Overview

Since the spectrum of the correlation function has the form, Q( k) = c/ k ² where c is a constant [Field, 1987], Q( r) satisfies

This can be easily shown by taking Fourier transformation of both sides of this equation. In principle, a gauge g( r) which satisfies ? ² g( r) = 0, can be added to Q( r). But the only such gauge, which is radially symmetric, is a constant (denoted as b).

A representation for the radially symmetric form of the ?( r) function in terms of Bessel functions, is

where J _t is the lth order Bessel function, k _i ⁰ is the ith zero of J ₀, r and ? are the polar coordinates of r, and N ₀( k _j ⁰) = ? ¹ ₀ rdr[ J ₀( k _j ⁰ r)] ² is the normalization constant of J ₀( k _j ⁰ r). Therefore it can be seen that

since it solves Equation 5A.1. Using an addition theorem for Bessel functions, we obtain a representation for the correlation function Q( r ? r ?)

We shall rewrite this correlation function in terms of the normalized Bessel-Fourier basis W _mi. These functions are zero on the boundary and take the form [Jackson, 1975]

In which N _m