  We study models of the gamma-ray emission of Cyg X-3 observed by Fermi. We
calculate the average X-ray spectrum during the gamma-ray active periods. Then,
we calculate spectra from Compton scattering of a photon beam into a given
direction by isotropic relativistic electrons with a power-law distribution,
both based on the Klein-Nishina cross section and in the Thomson limit.
Applying the results to scattering of stellar blackbody radiation in the inner
jet of Cyg X-3, we find that a low-energy break in the electron distribution at
a Lorentz factor of ~ 300--1000 is required by the shape of the observed
X-ray/gamma-ray spectrum in order to avoid overproducing the observed X-ray
flux. The electrons giving rise to the observed \g-rays are efficiently cooled
by Compton scattering, and the power-law index of the acceleration process is ~
2.5--3. The bulk Lorentz factor of the jet and the kinetic power before the
dissipation region depend on the fraction of the dissipation power supplied to
the electrons; if it is ~ 1/2, the Lorentz factor is ~ 2.5, and the kinetic
power is ~ 10^38 erg/s, which represents a firm lower limit on the jet power,
and is comparable to the bolometric luminosity of Cyg X-3. Most of the power
supplied to the electrons is radiated. The broad band spectrum constrains the
synchrotron and self-Compton emission from the gamma-ray emitting electrons,
which requires the magnetic field to be relatively weak, with the magnetic
energy density < a few times 10^-3 of that in the electrons. The actual value
of the magnetic field strength can be inferred from a future simultaneous
measurement of the IR and gamma-ray fluxes.
