  Gravitational lenses on galaxy scales are plausibly modelled as having
ellipsoidal symmetry and a universal dark matter density profile, with a Sersic
profile to describe the distribution of baryonic matter. Predicting all lensing
effects requires knowledge of the total lens potential: in this work we give
analytic forms for that of the above hybrid model. Emphasising that complex
lens potentials can be constructed from simpler components in linear
combination, we provide a recipe for attaining elliptical symmetry in either
projected mass or lens potential. We also provide analytic formulae for the
lens potentials of Sersic profiles for integer and half-integer index. We then
present formulae describing the gravitational lensing effects due to
smoothly-truncated universal density profiles in cold dark matter model. For
our isolated haloes the density profile falls off as radius to the minus fifth
or seventh power beyond the tidal radius, functional forms that allow all
orders of lens potential derivatives to be calculated analytically, while
ensuring a non-divergent total mass. We show how the observables predicted by
this profile differ from that of the original infinite-mass NFW profile.
Expressions for the gravitational flexion are highlighted. We show how
decreasing the tidal radius allows stripped haloes to be modelled, providing a
framework for a fuller investigation of dark matter substructure in galaxies
and clusters. Finally we remark on the need for finite mass halo profiles when
doing cosmological ray-tracing simulations, and the need for readily-calculable
higher order derivatives of the lens potential when studying catastrophes in
strong lenses.
