AbstractBesides the general formula for the intensity of synchrotron radiation which is however not suitable for explicite calculations there exist a number of approximations valuable in certain frequency and energy regions. These approximative formulas together with a new approximation for relativistic electrons are discussed. It is shown how they fit to solve an integral equation for the energy distribution of the radiant electrons if the frequency distribution of their synchrotron radiation is given. In some cases our new approximation leads to a solution by Laplace transformation. This turns out convenient especially when investigating discrete cosmic radio sources. The received results are applicated to the optical and radio emission of the Crab Nebula. The influence of plasma on generating and propagating of synchrotron radiation is estimated.