## Encyclopedia > Beta function

Article Content

# Beta function

The Beta function, also called Euler integral of the first kind, is a special function defined by
$B(x,y) = \int_0^1t^{x-1}(1-t)^{y-1}\,dt$

The Beta function is symmetric, meaning $B(x,y) = B(y,x)$.

It has many other forms, including:

$\begin{matrix}B(x,y)&=&\frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)} \\  &=& 2\int_0^\frac{\pi}{2}\sin^{2x-1}\theta\cos^{2y-1}\theta\,d\theta, & {\mathcal Re}(x)>0, {\mathcal Re}(y)>0\\  &=&\int_0^\infty\frac{t^{x-1}}{(1+t)^{x+y}}\,dt, & {\mathcal Re}(x)>0, {\mathcal Re}(y)>0 \\ &=&\frac{1}{y}\sum_{n=0}^\infty(-1)^n\frac{(x)_{n+1}}{n!(x+n)}, \end{matrix}$

where $(x)_{n}$ is the falling factorial.

All Wikipedia text is available under the terms of the GNU Free Documentation License

Search Encyclopedia
 Search over one million articles, find something about almost anything!

Featured Article
 Springs, New York ... (0.8 mi²) of it is water. The total area is 8.24% water. Demographics As of the census of 2000, there are 4,950 people, 1,924 households, and 1,252 famil ...