We demonstrate that the exact solution for the spectrum of synchrotron radiation from an isotropic population of mono-energetic electrons in turbulent magnetic field with Gaussian distribution of local field strengths can be expressed in the simple analytic form : \left ( \frac { { d } \dot { N } } { { 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 expression to find approximate synchrotron spectra for power-law electron distributions with \propto \exp \left ( - \left [ \gamma / \gamma _ { 0 } \right ] ^ { \beta } \right ) type high-energy cut-off ; the resulting synchrotron spectrum has the exponential cut-off factor with frequency raised to 2 \beta / ( 3 \beta + 4 ) power in the exponent .
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 , which results in exact expression for the synchrotron spectrum when using monochromatic ( i.e. , each electron radiates at frequency \omega _ { m } = a _ { m } \gamma ^ { 2 } \frac { eB _ { 0 } } { m _ { e } c } ) approximation .