Euler's gamma function

Definition: The gamma function Γ\Gamma is defined as
Γ(x)=∫0∞tx−1e−t dt \Gamma(x) = \int_0^\infty t^{x - 1} e^{-t}\ \text{d}t

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

By the integral test the integral converges if and only if the sum
∑k=0∞kx−1e−k \sum_{k = 0}^\infty k^{x - 1} e^{-k}

converges.

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

Otherwise, the ratio between two terms is
(n+1)x−1e−(n+1)nx−1e−n=(n+1n)x−1⋅e−1 \frac{(n + 1)^{x - 1} e^{-(n + 1)}}{n^{x - 1} e^{-n}} = \left(\frac{n + 1}{n} \right)^{x - 1} \cdot e^{-1}

Now (n+1n)x−1(\frac{n + 1}{n})^{x - 1} goes to 11 as nn goes to infinity, so the ratio is certainly below 11. By the ratio test the sum converges. So the integral converges as well.

Theorem: When x∈Rx \in \mathbb{R} is not a nonpositive integer, the integral
∫0∞tx−1e−t dt \int_0^\infty t^{x - 1} e^{-t}\ \text{d}t

converges.

Now suppose that x=1x = 1. We have
Γ(1)=∫0∞e−t dt=[−e−t]t=0∞=1 \Gamma(1) = \int_0^\infty e^{-t}\ \text{d}t = [-e^{-t}]_{t = 0}^\infty = 1

Now suppose that x=nx = n for some n∈N+n \in \mathbb{N}_+. Then we have
Γ(n)=∫0∞tn−1e−t dt \Gamma(n) = \int_0^\infty t^{n - 1} e^{-t}\ \text{d}t

Using integration by parts we have
∫0∞tn−1e−t dt=[−tx−1e−t]t=0∞−∫0∞−(x−1)tx−2e−t dt=(x−1)⋅Γ(x−1) \int_0^\infty t^{n - 1} e^{-t}\ \text{d}t = [-t^{x - 1} e^{-t}]_{t = 0}^\infty - \int_0^\infty -(x - 1) t^{x - 2}e^{-t} \ \text{d}t = (x - 1) \cdot \Gamma(x - 1)

So we have Γ(1)=1\Gamma(1) = 1, Γ(n)=(n−1)Γ(n−1)\Gamma(n) = (n - 1) \Gamma(n - 1). This is analogous to the definition of n!n!
0!=1n!=n⋅(n−1)! \begin{aligned} 0! &= 1\\n! &= n \cdot (n - 1)! \end{aligned}

This implies the following result:

Theorem:
Γ(n)=(n−1)! \Gamma(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 zz, 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.