  We investigate the connection between the full- and flat-sky angular power
spectra. First, we revisit this connection established on the geometric and
physical grounds, namely that the angular correlations on the sphere and in the
plane (flat-sky approximation) correspond to each other in the limiting case of
small angles and a distant observer. To establish the formal conditions for
this limit, we first resort to a simplified shape of the 3D power spectrum,
which allows us to obtain analytic results for both the full- and flat-sky
angular power spectra. Using a saddle-point approximation, we find that the
flat-sky results are obtained in the limit when the comoving distance and wave
modes $\ell$ approach infinity at the same rate. This allows us to obtain an
analogous asymptotic expansion of the full-sky angular power spectrum for
general 3D power spectrum shapes, including the LCDM Universe. In this way, we
find a robust limit of correspondence between the full- and flat-sky results.
These results also establish a mathematical relation, i.e., an asymptotic
expansion of the ordinary hypergeometric function of a particular choice of
arguments that physically corresponds to the flat-sky approximation of a
distant observer. This asymptotic form of the ordinary hypergeometric function
is obtained in two ways: relying on our saddle-point approximation and using
some of the known properties of the hypergeometric function.
