results will apply for $\epsilon =1$ and $d_p=\hat d_p$, if $\hat d_p\ll 1$.)

B. Order $\epsilon $
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In linear order (i.e., $\mathcal{O}(\epsilon))$, by multiplying Eq. (11) by $\exp (-i\theta )d\theta $ and integrating over $\theta $, we have for the component of $f^{(1)}$ varying as $e^{i\theta }$, $$\frac{\partial f_1^{(1)}}{\partial t}+(i\omega +\xi
)f_1^{(1)}=\frac{K}{2} R^{(1)*}+ih_1\Delta (t) \ , \ \
R^{(1)*}(t)=\int f^{(1)}gd\omega \ ,$$ where $f_1^{(1)}(\omega ,t)=0$ for $t<0$ and $R^{(1)*}$ is the complex conjugate of $R^{(1)}$. Due to the delta function term on the right hand side of Eq. (15), $ih_1d_0\delta (t)$, at the instant just after the first delta function (denoted $t=0^+$), $f_1^{(1)}$ jumps from zero just before the delta function (denoted $t=0^-$) to the value $f_1^{(1)}(\omega ,0^+)=ih_1d_0$. Making use of this observation, in Appendix I we solve Eq. (15) for $0<t<\tau $, with the result that, for $K<K_c$, $$f_1^{(1)}(\omega ,t)=A(\omega )e^{-(i\omega +\xi )t}+\ \ ({\rm a\
more\ rapidly\  exponentially\  decaying\  component)} \ ,$$ where $$A(\omega )=ih_1d_0/D[-(i\omega +\xi )] \ ,$$ $$D(s)=1-\frac{K}{2}\int ^{+\infty}_{-\infty} \frac{g(\omega
)d\omega }{s+\xi +i\omega } \ , \ {\rm for} \ Re(s)>0 \ ,$$ and $D(s)$ for $Re(s)\leq 0$ is defined from Eq.(18) by analytic continuation. Since Eq. (16) applies for $0<t<\tau $, we have that 