\pm)}(k,r) =
      \bar{N}_{2}
        \biggl( 1 \mp \displaystyle\frac{iW(r)}{k\alpha} \biggr)
        e^{\pm ikr} 
\label{eq.3.2.1.3}$$ and $$\bar{N}_{2} = i\displaystyle\frac{\bar{N}_{1}}{k\alpha N_{2}}
\label{eq.3.2.1.4}$$

In accordance with main statements of quantum mechanics, for applying such form of the radial wave function to the description of scattering of the particle in the field of the potential $V_{2}(r)$, it needs to achieve a boundary requirement $\chi_{l=0}^{(2)}(k,r) \to 0$ at $r \to 0$, which gives a finiteness of the wave function (\[eq.2.1.1\]) at $r=0$ (and $S_{l=0}^{(2)}$ must have finite values and be not zero). One can see from (\[eq.3.2.1.2\]), that it is fulfilled only in case ($W(r)$ is real): $$\begin{array}{lcl}
  Re (S_{l=0}^{(2)}) =
    \displaystyle\frac{k^{2}\alpha^{2}-W^{2}(0)}
    {k^{2}\alpha^{2}+W^{2}(0)}, &
  Im (S_{l=0}^{(2)}) =
    \displaystyle\frac{2W(0)k\alpha}{k^{2}\alpha^{2}+W^{2}(0)},
\end{array}
\label{eq.3.2.1.5}$$ where $$W(0) =
    -\displaystyle\frac{\alpha}{r_{0} + \displaystyle\frac{1}{C\alpha}}.
\label{eq.3.2.1.6}$$ For the partial components of the S-matrix the following property $|S_{l=0}|^{2} = 1$ is fulfilled also. In limit $r \to 0$ we obtain the following expression for the radial wave function: $$\chi_{l=0}^{(2)}(k,r) =
    \bar{N}_{2} \biggl(1 + \displaystyle\frac{iW(0)}{k\alpha}\biggr)
    \biggl(e^{-ikr} - S_{l=0}^{(2)}
      \displaystyle\frac{k\alpha - iW(0)}{k\alpha + iW(0)}
      e^{ikr}\biggl),
\label{eq.3.2.1.7}$$ which in its form coincides with Exp. (\[eq.3.2.1.1\]) for the wave function for the potential $V_{1}(r)$ from (\[eq.3.1.4\]) with zero value.

In Fig. \[fig.3\] real and imaginary parts of the wave function near to the point $r=0$ are shown (here the starting formulas (\[eq.3.2.1.2\])–(\[eq.3.2.1.3\]) are taken). From Fig. \[fig.3\] (a, b) one can see a deformation of the imaginary part of this wave function with change of the wave vector $k$ and the parameter $C$. The real part of the wave function in its behavior looks lik