Abstract One frequently gains insight simply by changing perspective. The analysis of physical problems in terms of sines and cosines is appropriate if the problem has planar symmetries or no symmetry. In the latter case, introducing Christoffel symbols would afford no advantage and would complicate things substantially. This chapter demonstrates that problems involving point sources are almost always more easily analyzed in terms of a different sort of basic function, called Bessel functions. Hankel transforms (which use Bessel functions) are introduced, using a modernized version of the tutorial approach first used by Sommerfeld (1896, 1964). This means we express two-dimensional Fourier transforms in terms of polar coordinates such as those of Figure 33. If we adopt invariant transformation for functions and the increments of area, we stumble upon the definitions of Bessel functions and Hankel transforms. Remember these definitions are arbitrary. We will consider coordinates z i and k i because there is some indication we frequently will be dealing with physical problems that have some sort of point symmetry to them. Just as there is a nondenumerable set of spatial coordinates, so also is there a nondenumerable set of spectral coordinates; however, for the purposes of this chapter we need only two.

Abstract The preface of TOG1 states we would employ a theoretical approach based on three physical principles: conservation of momentum, conservation of energy, and Maxwell's equations. In addition, two mathematical principles were to be used: invariance of mathematical form and embedding an old calculus in a new calculus. Invariance of mathematical form implies the structure of tensor analysis, and that structure implies the Bessel functions of the previous two chapters; they were just the offset-dependent part of a set of invariants that satisfied the wave equation. In this chapter we adhere to the embedding strategy. Specifically, we embed the Bessel functions in a larger set of functions that satisfies the cylindrical view of the wave equation. In terms of specific goals, in this chapter we attempt to develop cylindrical analogs of exponentials, i.e., equation (664) and an expression of the Hankel transform that looks more like the Fourier transform, i.e., equation (698). As you study this chapter, observe that much of its benefit is a greater appreciation of chapter 7.