Paul Schechter

Paul Schechter introduced an analytic description of the galaxy luminosity function in 1976. The Schechter function combines a power law at low luminosity with an exponential high-luminosity cutoff:Here $L_{\ast }$ is the characteristic turnover luminosity, ${\varphi }_{\ast }$ a number-per-volume normalization, and $\alpha$ the faint-end exponent.^[ref]

Separating the turnover from the faint-end slope makes populations comparable. Extrapolating the fit to arbitrarily faint galaxies needs care: total number converges only for $\alpha >-1$, while total luminosity converges for $\alpha >-2$.