ABSTRACT
The Laplace transform is a useful tool because: it has formal similarities to the Fourier transform; it can be applied to a larger class of functions than the Fourier transform (since the factor e-st decays rapidly at infinity); and it is often straightforward to compute. It is sometimes useful to modify a familiar mathematical operation (the Fourier transform) by letting the variable be complex (in producing the Laplace transform). The chapter provides a set of examples to illustrate the utility of the Laplace transform. A differential equation is solved using the Laplace transform. The z-transform is more familiar in the engineering community than in the mathematics community. Mathematicians group this circle of ideas with the notion of generating function and with allied ideas from finite and combinatorial mathematics.
