MATH201/content/Laplace transform (lec 14).md

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