= Solution
Assume <local thermodynamic equilibrium>, negligible scattering of thermal radiation, and a plane-parallel atmosphere. With inward <optical depth> $\tau_\lambda$ and outward ray cosine $\mu$, the <radiative transfer equation> is
$$
\mu\frac{dI_\lambda}{d\tau_\lambda}=I_\lambda-B_\lambda[T(\tau_\lambda)].
$$
For a deep atmosphere, its <formal solution of the radiative transfer equation> is
$$
I_\lambda(0,\mu)=\int_0^\infty B_\lambda[T(t)]e^{-t/\mu}\frac{dt}{\mu}.
$$
The weighting samples $\tau_\lambda$ of order unity. The <Eddington-Barbier relation> makes this precise when the source function is nearly linear: $I_\lambda(0,\mu)\simeq B_\lambda[T(\tau_\lambda=\mu)]$.
A molecular band with greater <opacity> reaches unit <optical depth> higher than the nearby continuum. If <temperature> decreases upward, that band samples cooler gas and appears in absorption. If an <atmospheric thermal inversion> makes the upper gas hotter, the band appears in emission. If both depths have the same <temperature>, an opaque isothermal atmosphere has
$$
\boxed{I_\lambda=B_\lambda(T),}
$$
and no opacity-dependent features. This last conclusion assumes all wavelengths are opaque or the lower boundary emits the same <Planck function>; an optically thin isothermal layer over a different background need not be featureless.
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