Euler’s gamma function

Definition: The gamma function $Γ$ is defined as

Γ (x) = \int_{0}^{\infty} t^{x - 1} e^{- t} d t

First, do we even expect the integral in the definition of the gamma function to converge? The integrand $t^{x - 1} e^{- t}$ is the product of two factors. The first factor, $t^{x - 1}$ , goes to infinity as $t$ goes to infinity. The second factor, $e^{- t}$ , goes to zero as $t$ goes to infinity.

By the integral test the integral converges if and only if the sum

k = 0 \sum \infty k^{x - 1} e^{- k}

converges.

If $x$ is a nonpositive integer, $x - 1$ is a negative integer and the factor $k^{x - 1}$ is undefined at $k = 0$ .

Otherwise, the ratio between two terms is

\frac{( n + 1 ) ^{x - 1} e ^{- (n + 1)}}{n ^{x - 1} e ^{- n}} = (\frac{n + 1}{n})^{x - 1} \cdot e^{- 1}

Now $(\frac{n + 1}{n})^{x - 1}$ goes to $1$ as $n$ goes to infinity, so the ratio is certainly below $1$ . By the ratio test the sum converges. So the integral converges as well.

Theorem: When $x \in R$ is not a nonpositive integer, the integral

\int_{0}^{\infty} t^{x - 1} e^{- t} d t

converges.

Now suppose that $x = 1$ . We have

Γ (1) = \int_{0}^{\infty} e^{- t} d t = [- e^{- t}]_{t = 0}^{\infty} = 1

Now suppose that $x = n$ for some $n \in N_{+}$ . Then we have

Γ (n) = \int_{0}^{\infty} t^{n - 1} e^{- t} d t

Using integration by parts we have

\int_{0}^{\infty} t^{n - 1} e^{- t} d t = [- t^{x - 1} e^{- t}]_{t = 0}^{\infty} - \int_{0}^{\infty} - (x - 1) t^{x - 2} e^{- t} d t = (x - 1) \cdot Γ (x - 1)

So we have $Γ (1) = 1$ , $Γ (n) = (n - 1) Γ (n - 1)$ . This is analogous to the definition of $n!$

0! n! = 1 = n \cdot (n - 1)!

This implies the following result:

Theorem:

Γ (n) = (n - 1)!

In fact, the integral in the definition of the gamma function is well defined for any complex number that is not a nonpositive integer (which is why the wikipedia page writes the gamma function as a function of a parameter $z$ , which is usually used for complex variables).

Why do we care about the gamma function? For one, it is just a beautiful mathematical object that is worthy of study on it’s own. It is also related to the Riemann zeta function and the Beta function.

Other than that, I have to be honest and tell you I don’t really know what the significance of the gamma function is. It is commonly accepted to be one of the “special” functions which are of great importance, but I am struggling to give an application of the gamma function itself. It’s application seems to be mostly its relatedness to other “special” functions. If I were more familiar with the field of special functions I could probably give better examples here.