Synchrotron Radiation as the Source of GRB Spectra, Part I: Theory

We investigate synchrotron emission models as the source of gamma ray burst spectra. We show that allowing for synchrotron self absorption and a ``smooth cutoff'' to the electron energy distribution produces a wide range of low energy spectral behavior. We show that there exists a correlation between the value of the peak of the $\nu F_{\nu}$ spectrum, $E_{p}$, and the low energy spectral index $\alpha$ as determined by spectral fits over a finite bandwidth. Finally, we discuss the implications of synchrotron emission from internal shocks for GRB spectral evolution.