$dV_{HI}$ at $z$ due to the overlap that occurs in observed frequency for the two populations. For simplicity in this estimate, approximations for the luminosity distance $d_l\approx z_{OH}c/H_o$ and for the comoving volume $dV_{OH}\approx z_{OH}^2dz_{OH}d\Omega$ will be assumed, instead of the more rigorous forms used earlier in this paper in the discussion of OH masers at cosmological distances. These produce errors of less than 10 percent in $d_l$ and less than 40 percent in comoving volume.

The incremental frequency band $d\nu$ observed at the frequency $\nu$ is related to increments in redshift for the OH and HI lines: $$d\nu = - \frac{\nu^2}{\nu_{OH}}dz_{OH}= - \frac{\nu^2}{\nu_{HI}}dz_{HI}$$ implying that $dz_{OH}=(\nu_{OH}/\nu_{HI})dz_{HI}$. The flux $S$ measured in the line emission from the OH megamaser of luminosity $L_{OH}$ is falsely interpreted as HI emission from an object of $L_{HI}$: $$S =  \frac{ L_{OH}}{4\pi(z_{OH}c/H_o)^2}=  \frac{ L_{HI}}{4\pi(zc/H_o)^2}$$ so that $L_{HI}=(z/z_{OH})^2L_{OH}$. Since the redshift range being explored for imitation HI signals is small, $z<0.01$, the OH masers are drawn from a thin shell, allowing $z_{OH}$ to be considered constant at $z_{OH}\approx 0.17$ in what follows.

The goal is to derive an approximation for the imitation HI mass function $\phi_{im}$ provided by distant OH masers. This can be accomplished by 