We'll next build up a collection of Laplace transform pairs which we will include in a table. It's important to keep in mind that once the transform pair has been derived, the focus should be on utilizing the transform pair found in the table rather than in recalculating the transform.
Exponential Signal
Consider the Laplace transform of
where
Unit Step Function
Recall that the Fourier transform of
The region of convergence is
Ramp Signal
This signal is given by
The region of convergence is once again the righthalf plane,
Cosine Signal
Even though we computed the Fourier transform of the cosine signal,
Since each of the two terms is an exponential function we have
Here, the region of convergence corresponds to the righthalf plane,
More Transform Pairs
We can use the existing transform pairs along with the properties of the Laplace transform to derive many new transform pairs. Consider the exponentially weighted cosine signal. This signal is given by
We can use the
Another common signal is
We use the Laplace transform of the exponential signal "Exponential Signal" and the
Extending this idea one step further, we have
Here, multiplication by
Example 3.1 Consider the signal
Example 3.2 Consider the signal























