  Synchrotron self-Compton (SSC) scattering is an important emission mechanism
in many astronomical sources, such as gamma-ray bursts (GRBs) and active
galactic nuclei (AGNs). We give a complete presentation of the analytical
approximations for the Compton scattering of synchrotron emission with both
weak and strong synchrotron self-absorption. All possible orders of the
characteristic synchrotron spectral breaks ($\nu_{\rm a}$, $\nu_{\rm m}$, and
$\nu_{\rm c}$) are studied. In the weak self-absorption regime, i.e., $\nu_{\rm
a} < \nu_c$, the electron energy distribution is not modified by the
self-absorption process. The shape of the SSC component broadly resembles that
of synchrotron, but with the following features: The SSC flux increases
linearly with frequency up to the SSC break frequency corresponding to the
self-absorption frequency $\nu_{\rm a}$; and the presence of a logarithmic term
in the high-frequency range of the SSC spectra makes it harder than the
power-law approximation. In the strong absorption regime, i.e. $\nu_{\rm a} >
\nu_{\rm c}$, heating of low energy electrons due to synchrotron absorption
leads to pile-up of electrons, and form a thermal component besides the broken
power-law component. This leads to two-component (thermal + non-thermal)
spectra for both the synchrotron and SSC spectral components. For $\nu_{\rm c}
< \nu_{\rm a} < \nu_{\rm m}$, the spectrum is thermal (non-thermal) -dominated
if $\nu_a > \sqrt{\nu_m \nu_c}$ ($\nu_a < \sqrt{\nu_m \nu_c}$). Similar to the
weak-absorption regime, the SSC spectral component is broader than the simple
broken power law approximation. We derive the critical condition for strong
absorption (electron pile-up), and discuss a case of GRB reverse shock emission
in a wind medium, which invokes $\nu_{\rm a} > {\rm max} (\nu_{\rm m}, \nu_{\rm
c})$.
