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 No discussion involving an integral transform by faculty of an engineering school is complete without some discussion of convolution. This is as true in polar coordinates as it is in orthogonal Cartesian; but when using polar coordinates and Bessel functions, one must pay special attention to the symmetries involved. Recall that the only reason to adopt coordinates, other than the orthogonal Cartesian, is symmetry that exists in the boundary value problem. We can now be more specific about symmetry in the context of Flatland. We say that a , or whatever it represents, contains γ-fold symmetry if Note that here μ is not a tensor index but rather is a subscript denoting a term in the Hankel expansion of equation (468). The two-dimensional form of equation (145) provides where we are speaking of spatial convolution and spectral multiplication in orthogonal Cartesian coordinates. In the following we discuss spatial multiplication and spectral convolution in terms of polar coordinates. Hankel transforms evolved from the Hankel expansion [equations (468) or (478)]. We now coordinate these as the Fourier transform pair which tells us the Hankel expansion coefficients of the product of two polar views of Fourier transforms can be constructed by a digital convolution. [See TOG1 equation (400).] To drive this point home, we now consider a simple special case illustrated in Figure 38. From equation (596), if we have M nonzero terms in the Hankel expansion of a and N nonzero terms in the Hankel expansion of b , then we have N