Use of the Fourier transform to derive the gravitational lens deflection angle

Knowing that the gravitational lens deflection angle can be expressed as the convolution product between the dimensionless surface mass density κ(x) and a simple function of the scaled impact parameter vector x, we make use of the Fourier transform to derive its analytical expression for the case of mass distributions presenting a homoeoidal sym- metry. For this family of models, we obtain the expression of the two components of the deflection angle in the form of integrals performed over the radial coordinate ρ. In the limiting case of axially symmetric lenses, we obviously retrieve the well-known relation α(x)∝ M(≤ |x|)x/|x|^2. Furthermore, we derive explicit solutions for the deflection angle characterized by dimensionless surface mass density profiles such as κ ∝ (ρ^2c + ρ^2)^{−ν}; corresponding to the non-singular isothermal ellipsoid (NSIE) model for the particular case ν = 1/2. Let us insist that all these results are obtained without using the complex formal- ism introduced by Bourassa and Kantowski (1973,1975). Further straightforward applica- tions of this Fourier approach are suggested in the conclusions of the present work.