Enclosed Mass and Circular Velocity for the Einasto Dark Matter Halo Profile

This note presents an independent derivation of the enclosed mass M(r) and circular velocity v(r) for a spherical dark matter halo described by the Einasto density profile. Because the Einasto profile has no elementary antiderivative, the volume integral for the enclosed mass is reduced — via a fractional substitution — to the lower incomplete gamma function γ(s,x). The resulting closed-form expression avoids numerical integration when fitting rotation curves, and will be used as the dark-matter model in a forthcoming analysis comparing NFW and Einasto halo fits across the SPARC galaxy sample. The result itself is known in the literature (Mamon & Łokas 2005; Retana-Montenegro et al. 2012); this document is an independent derivation produced as part of that project.