# Laplace transform
From now on, LT is short for Laplace Transform
What is LT? It's denoted as $\mathcal{L}$ and it's an operator defined as the following:
$$\underset{ =F(s) }{ \mathcal{L}\{f(t)\}(s) }:=\int _{0}^\infty e^{-st}f(t) dt=\lim_{ T \to \infty }\int _{0} ^T e^{-st}f(t)\, dt$$
This doesn't look like anything useful, but later on we will learn how it is.

$\mathcal{L}\{0\}=0$ Look at your bank account, integrate 0 you still get 0 :D
$\mathcal{L}\{1\}=\int_{0}^\infty e^{-st} \, dt=-\frac{1}{s}e^{-st}|_{0}^\infty=\frac{1}{s}$ if s>0
$\mathcal{L}\{e^{at}\}$=$\int_{0}^{\infty} e^{at}e^{-st}\, dt=\int_{0}^{\infty}e^{-(s-a)t}dt=\frac{1}{s-a}$ if $s-a>0$
$\mathcal{L}\{\sin bt\}=\int _{0}^{\infty}e^{-st}\sin(bt) \, dt=\frac{b}{s^2+b^2}$ by integration by parts
similarly can be done for cos, but we have run out of time.
#end of lec 14