Schechter luminosity function

Remarks

In terms of the absolute magnitude $M$, with $L / L^* = 10^{0.4 (M^* - M)}$, the same law reads $$\phi(M) \, dM = 0.4 \ln 10 \; \phi^* \, 10^{0.4 (M^* - M)(\alpha + 1)} \exp\left( -10^{0.4 (M^* - M)} \right) dM.$$

Example: the total luminosity density is $$j = \int_0^\infty L \, \phi(L) \, dL = \phi^* L^* \, \Gamma(\alpha + 2),$$ by the substitution $x = L / L^*$ and the definition of the Gamma function. This is finite for $\alpha > -2$; for $\alpha = -1$ it gives $j = \phi^* L^*$.

Non-example: the number density $\int_0^\infty \phi(L) \, dL = \phi^* \int_0^\infty x^\alpha e^{-x} \, dx$ diverges at $x = 0$ when $\alpha \le -1$. So for such $\alpha$ the function is not a probability density of luminosities on $(0, \infty)$, and a lower cut-off is needed. Fitted values of $\alpha$ for galaxies are usually close to $-1$.