## ABSTRACT

Despite the claim in an earlier chapter that the Laplace transform is “particularly useful when dealing with nonhomogeneous equations in which the forcing functions are not continuous”, the only discontinuous functions dealt with in previous chapters were the step functions. In this chapter, that analysis begun with the step functions is expanded to include the sort of discontinuous (and, more generally, “piecewise-defined” functions) that often arise in applications. This analysis includes the development of another identity, the “translation along the T-axis” identity, along with illustrations of its applicability in computing both Laplace transforms of piecewise-defined functions and in computing certain inverse transforms involving exponentials.

In addition, the Laplace transforms of general periodic functions (which are often, themselves, piecewise defined) are discussed. This, in turn, leads to a return to the discussion of resonance in forced mass-spring systems, this time using fairly arbitrary periodic functions (instead of just sines and cosines) and Duhamel’s principle from the previous chapter.