H i mass function
Schechter function
The HIMF is usually fitted by a Schechter function that describes the space density of galaxies per H i mass interval:
\begin{equation*} \Theta(\mu) = \frac{\mathrm{d}n(\mu)}{\mathrm{d}\mu} = \Theta^{\ast} \mu^{\alpha} \exp(-\mu) \end{equation*}
with $\mu = M_{\rm HI} / M_{\rm HI}^{\ast}$, where $\Theta^{\ast}$ is the global normalisation factor, $\alpha$ is the exponent of the power-law component of the Schechter function, and $M_{\rm HI}^{\ast}$ is the characteristic turnover mass (or “knee”) of the function. Alternatively, by substituting $\mathrm{d}\mu = \ln(b) \, \mu \, \mathrm{d} \log_{b}(\mu)$, the Schechter function can be expressed per logarithmic mass interval:
\begin{equation*} \Phi(\mu) = \frac{\mathrm{d}n(\mu)}{\mathrm{d}\log_{b}(\mu)} = \ln(b) \, \Theta^{\ast} \mu^{\alpha + 1} \exp(-\mu) \end{equation*}
where $b = 10$ is the most common choice. To be continued...