been obtained: $$\begin{array}{ccc}
  m = -\mu = \pm 1,  &  l_{f} \ge 1, &
  n \ge |\mu - \mu^{\prime}| = |m + \mu^{\prime}|.
\end{array}
\label{eq.2.5.3}$$

Let’s introduce the following differential matrix elements $dp_{l}^{M}$ and $dp_{l}^{E}$ dependent on the angle $\theta$ : $$\begin{array}{lcl}
  \displaystyle\frac{d \,p_{l}^{M}}{\sin{\theta}\,d\theta} & = &
    i^{l_{f}+l_{ph}+1} \;
    J(l_{f},l_{ph})
    \displaystyle\sum\limits_{m = \pm 1}
    m \,h_{-m} \;
    \displaystyle\sum\limits_{\mu^{\prime} = \pm 1}
    C_{l_{f}l_{ph}l_{ph}}^{m \mu^{\prime}} f_{l_{f}l_{ph}}^{m \mu^{\prime}}(\theta), \\

  \displaystyle\frac{d \,p_{l}^{E}}{\sin{\theta}\,d\theta} & = &
    -i^{l_{f}+l_{ph}} \;
    \sqrt{\displaystyle\frac{l_{ph}+1}{2l_{ph}+1}} \, J(l_{f},l_{ph}-1)
    \displaystyle\sum\limits_{m = \pm 1}
    h_{-m} \;
    \displaystyle\sum\limits_{\mu^{\prime} = \pm 1}
      C_{l_{f},l_{ph},l_{ph}-1}^{m \mu^{\prime}} \: f_{l_{f},l_{ph}-1}^{m \mu^{\prime}}(\theta) \: - \\
  & - &
    i^{l_{f}+l_{ph}} \;
    \sqrt{\displaystyle\frac{l_{ph}}{2l_{ph}+1}} \, J(l_{f},l_{ph}+1)
    \displaystyle\sum\limits_{m = \pm 1}
    h_{-m} \;
    \displaystyle\sum\limits_{\mu^{\prime} = \pm 1}
      C_{l_{f},l_{ph},l_{ph}+1}^{m \mu^{\prime}} \: f_{l_{f},l_{ph}+1}^{m \mu^{\prime}}(\theta),
\end{array}
\label{eq.2.5.4}$$ where $$\begin{array}{lcl}
  C_{l_{f} l_{ph} n}^{m \mu^{\prime}} & = &
    (-1)^{l_{f}+n+1 - \mu^{\prime} + \frac{|m+\mu^{\prime}|}{2}} \;
    (n, 1, l_{ph} \big| -m-\mu^{\prime}, \mu^{\prime}, -m) \; \times \\
  & \times &
    \sqrt{\displaystyle\frac{(2l_{f}+1)\,(2n+1)}{32\pi}\;
          \displaystyle\frac{(l_{f}-1)!}{(l_{f}+1)!} \;
          \displaystyle\frac{(n-|m+\mu^{\prime}|)!}{(n+|m+\mu^{\prime}|)!}}, \\
  f_{l_{f} n}^{m \mu^{\prime}}(\theta) & = &
    P_{l_{f}}^{1}(\cos{\theta}) \; P_{1}^{1}(\cos{\theta}) \; P_{n}^{|m+\mu^{\prime}|} (\cos{\theta}).
\end{array}
\label{eq.2.5.5}$$ One can see that integration of functions (\[eq.2.5.4\]) by angle $\theta$ with limits from 0 to $\pi$ gives the total matrix elements $p_{l}^{M}$ and $p_{l}^{E}$ exactly for transition into superposition of all possible final 