Dispersion and birefringence in a synchrotron-emitting gas

The response tensor for a highly relativistic, magnetized electron gas is evaluated by performing the integrals over gyrophase and pitch angle using the method of stationary phase. Only the transverse components are considered explicitly. The anti-Hermitian part is expressed in terms of the Airy function $\mathrm{Ai}\mathrm{}\left(z\right)\mathrm{}$, and its derivative and integral, and is shown to reproduce the known formulas for synchrotron absorption. It is shown that the full tensor is obtained by the replacement $\mathrm{Ai}\mathrm{}\left(z\right)\mathrm{}\to \mathrm{Ai}\mathrm{}\left(z\right)+i\mathrm{Gi}\mathrm{}\left(z\right)\mathrm{}$, where $\mathrm{Gi}\mathrm{}\left(z\right)\mathrm{}$ is a generalized Airy function. A simple form for the high-frequency response is derived for a power-law distribution of particles.