Videos
The gamma function is defined by the following integral, which converges for real 
:
The function can also be extended into the complex plane, if you're familiar with that subject. I'll assume not and just let 
be real.
This function is like the factorial in the when 
is a positive integer, say 
, it satisfies 
. It generalizes the factorial in the sense that it is the factorial for positive integer arguments, and is also well-defined for positive rational (and even real) numbers. This is what it means to take a "rational factorial," but I would hesitate to call it that. Many functions have those two properties, and 
is chosen out of all of them because it is the most useful in other applications. Rather than the notation used in that article you refer to, it would be more accurate for you to say that "the gamma function takes these values for these arguments." Gamma is not a function that intends to generalize factorials; rather, generalizing factorials came along as something of an accident following the definition. Its true purpose is deeper.
As for why 
, this comes out of an interesting property of the 
function: some of them are here http://en.wikipedia.org/wiki/Gamma_function#Properties. The property you are interested in is the reflection formula: 
Set 
in the formula to get the desired identity.
If you want to learn more about the gamma function, the hard way is to learn a lot more math, in particular real and complex analysis. An easier way is to read this excellent set of notes: http://www.sosmath.com/calculus/improper/gamma/gamma.html.
The gamma function, shown with a Greek capital gamma 
, is a function that extends the factorial function to all real numbers, except to the negative integers and zero, for which it is not defined. 
is related to the factorial in that it is equal to 
. The function is defined as
Simply use this to compute factorials for any number. A handy way of calculating for real fractions with even denominators is:
Where n is an integer. But keep in mind that the gamma function is actually the factorial of 1 less than the number than it evaluates, so if you want 
use n = 2 instead of 1.
Or, you could just put the fraction into Google Calculator, which uses the gamma function to evaluate factorials of any number.
For some more examples of the gamma function's values, see here.
(If you don't understand this, don't worry, because I don't either, and the Wikipedia article on the function seems to lack a clear-cut definition of it or how it relates to 
.)
there (also 
and 
) is the Gamma function. Unfortunately the gamma function doesn't extend to negative integers, and that is why I guess people don't really care that much for defining them as they know that no "good" answer can be found. 
, then, near z = -a, we have · 
· and so it might be tempting to say that, in some sense, · 
· where the symbol 
represents the rate at which 
blows up near the pole at 
. That is, 
, and so on. · In particular, this interpretation might work in some formula in which 
evaluated at nonpositive integers appears in both the numerator and the denominator, and the symbol 
can be canceled to yield a real number.