What is the Gamma Function?
The gamma function is a generalization of the factorial to complex numbers. The factorial is function defined for positive natural numbers as the product of all the positive numbers preceding it.
$$
n! = n (n-1) (n-2) ...
$$
The factorial is also defined by the recurrent relation: $n! = n (n-1)!$. The gamma function is based on this recurrent relation. The gamma function is a complex valued function that also has this same feature, i.e.
$$
\Gamma(z) = z \Gamma(z - 1)
$$
This was derived by Daneil Bernoulli to be the improper integeral:
$$
\Gamma(z) = \int_0^\infty \frac{x^{z-1}}{e^x} dx \quad\text{ for all } \Re(z) > 0
$$
The function has some interesting structure when evaluated over the complex plane and this is examined in the referenced video.
References