Euler’s gamma function
Definition: The gamma function is defined as
First, do we even expect the integral in the definition of the gamma function to converge? The integrand is the product of two factors. The first factor, , goes to infinity as goes to infinity. The second factor, , goes to zero as goes to infinity.
By the integral test the integral converges if and only if the sum
If is a nonpositive integer, is a negative integer and the factor is undefined at .
Otherwise, the ratio between two terms is
Now goes to as goes to infinity, so the ratio is certainly below . By the ratio test the sum converges. So the integral converges as well.
Theorem: When is not a nonpositive integer, the integral
Now suppose that . We have
Now suppose that for some . Then we have
Using integration by parts we have
So we have , . This is analogous to the definition of
This implies the following result:
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 , 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.