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.