Need Help with Gamma Functions

The Gamma function is the generalization of factoral. The simplest representation is an integral:

Gamma(x)=int(0,oo)t^x-1e^-tdt, for x>0.

It can be extended by analytic continuation.

 

[tex]\Gamma\left(\frac{5}{4}\right) = \Gamma\left(1+\frac{1}{4}\right) = \frac{1}{4}\Gamma\left(\frac{1}{4}\right)[/tex]

[tex]\Gamma\left(\frac{1}{4}\right)[/tex] has no known basic expression, but is known to be transcendental.

 

I read that (1/q)! has no representation for integer q>2, except decimal form. But, if you are concerned with gamma(1/4), you can see it to 1,000,000 decimals at:

http://www.dd.chalmers.se/~frejohl/math/gamma14_1_000_000.txt

Working with Euler's reflection formula, I get

(1/4)!(3/4)! = [tex]\frac{3\pi\sqrt2}{16}=.833041[/tex]

 

mjk1: how about (0)

Well, when you have a form like[tex]\int_{0}^{\infty} \frac{e^-x}{x} dx[/tex], you have trouble.

If you check out the Euler's reflection formula, we have


[tex]\Gamma(Z)\Gamma(1-Z) = \frac{\pi}{sin\pi(x)} for Z=0.[/tex]