Analytical shear and flexion of Einasto dark matter haloes
  N-body simulations predict that dark matter haloes are described by specific
density profiles on both galactic- and cluster-sized scales. Weak gravitational
lensing through the measurements of their first and second order properties,
shear and flexion, is a powerful observational tool for investigating the true
shape of these profiles. One of the three-parameter density profiles recently
favoured in the description of dark matter haloes is the Einasto profile. We
present exact expressions for the shear and the first and second flexions of
Einasto dark matter haloes derived using a Mellin-transform formalism in terms
of the Fox H and Meijer G functions, that are valid for general values of the
Einasto index. The resulting expressions can be written as series expansions
that permit us to investigate the asymptotic behaviour of these quantities.
Moreover, we compare the shear and flexion of the Einasto profile with those of
different mass profiles including the singular isothermal sphere, the
Navarro-Frenk-White profile, and the S\'ersic profile. We investigate the
concentration and index dependences of the Einasto profile, finding that the
shear and second flexion could be used to determine the halo concentration,
whilst for the Einasto index the shear and first and second flexions may be
employed. We also provide simplified expressions for the weak lensing
properties and other lensing quantities in terms of the generalized
hypergeometric function.
