Beta function
In mathematics, the Beta function, also called the Euler integral of the first kind, is a special function defined by
- <math>\mathrm B(x,y) = \int_0^1t^{x-1}(1-t)^{y-1}\,dt<math>
The Beta function is symmetric, meaning that
- <math>\mathrm B(x,y) = \mathrm B(y,x).<math>
It has many other forms, including:
- <math>
\mathrm B(x,y)=\frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}
<math>
- <math>
\mathrm B(x,y)=2\int_0^{\pi/2}\sin^{2x-1}\theta\cos^{2y-1}\theta\,d\theta,
\qquad{\mathrm Re}(x)>0,\ {\mathrm Re}(y)>0
<math>
- <math>
\mathrm B(x,y)=\int_0^\infty\frac{t^{x-1}}{(1+t)^{x+y}}\,dt,
\qquad{\mathrm Re}(x)>0,\ {\mathrm Re}(y)>0
<math>
- <math>
\mathrm B(x,y)=\frac{1}{y}\sum_{n=0}^\infty(-1)^n\frac{(x)_{n+1}}{n!(x+n)}
<math>
where (x)n is the falling factorial.
See also: Euler integral, falling factorial, gamma function
