I''ve been looking through La Place transformations for a few days and I haven''t found a good example to understand them I only know that the transformation equals to the integration from zero to infinite of e^-st * f(t) dt I don''t know if anyone is willing to explain them to me... Im just a curious kid

Well, I don''t know anything about Laplace transformations, but mathworld.com is sometimes a good start. Do you know what they are meant to do?

Ummm, I don''t know how much math you have had, but I will try to explain it as simply as possible. But, you do know what a differential equation (DE) is, right? I will assume you do, otherwise you can complain and I will try to explain that to you, too.

Basically, a Laplace transform is a one-to-one mapping from one set of functions to another. So, you put a function in, say
f (t) = e^t, and you get a function out

F (s) = § e^(-st)e^t dt = § e^-(s-1)t dt
=1/(s-1) - lim N-> +00 (e^-(s-1)N/(s-1)) = 1/(s-1), s > 1.
Where the integrals are from 0 to +00.

So the Laplace transform of
f (t) = e^t is L {f (t)} = F (s) = 1/(s-1), s > 1.
Of course, t and s are just dummy variables, but for some reason people like to use them. There are big tables of Laplace tranforms for many different functions.

So what is the point of this? The beauty of the Laplace transform is the property it has with derivatives, i.e.:
L {f''} = s L {f} - f (0).
This allows some DEs to be "Laplaced" into new functions which don''t contain any derivatives . You only need to do some algebra on the new function to get it into a known form (like something you can look up in a table) and then "Deplace" (as we used to say) it back to the solution. In summary, Laplace transforms are a tool for solving DEs by changing it into an algebra problem. Very cool...

no wise fish would go anywhere without a porpoise - The Mock Turtle

Hi,

Further:
Laplacetransforms are widely used in control theory, where you steer dynamic systems, kind of.
Also, laplacetransformations are a generalization of Fouriertransforms. That is, fouriertransforms are like laplace except that you view the laplace on the imaginaryaxis. That is, instead of the complex ''s'' which Goat talked about, you replace that ''s'' with a pure complex one (Re{s}=0).

Fouriertransforms are widely used in signalanalysis (filtering signals...).

I hope this might get you some idea of what they are. There are plenty of good books on this.

/Mankind gave birth to God.