Uniform expansion near backward caustic

Zeroth order prefactor The derivation here closely follows that in /notes/aaf/aaf-rescat/uniform-aaf/, which follows brrq. We use the same definitions for the caustic, expect here we have \(m=1\) We start with Eq. /notes/aanf/aanf-rescat/ion-amplitude/#mjx-eqn-eq:ion-amplitude-integral, based on which the contribution to the ionization amplitude from near the caustic can be written as \begin{align} I_v^{r,c}(k) & = \int_{-\infty}^\infty \sum_{v'} \tilde{p}_{v'v}(t) e^{i\mathcal{S}_{r, v'v}(t, k)} dt \end{align} where \begin{align} \tilde{p}_{v'v}(t) & = \frac{\tilde{g}_{v'}(t_i)}{\abs{F(t_i)(t-t_i)}^{1/2}} j_{v',v}(x,t) |_{-a}^a \end{align} and