  We provide complete analytic expressions for the inclusive prompt photon
production cross section in hadronic final states of $e^+e^-$ annihilation
reactions through one-loop order in quantum chromodynamics perturbation theory.
Computed explicitly are direct photon production through first order in the
electromagnetic strength $\alpha_{em}$ and the quark-to-photon and
gluon-to-photon fragmentation contributions through first order in the strong
coupling $\alpha_s$. The full angular dependence of the cross sections is
displayed, separated into transverse $(1 +\cos ^2\theta _\gamma)$ and
longitudinal $(\sin ^2\theta_\gamma)$ components, where $\theta_\gamma$
specifies the direction of the photon with respect to the $e^+e^-$ collision
axis. We discuss extraction of fragmentation functions from $e^+e^-$ data.
