d^3q \
\underline{\Lambda}^{(1)}(\vec{k},\vec{q},s)\ \cdot\
\underline{G}^{(n-1)}(\vec{q}, \vec{p},s)\nonumber\\
&&\Longrightarrow \underline{G}^{(n)}(\vec{k},
\vec{p},s)=-L^{(-1)}(\vec{k},s)\int d^3q\
\underline{\Lambda}^{(1)}(\vec{k},\vec{q},s)\ \cdot\
\underline{G}^{(n-1)}(\vec{q}, \vec{p},s)\end{aligned}$$ where we have used Eq.(\[c50\]). Using the series (\[c52\]) and the recurrence relation (\[c54\]) one can obtain the Green function $\underline{G}$ up to an arbitrary accuracy. In n’th order approximation it can be neglected from the terms $\underline{G}^{(j)}\hspace{00.50 cm}j\geq n+1$ in the series (\[c52\]) which may be useful for media with considerably weak polarizability.\
Now, having the Green function $\underline{G}$, the time dependence of electric field operator for negative and positive times can be obtained by inverse Laplace transformations of $
\underline{\vec{E}}^b$ and $ \underline{\vec{E}}^f$ , respectively. Carrying out the inverse Laplace transformation of Eq. (\[c46\]) for $ t<0$ we deduce $$\label{c55}
\underline{\vec{E}}(\vec{k},t)=\frac{1}{2\pi}\int_{-\infty}^{+\infty}
d\omega\ e^{-\imath\omega t}\int d^3p\ \underline{G}(\vec{k},\
\vec{p},\ \imath\omega+0^+) \cdot\
\underline{\vec{J}}^b_N(\vec{p},\ \imath\omega+0^+)$$ where $0^+$ is an arbitrarily small positive number. For $t>0$ the inverse Laplace transformation of $ \underline{\vec{E}}^f$ is $$\label{c56}
\underline{\vec{E}}(\vec{k},t)=\frac{1}{2\pi}\int_{-\infty}^{+\infty}
d\omega\ e^{-\imath\omega t}\int d^3p\ \underline{G}(\vec{k},\
\vec{p},\ -\imath\omega+0^+) \cdot\
\underline{\vec{J}}^f_N(\vec{p},\ -\imath\omega+0^+)$$ Because the Laplace transformations of the susceptibility tensors are analytic functions respect to the variable $s$ in any point of the half-plane $ Re[s]\geq 0$ then, clearly the $\omega$-dependence integrand in Eq. (\[c55\]) also is analytic in the half-plane $Im
[\omega] \leq 0 