ABSTRACT
In the last chapter, it was shown how to find the Laplace transforms of the “unknown” solutions to various initial-value problems. Of course, the main interest is not in those Laplace transforms, but in the actual solutions. Thus, it is necessary to be able to extract a function from its Laplace transform. This extraction process is, naturally, called the “inverse Laplace transform”.
This chapter begins with a simple definition of the inverse Laplace transform and a uniqueness theorem ensuring the validity of this definition. The linearity of the inverse Laplace transform is then derived from the linearity of the Laplace transform, and it is noted that all previous Laplace transform computations and identities can also be viewed as corresponding computations and identities for the inverse Laplace transform. It is further observed that partial fraction expansions can reduce the transforms of solutions to many differential equations to more manageable forms. Accordingly, the methods of partial fractions are briefly reviewed and illustrated.
