I am currently trying to implement the gamma function in a program I am writing. I know that the gamma function is: $$\Gamma(\alpha) = \int\limits_{0}^{\infty}x^{\alpha - 1}e^{-x}\text{ d}x$$
However, I do not know what the $\int$, the ${x}$, and the ${d}$ are supposed to stand for.
I know that you can do it quite easily with positive integers like this:
$$\Gamma(\alpha) = (a -1)!$$ How would I go about implementing that into a program for numbers other than positive integers?
My suggestion would be to take a look at SymPy's implementation of the Gamma Function:
The full implementation considers numerical integral arguments, half-integral values and it is publicly available at this GitHub Page.