  We present analytic approximations to the optically thin synchrotron and
synchrotron self-Compton (SSC) spectra when Klein-Nishina (KN) effects are
important and pair production and external radiation fields can be neglected.
This theory is useful for analytical treatment of radiation from astrophysical
sources, such as gamma-ray bursts (GRBs), active galactic nuclei and pulsar
wind nebula, where KN effects may be important. We consider a source with a
continuous injection of relativistic electrons with a power-law energy
distribution above some typical injection energy. We find that the
synchrotron-SSC spectra can be described by a broken power-law, and provide
analytic estimates for the break frequencies and power-law indices. In general,
we show that the dependence of the KN cross-section on the energy of the
upscattering electron results in a hardening of the energy distribution of fast
cooling electrons and therefore in a hardening of the observed synchrotron
spectrum. As a result the synchrotron spectrum of fast cooling electrons, below
the typical injection energy, can be as hard as $F_\nu \propto \nu^0$, instead
of the classical $\nu^{-1/2}$ when KN effects are neglected. The synchrotron
energy output can be dominated by electrons with energy above the typical
injection energy. We solve self-consistently for the cooling frequency and find
that the transition between synchrotron and SSC cooling can result in a
discontinuous variations of the cooling frequency and the synchrotron and SSC
spectra. We demonstrate the application of our results to theory by applying
them to prompt and afterglow emission models of GRBs.
