  For the super-hyperbolic space in any dimension, we introduce the
non-Euclidean Helgason--Fourier transform. We prove an inversion formula
exhibiting residue contributions at the poles of the Harish-Chandra c-function,
signalling discrete parts in the spectrum. The proof is based on a detailed
study of the spherical superfunctions, using recursion relations and
localization techniques to normalize them precisely, careful estimates of their
derivatives, and a rigorous analysis of the boundary terms appearing in the
polar coordinate expression of the invariant integral
