  In this paper we consider the Sturm-Liouville equation -y"+qy = lambda*y on
the half line (0,infinity) under the assumptions that x=0 is a regular singular
point and nonoscillatory for all real lambda, and that either (i) q is L_1 near
x=infinity, or (ii) q' is L_1 near infinity with q(x) --> 0 as x --> infinity,
so that there is absolutely continuous spectrum in (0,infinity).
Characterizations of the spectral density function for this doubly singular
problem, similar to those obtained in [12] and [13] (when the left endpoint is
regular) are established; corresponding approximants from the two algorithms in
[12] and [13] are then utilized, along with the Frobenius recurrence relations
and piecewise trigonometric - hyperbolic splines, to generate numerical
approximations to the spectral density function associated with the doubly
singular problem on (0,infinity). In the case of the radial part of the
separated hydrogen atom problem, the new algorithms are capable of achieving
near machine precision accuracy over the range of lambda from 0.1 to 10000,
accuracies which could not be achieved using the SLEDGE software package.
