Analytical expressions for the deprojected Sérsic model

The Sérsic model has become the standard for parametrizing the surface brightness distribution of early-type galaxies and bulges of spiral galaxies. A major problem is that the deprojection of the Sérsic surface brightness profile to a luminosity density cannot be executed analytically for general values of the Sérsic index. We use Mellin integral transforms to derive an analytical expressions for the luminosity density in terms of the Fox H function for all values of the Sérsic index. We derive simplified expressions for the luminosity density, cumulative luminosity, and gravitational potential in terms of the Meijer G function for all rational values of the Sérsic index, and we investigate their asymptotic behaviour at small and large radii. As implementations of the Meijer G function are currently available both in symbolic computer algebra packages and as high-performance computing code, our results open up the possibility to calculate the density of the Sérsic models to arbitrary precision.