Second order singular pertubative theory for gravitational lenses
  The extension of the singular perturbative approach to the second order is
presented in this paper. The general expansion to the second order is derived.
The second order expansion is considered as a small correction to the first
order expansion. Using this approach it is demonstrated that the second order
expansion is reducible to a first order expansion via a re-definition of the
first order pertubative fields. Even if in practice the second order correction
is small the reducibility of the second order expansion to the first order
expansion indicates a degeneracy problem. In general this degeneracy is hard to
break. A useful and simple second order approximation is the thin source
approximation which offers a direct estimation of the correction. The practical
application of the corrections derived in this paper are illustrated by using
an elliptical NFW lens model. The second order pertubative expansion provides a
noticeable improvement, even for the simplest case of thin source
approximation. To conclude it is clear that for accurate modelisation of
gravitational lenses using the perturbative method the second order
perturbative expansion should be considered. In particular an evaluation of the
degeneracy due to the second order term should be performed, for which the thin
source approximation is particularly useful.
