 Solution

A shorter radiation flash has a broader Fourier transform. From iii, its characteristic angular-frequency bandwidth isAt fixed bending radius this scales as ${\gamma }^3$ as $v\to c$. The standard synchrotron critical angular frequency makes the same scaling explicit:A periodically circulating electron produces harmonics of its revolution frequency. The very large harmonic range, of order ${\gamma }^3$, forms the familiar effectively continuous broad synchrotron spectrum at ordinary detector resolution. The result concerns the intrinsic spectrum of the radiation, not merely the slow envelope imposed by a long incoherent bunch. Ordinary frequency is $\nu =\omega /(2\pi )$; mixing cycles per second with radians per second would introduce a spurious factor of $2\pi$ into photon-energy estimates.^[ref]