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We calculate the bound state spectrum of the highly excited valence electronin the heavy alkali atoms solving the radial Schr\"odinger eigenvalue problemnumerically with an accurate spectral collocation algorithm that applies alsofor a large principal quantum number $n\gg1$. As an effective single-particlepotential we favor the reputable potential of Marinescu \emph{et al}., {[}Phys.Rev.A \textbf{49}, 982(1994){]}. Recent quasiclassical calculations of thequantum defect of the valence electron agree for orbital angular momentum$l=0,1,2,...$ overall remarkably well with the results of the numericalcalculations, 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 potentialacquires for $l=3$ deep inside the ionic core a tiny second classical region,thus invalidating a standard WKB calculation with two widely spaced turningpoints. Comparing then our numerical solutions of the radial Schr\"odingereigenvalue problem with the uniform analytic WKB approximation of Langer weobserve everywhere a remarkable agreement, apart from a tiny region around theinner 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 wavefunction complementary to the WKB approximation of Langer, which is exact for$r\to0^{+}$. The value of the radial \emph{s}-wave function at $r=0$ isanalytically found, thus validating the Fermi-Segr\`e formula for the magneticdipole interaction constant $A_{n,j,0}^{\left(\mathrm{HFS}\right)}$.

On the spectrum, radial wave functions, and hyperfine splittings of the Rydberg states in heavy alkali atoms