Beta function
From Freepedia
- A separate article treats the beta-function (written with a hyphen) of physics.
In mathematics, the beta function (occasionally written as Beta function), also called the Euler integral of the first kind, is a special function defined by
- <math>
\mathrm{\Beta}(x,y) = \int_0^1t^{x-1}(1-t)^{y-1}\,dt
\!</math>
for Re(x), Re(y) > 0. The beta function is symmetric, meaning that
- <math>
\mathrm{\Beta}(x,y) = \mathrm{\Beta}(y,x).
\!</math>
It has many other forms, including:
- <math>
\mathrm{\Beta}(x,y)=\frac{\Gamma(x)\,\Gamma(y)}{\Gamma(x+y)}
\!</math>
- <math>
\mathrm{\Beta}(x,y) =
2\int_0^{\pi/2}\sin^{2x-1}\theta\cos^{2y-1}\theta\,d\theta,
\qquad \Re(x)>0,\ \Re(y)>0
\!</math>
- <math>
\mathrm{\Beta}(x,y) =
\int_0^\infty\frac{t^{x-1}}{(1+t)^{x+y}}\,dt,
\qquad \Re(x)>0,\ \Re(y)>0
\!</math>
- <math>
\mathrm{\Beta}(x,y) =
\frac{1}{y}\sum_{n=0}^\infty(-1)^n\frac{(y)_{n+1}}{n!(x+n)}
\!</math>
where <math>\Gamma(x)</math> is the gamma function and (x)n is the falling factorial. Euler's beta function was the first known scattering amplitude in string theory, first conjectured by Gabriele Veneziano.
See also
- Beta_distribution
- binomial coefficient
- Euler integral
- falling factorial
- gamma function
- incomplete beta function
References
- Milton Abramowitz and Irene A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972. (See §6.2)
