Luminosity Schechter function for galaxies
Total Luminosity with $\alpha=-1$:
$$
\begin{aligned}
L_{tot}&amp;=\int_0^{\infty}L \Phi_0 \frac{L^*}{L}e^{-\frac{L}{L^*}}\frac{dL}{L^*}\\
&amp;=\int_0^{\infty}\Phi_0 e^{-\frac{L}{L^*}}dL
\end{aligned}
$$
Now put $L'=L/L^*$ and you get:
$$
L_{tot}=\Phi_0L^* \int_0^{\infty}e^{-L'}dL'=\Phi_0L^*
$$
Total number:
$$
\begin{aligned}
N_{tot}&amp;=\int_0^{\infty} \Phi_0 \frac{L^*}{L}e^{-\frac{L}{L^*}}\frac{dL}{L^*}\\
&amp;=\int_0^{\infty} \Phi_0 \frac{1}{L}e^{-\frac{L}{L^*}}dL
\end{aligned}
$$
which diverges at the lower limit.

Observe with $\alpha=-1$ we have no gamma functions.

Also IIRC with $\alpha \in (-1,0)$ when computing $L_{tot}/N_{tot}$ the gamma functions cancel leaving $(1+\alpha)L^*$