Luminosity Schechter function for galaxies Just a question I am having trouble understanding. I have the Schechter luminosity function for galaxies, given as:

$$\Phi(L)dL=\Phi_{0}\left({\frac{L}{L_{\star}}}\right)^{\alpha}e^{-\frac{L}{L_{\star}}}\frac{dL}{L_{\star}}$$

I need to consider the case when $\alpha=-1$. And then show that the average luminosity of a galaxy is exactly $L_{\star}$. Could somebody explain how I could go about doing this and perhaps a hint or some part of a setup would be excellent. I really need to understand this. 

Another part, which is related to the above question, is asking me to explain why the total luminosity is a finite number, whereas the total number of galaxies diverges. By this does it mean that the total number is infinite? Any extra comments on this would also be really appreciated.
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 $(1+\alpha)L^*$