k, \quad p' = p - k \cos \theta.$$ We use the symbol $ \left( \frac{\partial E'}{\partial k} \right)_{n'} $ to denote a derivative of the energy $ E' $ considered as a function of $ p' $, where $ p' $ is defined through (\[rad\_conserv\_cond\]), with respect to $ k $, with $ n' $ being fixed. It is important that since there exists the relation (see [@book_hep_rad]): $$\sum_{s' = 0}^{\infty} I_{s,s'}(x) I_{m,s'}(x) = \delta_{sm} \qquad
    \text{(for all allowed $ x $)},$$ the summation has taken the numbers $ s $, $ s' $ out of consideration; this is closely connected with the degeneracy and invariance existing in our problem (see Section \[sol\]). Thus, the initial quantum number $ s $ may be arbitrary.

Note that we are still considering the electron initial and final states with the definite spin quantum numbers $ \zeta $ and $ \zeta' $ respectively, i.e., we do not make any averaging or summation over them.

Radiation of an ultra-relativistic electron
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Let us now consider the most interesting case of a high-energy particle ($ m / E \equiv \lambda \ll 1 $) in a comparatively weak magnetic field ($ H \ll H_{c} = 