  Synchrotron radiation of relativistic electrons is an important radiation
mechanism in many astrophysical sources. In the sources where the synchrotron
cooling time scale $t_c$ is shorter than the dynamical time scale $t_{dyn}$,
electrons are cooled down below the minimum injection energy. It has been
believed that such "fast cooling" electrons have an energy distribution $dN_e
/d\gamma_e \propto \gamma_e^{-2}$, and their synchrotron radiation flux density
has a spectral shape $F_\nu \propto \nu^{-1/2}$. On the other hand, in a
transient expanding astrophysical source, such as a gamma-ray burst (GRB), the
magnetic field strength in the emission region continuously decreases with
radius. Here we study such a system, and find that in a certain parameter
regime, the fast cooling electrons can have a harder energy spectrum, and the
standard $d N_e / d \gamma_e \propto \gamma_e^{-2}$ spectrum is achieved only
in the deep fast cooling regime when $t_c \ll t_{dyn}$. We apply this new
physical regime to GRBs, and suggest that the GRB prompt emission spectra whose
low-energy photon index $\alpha$ has a typical value -1 could be due to
synchrotron radiation in this moderately fast cooling regime.
