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We demonstrate that the exact solution for the spectrum of synchrotronradiation from an isotropic population of mono-energetic electrons in turbulentmagnetic field with Gaussian distribution of local field strengths can beexpressed in the simple analytic form: $\left( \frac{{\rm d} \dot{N}}{{\rm d}\omega} \right)_t  = \frac{\alpha}{3} \frac{1}{\gamma^2} \left( 1 + \frac{1}{x^{2/3}} \right)\exp \left( - 2 x^{2/3} \right)$, where $x = \frac{\omega}{\omega_0}\, ;\omega_0 = \frac{4}{3} \gamma^2 \frac{eB_0}{m_e c}\, .$ We use this expressionto find approximate synchrotron spectra for power-law electron distributionswith $\propto \exp\left( -\left[ \gamma/\gamma_0 \right]^\beta\right)$ typehigh-energy cut-off; the resulting synchrotron spectrum has the exponentialcut-off factor with frequency raised to $2\beta/(3\beta+4)$ power in theexponent. For the power-law electron distribution without high-energy cut-off,we find the coefficient $a_m$ as a function of the power-law index, whichresults in exact expression for the synchrotron spectrum when usingmonochromatic (i.e., each electron radiates at frequency $\omega_m = a_m\gamma^2 \, \frac{e B_0}{m_e c}$) approximation.

Exact analytical expression for the synchrotron radiation spectrum in the Gaussian turbulent magnetic field