In the external shock model, gamma-ray burst (GRB) emissions are produced by the energization and deceleration of a thin relativistic blast wave due to its interactions with the circumburst medium (CBM). We study the physical properties of an analytic function which describes temporally-evolving GRB spectra in the limit of a smooth CBM with density n(x) ∝ x−η, where x is the radial coordinate. ...