On the accuracy of the Perturbative Approach for Strong Lensing: Local
  Distortion for Pseudo-Elliptical Models
  The Perturbative Approach (PA) introduced by \citet{alard07} provides
analytic solutions for gravitational arcs by solving the lens equation
linearized around the Einstein ring solution. This is a powerful method for
lens inversion and simulations in that it can be used, in principle, for
generic lens models. In this paper we aim to quantify the domain of validity of
this method for three quantities derived from the linearized mapping: caustics,
critical curves, and the deformation cross section (i.e. the arc cross section
in the infinitesimal circular source approximation). We consider lens models
with elliptical potentials, in particular the Singular Isothermal Elliptic
Potential and Pseudo-Elliptical Navarro--Frenk--White models. We show that the
PA is exact for this first model. For the second, we obtain constraints on the
model parameter space (given by the potential ellipticity parameter
$\varepsilon$ and characteristic convergence $\kappa_s$) such that the PA is
accurate for the aforementioned quantities. In this process we obtain analytic
expressions for several lensing functions, which are valid for the PA in
general. The determination of this domain of validity could have significant
implications for the use of the PA, but it still needs to be probed with
extended sources.
