We present a complete solution to the cyclotron-synchrotron radiation due to an isotropic distribution of electrons moving in a magnetic field .
We make no approximations in the calculations other than artificially broadening the harmonics by a small amount in order to facilitate the numerics .
In contrast to previous calculations , we sum over all relevant harmonics and integrate over all particle and observer angles relative to the magnetic field .
We present emission spectra for electron temperatures T = 5 \times 10 ^ { 8 } K , 10 ^ { 9 } K , 2 \times 10 ^ { 9 } K to 3.2 \times 10 ^ { 10 } K , and provide simple fitting formulae which give a fairly accurate representation of the detailed results .
For T \geq 3.2 \times 10 ^ { 10 } K , the spectrum is represented well by the asymptotic synchrotron formula , which is obtained by assuming that the radiating electrons have Lorentz factors large compared to unity .
We give an improved fitting formula also for this asymptotic case .