The Beta Function

The beta function is defined by B(x,y) = I0,1(t^(x-1)(1-t)^(y-1) dt, where the real parts of x and y are greater than 0.
This function can also be expressed in terms of the gamma function, through the formula B(x,y) = G(x)G(y)/G(x+y).

Other important relationships include:
B(x,y) = B(y,x)
B(x,y) = 2I0,pi/2((sin(t))^(2x-1)(cos(t))^(2y-1) dt
B(x,y) = I0,infinity(t^(x-1)/(1+t)^(x+y)
B(x,y)B(x+y,1-y) = pi/(x sin(pi y)) 

Also, the binomial coefficient can be defined in terms of the beta function through the formula (n,k) = 1/((n+1)B(n-k+1,k+1)), where (n,k) is the binomial coefficient of n and k.

The Gamma Function

The gamma function of z.  In my discussion I will use G(z) to represent it.

The absolute value of the gamma function on the complex plane.

The gamma function, G(x) can be seen as a line connecting points given by the equation y = (x-1)! for integer values of x, where ! is the factorial function.  Thus, for integers, G(x) = (x-1)!.  Other important properties of the gamma function include:
G(1-x)G(x) =  pi/sin(pi x)
G(x)G(x+.5) = 2^(1-2x) pi^.5 G(2x)
G(x)G(x+1/m)G(x+2/m)..G(x+(m-1)/m) = (2pi)^((m-1)/2) m^(.5-mx) G(mx)

The derivative of the gamma function can be shown in terms of the polygamma function, P0(x), with the relationship G`(x) = G(x)P0(x).  This can be generalized into the form d^n/dx^n(G(x)) = I0,infinity(t^(x-1)e^(-t)(ln t)^n dt, where I0,infinity is the definite integral from 0 to infinity.