fourier transforms, which have the theory that any periodic function is an infinite sum of sines and cosines), or is it just a (mundane) trick that happens to work?

Note that I know almost nothing of Laplace transforms, what they are used for (specifically, I know they're useful in differential equations).

Question

Is there a deeper explanations for why Laplace transforms work (like Fourier transforms, which have the theory that any periodic function is an infinite sum of sines and cosines), or is it just a (mundane) trick that happens to work?

Note that I know almost nothing of Laplace transforms, what they are used for (specifically, I  know they're useful in differential equations).

zepdrix · Answer

I don't have a great understanding of the Laplace Transform.
But I think it's mostly just a trick that happens to work `most of the time`.

Using the familiar notation, it's often used to transform a function from the time domain $\large f(t)$ to something else, often frequency $\large F(s)$.

The reason I say it only works `most of the time` is because it can't be applied to all functions.  Take for example:
$$\large f(t)=e^{t^2}$$

$\large \int_0^{\infty} e^{t^2}e^{-st}\;dt$

Our function f(t) is increasing a lot faster than the e^{-st} is decreasing, so this integral will not converge.

I dunno. Like I said, I don't have a lot of great info on the topic, just adding my two cents. :)