Hi all,

I am reviewing a section of my device physics textbook which covers the effective density of states. In the books' derivation, they reach a point where they say this:

The integral is the gamma function, with a value of

$\displaystyle \int_0^\infty \eta ^\frac{1}{2} exp(-\eta ) d\eta = \frac {1}{2} \sqrt{\pi}$

...and then jump back into the full derivation of the effective density of states (using other factors not shown here).

My Question:

Can someone please show me (in a detailed way) how they arrived at $\displaystyle \frac {1}{2} \sqrt{\pi}$? Also, in laymen's terms... what is a gamma function, and how is it advantageous? I tried looking this up, and could not make sense of what was being explained.

Thank you!