Gravitational lensing by a generalised-NFW halo via the Fox H-function and its application to the super-NFW

For a particular generalised-NFW (gNFW) mass distribution, we derive analytical expressions for its surface mass density, projected mass, and deflection potential, given in terms of three representations of the Fox H-function, for which we provide their corresponding power (logarithmic) series expansions. They are handy in computationally intensive tasks. From these results, we obtain closed-form expressions for the super-NFW (sNFW) in terms of complete elliptic functions, which have not yet been reported in the literature. Additionally, we find that, for a fixed κ₀ (characteristic convergence), when the number of images is maximal (three for the sNFW), the sum of their signed magnification, named IsNFW, varies with the position of the source y inside the radial caustic yc2. The higher variations occur for κ₀ 1, while IsNFW can be considered as constant when κ₀ is approximately within the range 2-10. This range can be extended depending on the observational uncertainties, since for higher values of κ₀, the variations in IsNFW are relatively small. This behaviour is shared with the NFW and Hernquist lenses.