Looking at Gamma and Beta Functions

Gamma functions display a wide usage in many fields, from cosmology to astrophysics, and even have a role in intensive investigations - say to determine whether an image is photo-shopped or otherwise artificial- as opposed to being a faithful representation. For example, in the Appendix of my recent book on the JFK assassination, I employed equations of the form:


D_t(x^b) = G(b + 1) / G[b – t + 1]^{x(b – t)}

and

D_t(y^b) = G(b + 1) / G[b – t + 1]^{y(b – t)}

To assess the degree of drift in x and y-dimensions for silver iodide -based emulsions, e.g.  in the Oswald 'LIFE' photos (where he's shown holding a rifle) and the G-functions are Euler Gamma functions. 

We begin with the claim that there is the Gamma function (G) for which:

G (a) = (a - 1)!

where 'a' is a positive integer.   (Note: for a detailed derivation of the Gamma function, see: http://www.frm.utn.edu.ar/analisisdsys/material/funcion_gamma.pdf )

Thus, for a = 3:

G (3) = (3 - 1)! = 2! = 2·1 = 2

 
One can also make use of a recursion formula:

G(a + 1) = a G(a)

 
For example: G (4) = G (3 + 1) = 3 G (3) = 3 (2) = 6

Check this from the earlier formula: G (a) = (a - 1)!

G (4) = (4 - 1)! = 3! = 3·2·1 = 6

 
Now, there is also the Beta function, call it b(u,v)  which can be expressed in terms of the Gamma functions

G(u),  G (v).

Thus:

b(u,v) = G (u) G (v)/ G (u + v)

 
Some alert readers may recall this was the form used by cosmologist Brian Greene, when he sought to explicate string theory on the PBS special, The Elegant Universe.  The specific form Greene used in that chalk board segment was:

b(p, q) = G (p) G (q)/ G (p + q)

and arriving at the unique string theory form:

b([1- a(s)][1 - a(t)] =

G (1 - a(s)) G (1 - a(t))/ G (2 - a(s) - a(t))

 
Now, to fix ideas, consider the Beta function b(3, 4):

b(3, 4) = G (3) G (4)/ G (3 + 4) = (2) (6)/ G (7) = 12 / G (7)

 
where: G (7) = (7 - 1)! = 6! = 6·5·4·3·2·1 = 720

so b(3,4) = 12/ 720 = 1/60

 Problems for Math Mavens:

1. Find:  G (6)

2. Find:  b(3,6)

3. Using the link given earlier, for the detailed Gamma fn. derivation, show how:

G (½) = Öp