evaluating the number of signals that appear to come from the volume $dV_{HI}$ from emitters of strength $L_{HI}$ but which were actually produced in $dV_{OH}$ from emitters of $L_{OH}$. The relation between the luminosity functions then is $\Theta(L_{OH}) dL_{OH}dV_{OH}=\phi_{im}(L_{HI}) dL_{HI}dV_{HI}$. Solving for $\phi_{im}$, substituting from the above expressions as necessary and then including the approximate relation for the OH luminosity function from Eq.(\[approxtheta.eqn\]), which is valid here since all the OH signals detected will originate in emitters much brighter than $L_{OH}^*$: $$\begin{aligned}
\label{imhisteps.eqn}
\phi_{im} &=& \Theta(L_{OH})\left(\frac{dL_{OH}}{dL_{HI}}\right)
                \left(\frac{dV_{OH}}{dV_{HI}}\right) 
\\
&=& \Theta\left(\left(\frac{z_{OH}}{z}\right)^2L_{HI}\right)
                  \left(\frac{z_{OH}}{z}\right)^4 
                 \frac{\nu_{OH}}{\nu_{HI}}
\nonumber \\
&=& \frac{C(\alpha+\beta)f_m}{2L_{OH}^*}
     \frac{\nu_{OH}}{\nu_{HI}}
     \left(\frac{L_{HI}}{2L_{OH}^*}\right)^{-\frac{\alpha+\beta+2}{2}}
      \left(\frac{z_{OH}}{z}\right)^{2-\alpha-\beta}
\nonumber\end{aligned}$$ This result can then be recast as a dependence on $M_{HI}$: $$\begin{aligned}
\phi_{im}(M_{I})dM_{HI} &=& Cf_m\frac{\alpha+\beta}{2}
         \left(\frac{L_{OH}^*}{L_{HI}^*}\right)^{\frac{\alpha+\beta}{2}}
\nonumber \\
  & &  {\times} \,\,\, \frac{\nu_{OH}}{\nu_{HI}}
\left(\frac{M_{HI}}{M_{HI}^*}\right)^{-\frac{\alpha+\beta+2}{2}}
\nonumber \\
  & &   {\times} \,\,  \left(\frac{z}{z_{OH}}\right)^{\alpha+\beta-2}
      \frac{dM_{HI}}{M_{HI}^*}
\label{imhifin.eqn}\end{aligned}$$ After substitution for the constants in Eq.(\[imhifin.eqn\]), $\phi_{im}$ has the functional dependence $$\begin{aligned}
\phi_{im}(M_{HI})dM_{HI} &\propto&  
\left(\frac{M_{HI}}{M_{HI}^*}\right)^{-2.15}
%\nonumber \\  & &   {\times}   
\left(\frac{z}{z_{OH}}\right)^{0.3}
      \frac{dM_{HI}}{M_{HI}^*}
\label{imhiimp.eqn}\end{aligned}$$ producing a form with that mimics a diverging low luminosity tail, since $(\alpha+\beta+2)/2 = 2.15$ is greater than 2. The dependence on $z$ does not affect the power law shape of the imitation mass function but does weakly affect the normalization that would 