  We calculate the bound state spectrum of the highly excited valence electron
in the heavy alkali atoms solving the radial Schr\"odinger eigenvalue problem
numerically with an accurate spectral collocation algorithm that applies also
for a large principal quantum number $n\gg1$. As an effective single-particle
potential we favor the reputable potential of Marinescu \emph{et al}., {[}Phys.
Rev.A \textbf{49}, 982(1994){]}. Recent quasiclassical calculations of the
quantum defect of the valence electron agree for orbital angular momentum
$l=0,1,2,...$ overall remarkably well with the results of the numerical
calculations, but for the Rydberg states of rubidium and also cesium with $l=3$
this agreement is less fair. The reason for this anomaly is that the potential
acquires for $l=3$ deep inside the ionic core a tiny second classical region,
thus invalidating a standard WKB calculation with two widely spaced turning
points. Comparing then our numerical solutions of the radial Schr\"odinger
eigenvalue problem with the uniform analytic WKB approximation of Langer we
observe everywhere a remarkable agreement, apart from a tiny region around the
inner turning point. With help of an ansatz proposed by Fock we obtain for the
\emph{s}-states a second uniform analytic approximation to the radial wave
function complementary to the WKB approximation of Langer, which is exact for
$r\to0^{+}$. The value of the radial \emph{s}-wave function at $r=0$ is
analytically found, thus validating the Fermi-Segr\`e formula for the magnetic
dipole interaction constant $A_{n,j,0}^{\left(\mathrm{HFS}\right)}$.
