Question regarding gamma function

[Denver Dang]

Homework Statement

I have a quick question about the gamma function.

According to my textbook it says:
$$\int_{0}^{\infty }{{{x}^{n}}{{e}^{-ax}}dx}=\frac{\Gamma \left( n+1 \right)}{{{a}^{n+1}}}$$
where $\Gamma$ is the gamma function. My question is, do n has to be an integer number, or can it also be a half-integer number for the gamma function to work ?

Thanks in advance.

 

[Ray Vickson]

Homework Statement

I have a quick question about the gamma function.

According to my textbook it says:
$$\int_{0}^{\infty }{{{x}^{n}}{{e}^{-ax}}dx}=\frac{\Gamma \left( n+1 \right)}{{{a}^{n+1}}}$$
where $\Gamma$ is the gamma function. My question is, do n has to be an integer number, or can it also be a half-integer number for the gamma function to work ?

Thanks in advance.

n is any real number > -1.

 

[Denver Dang]

Ok, thank you very much :)