Appendix C: Computation of the circular prolate functions

The circular prolate functions used in this paper can be defined in terms of Slepian's "generalized prolate spheroidal functions'' $\varphi_{N,n}(c, x)$ (Slepian 1964):

$$\Theta_{N,n}(c,r) = \sqrt{\frac{2 \Lambda_{N,n}(c)}{D}} \frac{\varphi_{N,n}\left(c, \frac{2r}{D}\right)}{\sqrt{r}}\cdot$$ (C.1)

Here N is the order of the Hankel transform, n the order of the prolate function, and $c=\pi D a/2$ the prolateness parameter. In this paper we are primarily interested in the case N=n=0, and for convenience have generally omitted these labels.

The functions $\varphi_{N,n}(c, x)$ are solutions of a generalized spheroidal differential equation,

$$(1-x^{2}) \varphi''(x) - 2x \varphi'(x) + \left(\chi - c^{2}x^{2} + \frac{\frac{1}{4}-N^{2}}{x^{2}}\right) \varphi(x) = 0,$$ (C.2)

with eigenvalues $\chi=\chi_{N,n}(c)$. They are conventionally normalized so that

$$\int_{0}^{1}\varphi_{N,n}(c,t)^{2}{\rm d}t = 1.$$ (C.3)

When c=0, Eq. (C.2) is satisfied by the functions T_N,n(x), defined by

$$T_{N,n}(x) = x^{N+\frac{1}{2}}\ _{2}F_{1}\left(-n,n+N+1;N+1;x^{2}\right),$$ (C.4)

where ₂F₁(a,b;c;z) is Gauss' hypergeometric function (Abramowitz & Stegun 1965), and the eigenvalue is $\chi_{N,n}(0)=(N+2n+1/2)(N+2n+3/2)$. The T_N,n(x) satisfy the three-term recurrence relation

$$x^{2} T_{N,n} = \gamma_{N,n}^{1} T_{N,n+1}+\gamma_{N,n}^{0} T_{N,n}+\gamma_{N,n}^{-1} T_{N,n-1},$$ (C.5)

where

$$\gamma_{N,n}^{1}=-\frac{(n+N+1)^{2}}{(2n+N+1)(2n+N+2)},$$ (C.6a)

$$\gamma_{N,n}^{0}=\frac{1}{2}(1+\frac{N^{2}}{(2n+N)(2n+N+2)}),$$ (C.6b)

$$\gamma_{N,n}^{-1}=-\frac{n^{2}}{(2n+N)(2n+N+1)}\cdot$$ (C.6c)

For general c, the functions $\varphi_{N,n}(c, x)$ are expanded in a series of the T_N,k(x):

$$\varphi_{N,n}(c,x) = \sum_{k=0}^{\infty}d_{k}^{N,n}(c) T_{N,k}(x).$$ (C.7)

Substituting this expansion into Eq. (C.2) and using Eq. (C.5), we obtain a three-term recurrence for the series coefficients d_k^N,n(c):

$$c^{2} \gamma_{N,k-1}^{1} d_{k-1}^{N,n} + (c^{2} \gamma_{N,k}^{0}+... ...k}(0)-\chi_{N,n}(c)) d_{k}^{N,n} + c^{2} \gamma_{N,k+1}^{-1} d_{k+1}^{N,n} = 0.$$ (C.8)

The eigenvalues $\chi_{N,n}(c)$ and the coefficients d_k^N,n(c) can be computed from this equation using the standard tridiagonal matrix or continued fraction methods (Falloon 2001). For x > 1, $\varphi_{N,n}(c, x)$ can be computed using the series

$$\varphi_{N,n}(c,x) = \frac{1}{\gamma_{N,n}(c)} \sum_{k=0}^{\i... ...left(\begin{array}{c} k+N \\ k \end{array} \right) \sqrt{cx}},$$ (C.9)

where

$$\gamma_{N,n}(c) = \frac{c^{N+\frac{1}{2}}d_{0}^{N,n}}{2^{N+1}... ...mma(N+2)} \left(\sum_{k=0}^{\infty}d_{k}^{N,n}(c)\right)^{-1}.$$ (C.10)

The eigenvalues of the integral equation (Eq. (B.3)), $\Lambda_{N,n}(c)$, can be found using

$$\Lambda_{N,n}(c) = c \gamma_{N,n}(c)^{2}.$$ (C.11)