The Inverse Laplace Transform
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:
