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.


[1] https://www.youtube.com/watch?v=dGnIJFzkLI4


  • Mathematics