Charm production in deep-inelastic e$\gamma$ scattering to next-to-leading order in QCD

We discuss the calculation of F_2^{\gamma}({\rm charm}) to next-to-leading order (NLO) in QCD, including contributions from both hadronlike and pointlike photons. We show that the former dominates strongly below x\simeq 0.01, and the latter above this value. This fact makes F_2^{\gamma}({\rm charm}) for x \geq 0.01 calculable, whereas for x \leq 0.01 it serves to constrain the small-x gluon density in the photon. Both ranges in x are accessible at LEP2. Theoretical uncertainties are well under control. We present rates for single-tag events for the process for e^+e^- \rightarrow e^+e^- c X for LEP2. Although these event rates are small, we believe a measurement of F_2^{\gamma}({\rm charm}) is feasible.