\biggl| \,  e^{-i\mathbf{kr}} \displaystyle\frac{\partial}{\partial \mathbf{r}} \,
    \biggr| \,k_{i} \biggr> =
    \int
      \psi^{*}_{f}(\mathbf{r}) \:
      e^{-i\mathbf{kr}} \displaystyle\frac{\partial}{\partial \mathbf{r}}\:
      \psi_{i}(\mathbf{r}) \;
      \mathbf{dr}
\end{array}
\label{eq.2.1.2}$$ and $\psi_{i}(\mathbf{r}) = |k_{i}\bigr>$ and $\psi_{f}(\mathbf{r}) = |k_{f}\bigr>$ are stationary wave functions of the decaying system in the initial $i$-state and final $f$-state which do not contain number of photons emitted, $Z_{\rm eff}$ and $m$ are effective charge and reduced mass of this system. $\mathbf{e}^{(\alpha)}$ are unit vectors of polarization of the photon emitted, $\mathbf{k}$ is wave vector of the photon and $w = k = \bigl| \mathbf{k}\bigr|$. Vectors $\mathbf{e}^{(\alpha)}$ are perpendicular to $\mathbf{k}$ in Coulomb calibration. We have two independent polarizations $\mathbf{e}^{(1)}$ and $\mathbf{e}^{(2)}$ for the photon with impulse $\mathbf{k}$ ($\alpha=1,2$). One can develop formalism simpler in the system of units where $\hbar = 1$ and $c = 1$, but we shall write constants $\hbar$ and $c$ explicitly. Let’s find also square of the matrix element $a_{fi}$ used in definition of *probability of transition*. Using the *formula of power reduction of $\delta$-function* (see [@Bogoliubov.1980], § 21, p. 169): $$[\delta(w)]^{2} = \delta(w)\: \delta(0) = \delta(w) \: (2\pi)^{-1} \int dt =
  \delta(w)\: (2\pi)^{-1}\, T,
\label{eq.2.1.3}$$ we find ($T\to +\infty$ is higher time limit): $$|a_{fi}|^{2} = 2\pi\: T\: |F_{fi}|^{2} \cdot \delta(w_{f}-w_{i}+w),
