= Stellar mass density from a square-root exponential cutoff
{title2=$\rho_*=2\sqrt\pi\,\phi_{\rm ch}M_{*,\rm ch}$}
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.
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