||Message]
\begin{eqnarray} \frac{\mu V_0}{\hbar^2 k}\left|\int_0^{R_0} \left( e^{2ikr}-1\right) dr\right| &=& \frac{\mu V_0}{\hbar^2 k}\left|\frac{e^{2ikR_0}-1}{2ik} - R_0 \right|\label{E2}\\ &=& \frac{\mu V_0}{2\hbar^2 k^2}\left|e^{2ikR_0}- 2ik R_0 -1 \right|\label{E3} \end{eqnarray}
\begin{eqnarray}\psi_2(\vec{r}) &=&e^{i\vec{k_i}.\vec{r}}-\frac{1}{4\pi} \int\frac{e^{ik|\vec{r}-\vec{r}^{'}|}}{|\vec{r}-\vec{r}^{'}|} U(\vec{r}^{'})\psi_1(\vec{r}^{'}) d^3r{'} \\
&=& e^{i\vec{k}_i\cdot\vec{r}} - \frac{1}{4\pi} \int\frac{e^{ik|\vec{r}-\vec{r}^{'}|}}{|\vec{r}-\vec{r}^{'}|} U(\vec{r}^{'})e^{(i\vec{k}_i\cdot\vec{r}^{'})} d^3 r{'}\\
&& + \frac{1}{(4\pi)^2}\int \frac{e^{ik|\vec{r}-\vec{r}^{'}|}}{|\vec{r}-\vec{r}^{'}|} U(\vec{r^{'}})\int \exp(i\vec{k}_i\cdot\vec{r^{'}}) \frac{e^{ik|\vec{r}^{'}-\vec{r}^{{''}}|}}{|\vec{r^{'}}-\vec{r^{{''}} }|} U(\vec{r}^{''})e^{i\vec{k}_i\cdot(\vec{r}^{'}+\vec{r}^{''})}d^3r^{'} d^3r^{''}. \label{E4}
\end{eqnarray}
