The Laplace Transform

I introduced the Laplace Transform which takes a function f(t) in the time domain and transforms it to another function, F(s), in the Laplace domain. The variable "s" used in the Laplace domain will be referred to as the "Laplace Variable."

The power of the Laplace Transform, for our purposes, is that it will turn differential equations into algebraic equations.  We won't need to guess at the form of the solution anymore either.  The Laplace Transform also obviates the need to find initial conditions at t=0+, you only need the conditions at t=0-.

I showed the definition of the Laplace Transform, and derived the transforms of a few functions.   I have a small table posted on the web as well (ignore the Z-Transform entries for now). Note particularly that the Laplace Transform of an impulse is unity - this will make working with the impulse response very easy.

Properties of the Laplace Transform

Several properties of the Laplace Transform were developed., these and others are listed in a table of properties.  There are several properties that will be the most immediately useful to us, and that you should easily understand how to apply:

Other theorems you should feel comfortable with (though we will use them less) are: