Extensive air showers are one of the most important phenomena to study in the context of ultrahigh-energy cosmic rays. Their longitudinal development can be directly observed using fluorescence detector telescopes, such as those employed at the Pierre Auger Observatory or the Telescope Array. In this work, we discuss the properties of the Greisen function, originally introduced to describe the longitudinal shower profiles of electromagnetic air showers, and demonstrate that, after appropriate modification, it can be used to describe longitudinal air-shower profiles, even for hadronic air showers. Furthermore we discuss the possibility to discriminate between hadrons and photons from the shape of air-shower profiles using the Greisen function.