m\Omega\frac{\cos{(\omega\tau)}}{\sin{(\Omega\tau)}}x^{\prime}_j
 \nonumber \\
 &-&
 m\Omega\frac{\sin{(\omega\tau)}}{\sin{(\Omega\tau)}}\epsilon_{jk3}x^{\prime}_k
 - e A_j(\textbf{r}) \Big\} \langle\textbf{r},\tau|\textbf{r}^{\prime},0\rangle
\label{Eq77}\end{aligned}$$ and $$\begin{aligned}
\Big\{
 \!\!\!\!\!&{\,}&\!\!\!\!\!
 \frac{i\hbar}{C(\textbf{r},\textbf{r}^{\prime})}
 \frac{\partial C(\textbf{r},\textbf{r}^{\prime})} {\partial x^{\prime}_j}
  m\Omega\frac{\cos{(\Omega\tau)}}{\sin{(\Omega\tau)}}x^{\prime}_j
 %\nonumber \\
 +  m\Omega\frac{\cos{(\omega\tau)}}{\sin{(\Omega\tau)}}x_j
  \nonumber\\
 &-&
 m\Omega\frac{\sin{(\omega\tau)}}{\sin{(\Omega\tau)}}\epsilon_{jk3}x_k
 -e A_j(\textbf{r}^{\prime})\Big\} \langle\textbf{r},\tau|\textbf{r}^{\prime},0\rangle.
\label{Eq78}\end{aligned}$$ Equating (\[Eq74\]) and (\[Eq77\]), and also (\[Eq75\]) and (\[Eq78\])), we get the system of differential equations for $C(\textbf{r},\textbf{r}^{\prime})$ $$\begin{aligned}
i\hbar \frac{\partial C(\textbf{r},\textbf{r}^{\prime})}{\partial
 x_j}
 &+&
 e\Big[A_j(\textbf{r})+\frac{F_{jk}}{2}x_k \Big] C(\textbf{r},\textbf{r}^{\prime})=0\, , \label{Eq79} \\
i\hbar \frac{\partial C(\textbf{r},\textbf{r}^{\prime})}{\partial x^{\prime}_j}
 &-&
 e\Big[A_j(\textbf{r}^{\prime})+\frac{F_{jk}}{2}x^{\prime}_k \Big] C(\textbf{r},\textbf{r}^{\prime})=0.
\label{Eq80}\end{aligned}$$ Proceeding as in the previous example, we first integrate (\[Eq79\]). With this goal, we multiply it by $dx_j$, sum in $j$ and integrate it to obtain $$C({\bf r},{\bf r}^{\;\prime})= C({{\bf r}}^{\;\prime},{\bf
r}^{\;\prime})
 \exp\left\lbrace {ie\over \hbar}_{\;\;\Gamma}\!\!
\int_{{\bf r}^{\prime}}^{{\bf r}}
\Big[ A_j(\mbox{\mathversion{bold}${\xi}$})+{F_{jk}\over 2}\xi_k\Big] d\xi_j \right\rbrace\, , \\
\label{CrrLinha1}$$ where the path of integration $\Gamma$ will be specified in a moment. Inserting expression (\[CrrLinha1\]) into the second differential equation (\[Eq80\]), we get $$\frac{\partial}{\partial x_j^{\prime}}
 C({{\bf r}}^{\;\prime},{{\bf r}}^{\;\prime}) = 0
 \;\;\Longrightarrow\;\;
 C({{\bf r}}^{\;\prime},{{\bf r}}^{\;\prime}) =C_0\, ,$$ where $C_0$ is a constant independent of ${{\bf r}}^{\;\prime}$, so that equation (\[CrrLinha1\]) can be cast, after some convenient rearrangements, into the form $$C({\bf r},{\bf r}^{\prime})=C_{0}\exp{\left\lbrace{ie\over\hbar}
 _{\;\;\Gamma}\!\!
\int_{{\bf r}^{\prime}}^{{\bf r}}
\Big[\textbf{A}(\mbox{\mathversion{bold}${\xi}$})
 - \frac{1}{2}\textbf{B}\times\mbox{\mathversion{bold}${\xi}$}\Big]\cdot
d\mbox{\mathversion{bold}${\xi}$}\right\rbrace }.
 \label{CrrLinha2}$$ Note that the integrand has a vanishing curl so that we 