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.
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.
This was derived by Daneil Bernoulli to be the improper integeral:
The function has some interesting structure when evaluated over the complex plane and this is examined in the referenced video.