I'm trying to get the total luminosity of a total population of galaxies. I have the Schechter Function like

$$\Phi(L) = \frac{\Phi^{*}}{L^{*}} \left(\frac{L}{L^{*}}\right)^{\alpha} \exp \left(\frac{L}{L^{*}}\right)$$

and I need to the total luminosity:

$$L_{tot} = \int_0^\infty L\Phi(L)dL = \Phi_{0}L^{*}\Gamma(2+\alpha)$$

but when I substitute the Schecter functions I don't get the $\Gamma$ function form. Could you have any tips for that?

And I need to show that the galaxies that more contributed to the luminosity are those have $L^{*}$

Thank you in advance.


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