Explicit expression for the photon number emission in synchrotron radiation

An explicit and remarkably simple one-dimensional integral expression is derived for the mean number $<N>$ of photons emitted per revolution in synchrotron radiation. The familiar high-energy expression $5\pi\alpha/\sqrt{3(1-\beta^2)}$, printed repeatedly in the literature, is found to be inaccurate and only truly asymptotic with relative errors of 160%, 82% for $\beta=0.8$, 0.9, respectively. A new improved high-energy expression for $<N>$ is given.