The inverse Laplace transform of the function Y(s) is the unique
function y(t) that is continuous on [0,infty) and satisfies
L[y(t)](s)=Y(s). If all possible functions y(t) are discontinous one
can select a piecewise continuous function to be the inverse transform.

We will use the notation

or Li[Y(s)](t) to denote the inverse Laplace transform of Y(s).

Determining the Inverse Laplace Transform

The Laplace transform Y(s) of a function y(t) defined on [0,infty) is
defined by an integral. It turns out that formula for determining y(t)
given Y(s) also involves an integral. The integral is complex valued
integral, and its evaluation is beyond the scope of this course. For
this reason we take a more pedestrian approach in computing the
inverse transform. We will use tables and a few tricks.

There are only small number of functions listed in the table. The set
of invertible functions can be increased by exploiting linearity.

Linearity of the Inverse Transform

Let y_1(t) and y_2(t) be the inverse Laplace transforms of Y_1(s)
and Y_2(s), respectively, and let c be a constant. We have

As a corollary, we have a third formula:

Here are several examples. What is the inverse transform of 7/s^3?
From the table above we know that the inverse of 2/s^3 is y(t)=t^2.
Hence

Consider the Laplace transform Y(s)=(3s+4)/(s^2+4). Using the table
above, we know that the inverse of s/(s^2+4) is cos(2t) and that the
inverse of 2/(s^2+4) is sin(2t). Hence: