# Euler’s gamma function

**Definition**: *The gamma function $Γ$ is defined as*

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

$k=0∑∞ k_{x−1}e_{−k}$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

$n_{x−1}e_{−n}(n+1)_{x−1}e_{−(n+1)} =(nn+1 )_{x−1}⋅e_{−1}$Now $(nn+1 )_{x−1}$ 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 $x∈R$ is not a nonpositive integer, the integral*

*converges.*

Now suppose that $x=1$. We have

$Γ(1)=∫_{0}e_{−t}dt=[−e_{−t}]_{t=0}=1$Now suppose that $x=n$ for some $n∈N_{+}$. Then we have

$Γ(n)=∫_{0}t_{n−1}e_{−t}dt$Using integration by parts we have

$∫_{0}t_{n−1}e_{−t}dt=[−t_{x−1}e_{−t}]_{t=0}−∫_{0}−(x−1)t_{x−2}e_{−t}dt=(x−1)⋅Γ(x−1)$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⋅(n−1)! $This implies the following result:

**Theorem**:

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.