= Solution
Use $u$ as a spectroscopic <wavenumber>, in the same units throughout. Write the mass <absorption coefficient> of the absorbing species as
$$
\boxed{\kappa_\nu=S\phi(\nu-\nu_0),\qquad
\int_{-\infty}^{\infty}\phi(\delta\nu)\,d\delta\nu=1,\qquad
S=\int\kappa_\nu\,d\nu.}
$$
Here $S$ is the <spectral line strength>, the integrated mass <absorption coefficient>; it is not the line-centre coefficient. If $\kappa_\nu$ has units $\mathrm{m^2\,kg^{-1}}$ and $\nu$ is in $\mathrm{cm^{-1}}$, then $S$ has units $\mathrm{m^2\,kg^{-1}\,cm^{-1}}$. For absorber <mass density> $\rho_a$, the volume coefficient in <Beer-Lambert law> is $\rho_a\kappa_\nu$.
\b[<Lifetime broadening>.] Finite lifetimes of the upper and lower states destroy optical coherence, even for an isolated molecule. Exponential coherence decay gives a <Lorentzian line shape>. For population decay rates $\Gamma_u,\Gamma_l$ and no extra dephasing, its <wavenumber> half-width is $(\Gamma_u+\Gamma_l)/(4\pi c)$, with $c$ expressed in the length unit reciprocal to $\nu$. This sets a natural linewidth and can matter at extremely low collision rates; it is generally much smaller than atmospheric collisional or thermal widths.
\b[<Thermal Doppler broadening>.] A <Maxwell-Boltzmann distribution> of molecular line-of-sight velocities gives
$$
\phi_D(\delta\nu)=\frac{e^{-(\delta\nu/\Delta_D)^2}}{\sqrt\pi\Delta_D},\qquad
\Delta_D=\frac{\nu_0}{c}\sqrt{\frac{2k_BT}{m_a}}.
$$
This <Gaussian distribution> has a $1/e$ half-width $\Delta_D$, determined by absorber <molecular mass> $m_a$ and <temperature>. It is especially relevant in the rarefied upper atmosphere, where collisions are infrequent.
\b[<Pressure broadening>.] Collisions perturb transition <Frequency>[frequencies] and phases. In the impact regime, exponential phase decorrelation gives
$$
\phi_L(\delta\nu)=\frac1\pi\frac{\Delta_L}{\delta\nu^2+\Delta_L^2}.
$$
Here $\Delta_L$ is the half-width at half maximum, proportional to the coherence-destroying collision rate. It depends on perturber <density>, molecular collision properties, and <temperature>; it is often parameterized by $\Delta_L\propto p(T_*/T)^{n_T}$ for a specified broadener. It dominates much of the dense lower atmosphere. The <Lorentzian line shape> has algebraic $\delta\nu^{-2}$ wings, whereas the <Thermal Doppler broadening>[Doppler profile] has rapidly decreasing <Gaussian distribution>[Gaussian] wings. Where both mechanisms matter, their convolution is a <Voigt profile>. A pure pressure-broadened profile should not be extrapolated to arbitrarily low <pressure> while discarding the finite Doppler core.
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