$. Accordingly, the integral over $\omega$ in (\[c55\]) vanishes for the positive times . Likewise the integral in (\[c56\]) is zero for $t<0$. Therefore we can combine the two expressions (\[c55\]) and (\[c56\]) into a single one and write the full time-dependence of electric field for all times as $$\begin{aligned}
\label{c57}
\underline{\vec{E}}(\vec{k},t)&=&\frac{1}{2\pi}\int_{-\infty}^{+\infty}
d\omega\ e^{-\imath\omega t}\int d^3p\
\left[\underline{G}(\vec{k},\ \vec{p},\ \imath\omega+0^+) \cdot\
\underline{\vec{J}}^b_N(\vec{p},\ \imath\omega+0^+)\right.\nonumber\\
&+& \left.\underline{G}(\vec{k},\ \vec{p},\ -\imath\omega+0^+)
\cdot\ \underline{\vec{J}}^f_N(\vec{p},\ -\imath\omega+0^+)\right]\end{aligned}$$ It is remarkable that using the asymptotic behavior of the Green function $\underline{G}$ and the susceptibility tensors for large $|\omega|$ it can be verified straightforwardly, the integral in (\[c57\]) reduces to $\underline{\vec{E}}(\vec{k},0)$ at $t=0$ and hence this equation give us correctly the time dependence of the electric field for all times. Finally having the electric field operator, we can obtain the magnetic field using $
\imath\vec{k}\times\underline{\vec{E}}(\vec{k},t)=\underline{\dot{\vec{B}}}(\vec{k},t)$ and then the polarization densities employing the constitutive Eqs.(\[c31\]) and (\[c36\]).

The magnetodielectric media with only one response equation
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In the previous sections we have assumed the magnetodielectric medium under consideration is one for which the distinction between polarization and magnetization physically is possible and the total current induced in medium can be separated into electric and magnetic parts. It is remarkable that the 