ABSTRACT
Most of this chapter is devoted to examining those functions commonly known as Gaussians. These are functions that naturally arise in many applications. In part, this is because they describe some of the most commonly expected probability distributions, and consequently, are used to model such diverse phenomena as the “noise” in electronic and optical devices, the likelihood of a missile hitting its target, and the distribution of grades in a large class. In addition, they arise as fundamental solutions to the differential equations describing heat flow and diffusion problems. We will also find Gaussian functions invaluable in further developing the mathematics used in everyday applications in science and engineering (and mathematics). In particular, Gaussian functions make up the “identity sequences” that will play a major role in confirming the fundamental theorem on invertibility and the more general theorems on the differential identities in chapter 22, and, in part IV of this text, will serve as the basic “test functions” on which the generalized theory of functions and transforms will be developed.
