Heavy-quark production in e$\gamma$ scattering

Open heavy flavour production at e^+e^- colliders in deeply inelastic e\gamma scattering has an interesting feature: the structure function F_2(x,Q^2) for this process is calculable for x>0.01, and is essentially proportional to the gluon density in the photon for smaller x values. We give estimates for event rates at LEP2 and a Next Linear Collider in x,Q^2 bins, and present differential distributions in the transverse momentum and rapidity of the heavy quark for the case of charm. We include all next-to-leading-order QCD corrections, and find theoretical uncertainties are well under control.