Beta function

The Beta function, also called Euler integral of the first kind, is a special function defined by

<math>B(x,y) = \int_0^1t^{x-1}(1-t)^{y-1}\,dt</math>

The Beta function is symmetric, meaning <math>B(x,y) = B(y,x)</math>.

It has many other forms, including:

<math>\begin{matrix}B(x,y)&=&\frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)} \\

 &=& 2\int_0^\frac{\pi}{2}\sin^{2x-1}\theta\cos^{2y-1}\theta\,d\theta, & {\mathcal Re}(x)>0, {\mathcal Re}(y)>0\\

&=&\int_0^\infty\frac{t^{x-1}}{(1+t)^{x+y}}\,dt, & {\mathcal Re}(x)>0, {\mathcal Re}(y)>0 \\ &=&\frac{1}{y}\sum_{n=0}^\infty(-1)^n\frac{(x)_{n+1}}{n!(x+n)}, \end{matrix}</math>

where <math>(x)_{n}</math> is the falling factorial.

See also: Euler integral, falling factorial, Gamma function