Gamma Function Explained for Upperclassmen Math Students

[rmiller70015]

I took differential equation over the last year and we talked about gamma functions in class but it wasn't in our books and I don't speak broken Russian so it was hard to understand what was going on. I'm wondering if the gamma function is similar to the dirac or heaviside functions and if anyone can point me to some textbooks or what have you that can explain it to someone who is only at an intermediate (upperclassmen undergrad) understanding of math.

 

[stevendaryl]

I'm not sure what you most want to know about it. The gamma function $\Gamma(x)$ is a generalization of the factorial function to the case where $x$ is not an integer. If $x$ is a positive integer, then $\Gamma(x) = 1 \cdot 2 \cdot ... \cdot (x-1) = (x-1)!$. The recursive equation for $n!$ is: $(n+1)! = (n+1) \cdot n!$. The gamma function has a similar defining equation: $\Gamma(x+1) = x \cdot \Gamma(x)$ (unless x=0).

For $x$ noninteger (but the real part of x is greater than zero), you can define $\Gamma(x)$ by:

$\Gamma(x) = \int_{t=0}^\infty e^{-t}t^{x-1} dt$

For the cases where the real part of $x$ is less than zero, you can define $\Gamma(x)$ by:

$\Gamma(x) = \frac{\pi}{sin(\pi x) \Gamma(1-x)}$

So that defines $\Gamma(x)$ everywhere, except that it blows up at $x=0, -1, -2, -3, ...$