  We present useful functions for the profiles of dark-matter (DM) haloes with
a free inner slope, from cusps to cores, where the profiles of density,
mass-velocity and potential are simple analytic expressions. Analytic velocity
is obtained by expressing the mean density as a simple functional form, and
deriving the local density by differentiation. The function involves four shape
parameters, with only two or three free: a concentration parameter $c$, inner
and outer asymptotic slopes $\alpha$ and $\bar{\gamma}$, and a middle shape
parameter $\beta$. Analytic expressions for the potential and velocity
dispersion exist for $\bar{\gamma}=3$ and for $\beta$ a natural number. We
match the models to the DM haloes in cosmological simulations, with and without
baryons, ranging from steep cusps to flat cores. Excellent fits are obtained
with three free parameters ($c$, $\alpha$, $\bar{\gamma}$) and $\beta=2$. For
an analytic potential, similar fits are obtained for $\bar{\gamma}=3$ and
$\beta=2$ with only two free parameters ($c$, $\alpha$); this is our favorite
model. A linear combination of two such profiles, with an additional free
concentration parameter, provides excellent fits also for $\beta=1$, where the
expressions are simpler. The fit quality is comparable to non-analytic popular
models. An analytic potential is useful for modeling the inner-halo evolution
due to gas inflows and outflows, studying environmental effects on the outer
halo, and generating halo potentials or initial conditions for simulations. The
analytic velocity can quantify simulated and observed rotation curves without
numerical integrations.
