Flexion is a non-linear gravitational lensing effect that arises from
gradients in the convergence and shear across an image. We derive a formalism
that describes non-linear gravitational lensing by a circularly symmetric lens
in the thin-lens approximation. This provides us with relatively simple
expressions for first- and second-flexion in terms of only the surface density
and projected mass distribution of the lens. We give details of exact lens
models, in particular providing flexion calculations for a Sersic-law profile,
which has become increasingly popular over recent years. We further provide a
single resource for the analytic forms of convergence, shear, first- and
second-flexion for the following mass distributions: a point mass, singular
isothermal sphere (SIS); Navarro-Frenk-White (NFW) profile; Sersic-law profile.
We quantitatively compare these mass distributions and show that the
convergence and first-flexion are better indicators of the Sersic shape
parameter, while for the concentration of NFW profiles the shear and
second-flexion terms are preferred.Comment: Accepted for publication in MNRA
