= Solution
For a vertical ray, the <transmittance> from $p_2$ to space is $\mathcal T_\nu(p_2)=e^{-\tau_\nu(p_2)}$. Direct integration of the full pressure-dependent <Lorentz line shape> gives, for $d>0$,
$$
\tau_\nu(p_2)=\frac qg\int_0^{p_2}\kappa_\nu(p)\,dp
=C\ln\left[1+\left(\frac{\Delta_s p_2}{p_s d}\right)^2\right].
$$
The exact Lorentz-model <transmittance> and its <strong-line limit> are therefore
$$
\boxed{\mathcal T_\nu(p_2)=\left[1+\left(\frac{\Delta_s p_2}{p_s d}\right)^2\right]^{-C}
\simeq\exp\left[-\left(\frac{wp_2}{p_s d}\right)^2\right].}
$$
The approximation is justified at the radiating level by $C\gg1$. At zenith-angle cosine $\mu$, replace $\tau$ by $\tau/\mu$; if a hemispheric flux transmission is desired, its normalized value is $2\int_0^1\mu e^{-\tau/\mu}d\mu=2E_3(\tau)$. The pure Lorentz model formally gives infinite line-centre <optical depth> from every nonzero $p_2$ to $p=0$, because $\kappa_{\nu_0}\propto1/p$. This is an unphysical central-core singularity, addressed in part (d), rather than a divergence of the integrated outgoing <flux>.
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