The Beta Function

Definition of the Beta Function B(m, n)

$B(m,n)=\int_0^1 t^{m-1}(1-t)^{n-1} dt$      m > 0, n > 0

Relationship of Beta Function to Gamma Function

$ B(m,n)=\frac{\Gamma(m)\Gamma(n)}{\Gamma(m+n)}$
Extensions of Β(m, n) to m < 0, n < 0 is provided using Γ(n) = Γ(n + 1)/n.

Some Important Results

Β(m ,n) = Β(n, m)

$B(m,n)=2\int_0^{\frac{\pi}{2}}\sin^{2m-1}\theta\cos^{2n-1}\theta d\theta$

$B(m,n)=\int_0^\infty \frac{t^{m-1}}{(1+t)^{m+n}} dt$

$B(m,n)=r^n(r+1)^m \int_0^1 \frac{t^{m-1}(1-t)^{n-1}}{(r+t)^{m+n}} dt$