Stellar mass density from a square-root exponential cutoff (${\rho }_{\ast }=2\sqrt{\pi }{\varphi }_{\mathrm{ch}}{M}_{\ast ,\mathrm{ch}}$)

Integrating galactic stellar mass against a $x^{-7/4}{e}^{-\sqrt{x}}$ mass function requires ${\int }_0^{\infty }{x}^{-3/4}{e}^{-\sqrt{x}}dx$. Substitution $u=\sqrt{x}$ turns it into $2\Gamma (1/2)=2\sqrt{\pi }$. Thus the stellar density is $2\sqrt{\pi }$ times characteristic number density times characteristic galactic stellar mass, in the same volume convention as the mass function.