Finding the Laplace and Inverse Laplace Transforms
Solving the Integral
Probably the most difficult and least used method for finding the Laplace transform of a signal is solving the integral. Although it is technically possible, it is extremely time consuming. Given how easy the next two methods are for finding it, we will not provide any more than this. The integrals are primarily there in order to understand where the following methods originate from.
Using a Computer
Using a computer to find Laplace transforms is relatively painless. Matlab has two functions, laplace and ilaplace, that are both part of the symbolic toolbox, and will find the Laplace and inverse Laplace transforms respectively. This method is generally preferred for more complicated functions. Simpler and more contrived functions are usually found easily enough by using tables.
When first learning about the Laplace transform, tables are the most common means for finding it. With enough practice, the tables themselves may become unnecessary, as the common transforms can become second nature. For the purpose of this section, we will focus on the inverse Laplace transform, since most design applications will begin in the Laplace domain and give rise to a result in the time domain.
(Copy and paste didn't work out so well on that...)
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