  We present a low energy expansion of the Kramers-Heisenberg formula for
atomic hydrogen in terms of $(\omega/\omega_l)$, where $\omega_l$ and $\omega$
are the angular frequencies corresponding to the Lyman limit and the incident
radiation, respectively. The leading term is proportional to
$(\omega/\omega_l)^4$, which admits a well-known classical interpretation. With
higher order terms we achieve accuracy with errors less than 4 % of the
scattering cross sections in the region $\omega/\omega_l\le 0.6$. In the
neighboring region around Ly$\alpha$ ($\omega/\omega_l >0.6$), we also present
an explicit expansion of the Kramers-Heisenberg formula in terms of
$\Delta\omega\equiv (\omega-\omega_{Ly\alpha})/\omega_{Ly\alpha}$. The accuracy
with errors less than 4 % can be attained for $\omega/\omega_l \ge 0.6$ with
the expansion up to the fifth order of $\Delta\omega$. We expect that these
formulae will be usefully applied to the radiative transfer in high neutral
column density regions, including the Gunn-Peterson absorption troughs and
Rayleigh scattering in the atmospheres of giants.
