= Paul Schechter
{c}
<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:
$$
\phi(L)\,dL=\phi_*\left(\frac L{L_*}\right)^\alpha e^{-L/L_*}\frac{dL}{L_*}.
$$
Here $L_*$ is the characteristic turnover <luminosity>, $\phi_*$ a number-per-volume normalization, and $\alpha$ the faint-end exponent.<An Analytic Expression for the Luminosity Function for Galaxies by Paul Schechter>[Section I, equation (1), and section II(a)]{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 $\alpha>-2$.
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