Introduction to Laplace Transform

I have been working on a few problems around Laplace Transform on Yahoo Answer (in fact, the one in Hong Kong/Taiwan). It would be a waste for those content to only post there, so I am posting them here. Feel free to checkout my profile there if you are interested

This post is about Laplace Transform, the Laplace Transform of a function can be simply written as $Y(s) = \mathcal{L}(y) = \int\limits_{0}^{\infty}{y(t)e^{-st}dt}$.

There are a few things to note:

1. $s$ is in general a complex number, $t$ is usually real.
   
2. The integral is improper, in particular, by writing $\int\limits_{0}^{\infty}{y(t)e^{-st}dt}$, we really mean $\lim_{T \to \infty}{\int\limits_{0}^{\infty}{y(t)e^{-st}dt}}$.
   
3. When necessary, we pick ranges of $s$ such that the limit converges and define $Y(s)$ there only.

That's it for this post. Next we will talk about some common Laplace Transform pair and properties.