We present the basic equations for calculating the time-dependent received spectral flux from the optically thin emissivities produced by temporally evolving, nonthermal electrons losing energy in a relativistically moving blob of plasma located at cosmological distances. Ionization, bremsstrahlung, Compton, and synchrotron energy losses of the electrons are considered. These equations are used to derive and calculate the received spectral fluence produced by instantaneously injected nonthermal electrons that also scatter photons from an isotropic external monochromatic radiation held in the Thomson regime of scattering. The derivation of the fluences is restricted to the regime where either synchrotron losses or Thomson losses with photons of the external radiation field dominate the electron energy loss rate. We also derive approximate expressions for the synchrotron and synchrotron self-Compton fluence. Beaming factors for thin-target and thick-target models of the electron distributions are derived. We show that ionization energy losses of relativistic electrons on the background thermal plasma of the blob can give Comptonized gamma-ray fluence spectra with spectral breaks Delta alpha much greater than 0.5. If only,Thomson and synchrotron losses are important, spectral breaks much greater than 0.5 are also possible if electrons are injected with a low-energy cutoff or if a ''snapshot'' of the spectral flux is measured over a time interval much less than the interval during which electrons cool. The important observational quantities for testing Compton-scattering models of radio galaxies and blazars are the bolometric luminosities in the radio/optical synchrotron component and the X-ray/gamma-ray Compton component, which approximately equal the peak values of the vF, spectral power flux or fluence. Numerical simulations of jet spectra are presented, showing the effects of different parameter values on the shapes of the synchrotron and Compton components. We derive straightforward predictions for the simplest external Compton and synchrotron self-Compton models from the analytic expressions for the spectra. The results are applied to quasi-simultaneous observations of 3C 279 at two different epochs, showing that this behavior can be understood if the blazar radiation results from electrons that Compton-scatter externally produced photons in outflowing blobs with different bulk Lorentz factors. Fits to quasi-simultaneous observations of 3C 273 and quiescent and flaring states of Mrk 421 are also presented.
