The Gamma function and factorials

The factorial of a positive integer is very well known for the general public. It is defined as

So



As it can be seen these numbers growth quite fast. By convention, we set 0!=1.

We would like to extend this concept to any positive real number, in a continuous manner. An answer is the Gamma function.

In the following lines, the reader should have basic knowledge of advanced calculus.

We define

It is an improper integral, so this means:



It can be easily check that both limits exist.

The Gamma function has the "factorial property", i.e. Γ(x+1)=xΓ(x). In fact, if we integrate by parts, we have:

After taking the limits of integration in the previous expression, we get



Which is exactly Γ(x)=(x-1)Γ(x-1).

If we set x=n a positive integer, we have:

And



So Γ(n)=(n-1)! if n is a positive integer.

This fact allow us to extend the notion of factorial for negative integer.

The computation of Γ(1/2) is quite nice:

After considering the substitution t=u², we have



This last integral is well known, and its computation is very beautiful. However we omit it in the present post. We say that the computation involves two variables calculus.



Therefore Γ(1/2)=√π. So we can compute the value of Γ(n+1/2), for any positive integer n.

The values of Γ(x), for x=1/6, 1/4, 1/3, 2/3, 3/4, 5/6 are transcendal numbers and algebraically independent of π.

I would like to mention that the gamma function is not the only differentiable function that extends the factorial. In fact there are infinitely many extensions of the factorial to the positive real numbers, for example consider Γ(x) + m sin (π x), for any integer m.

The identity Γ(x)=(x-1)Γ(x-1) can be used for extend the definition of the function to negative, non integer numbers. In fact, for example if -2<x<-1, so 0<x+2<1, then

Graph of the Gamma function -- Plot done in Mathematica

This function it can be extended to a complex variable function, i.e. the variable x is a complex number. The resulting function is meromorphic. But this topic is beyond the scope of the present post.

The Gamma function plays a very important role in different branches of mathematics, among them analysis, number theory, mathematical-physics. And moreover its history is very fascinating.

References:
https://en.wikipedia.org/wiki/Gamma_function
Courant R. and John F. "Introduction to Calculus and Analysis", vol 1, Springer, 1965.
Waldschmidt, M. "Transcendence of Periods: The State of the Art". Pure Appl. Math. Quart. 2 (2006): 435–463.