Incomplete beta function

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



\mathrm{\Beta}(x,y) = \int_0^1t^{x-1}(1-t)^{y-1}\,dt \!$$

for Re(x), Re(y) > 0.

The beta function was studied by Euler and Legendre and was given its name by Jacques Binet.

Properties
The beta function is symmetric, meaning that



\mathrm{\Beta}(x,y) = \mathrm{\Beta}(y,x). \!$$

It has many other forms, including:



\mathrm{\Beta}(x,y)=\frac{\Gamma(x)\,\Gamma(y)}{\Gamma(x+y)} \!$$



\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 \!$$



\mathrm{\Beta}(x,y) = \int_0^\infty\frac{t^{x-1}}{(1+t)^{x+y}}\,dt, \qquad \Re(x)>0,\ \Re(y)>0 \!$$



\mathrm{\Beta}(x,y) = \frac{1}{y}\sum_{n=0}^\infty(-1)^n\frac{(y)_{n+1}}{n!(x+n)} \!$$

where $$\Gamma(x)$$ is the gamma function and (x)n is the falling factorial; i.e., $$x(x - 1)(x - 2) ... (x - n + 1)$$. The second identity shows in particular $$\Gamma(1/2) = \sqrt \pi$$.

Like the gamma function for integers describes factorials, the beta function can define a binomial coefficient after adjusting indices:
 * $${n \choose k} = \frac1{(n+1) \mathrm{B}(n-k+1, k+1)}$$

The beta function was the first known scattering amplitude in string theory, first conjectured by Gabriele Veneziano.

Derivatives
The derivatives follow:


 * $${\partial \over \partial x} \mathrm{B}(x, y) = \mathrm{B}(x, y) \left( {\Gamma'(x) \over \Gamma(x)} - {\Gamma'(x + y) \over \Gamma(x + y)} \right) = \mathrm{B}(x, y) (\psi(x) - \psi(x + y))$$

where $$\psi(x)$$ is the digamma function.

Integrals
The Nörlund-Rice integral is a contour integral involving the beta function.

Incomplete beta function
The incomplete beta function is a generalization of the beta function that replaces the definite integral of the beta function with an indefinite integral. The situation is analogous to the incomplete gamma function being a generalization of the gamma function.

The incomplete beta function is defined as


 * $$ \mathrm{B}_x(a,b) = \int_0^x t^{a-1}\,(1-t)^{b-1}\,\mathrm{d}t \!$$

For x = 1, the incomplete beta function coincides with the complete beta function.

The regularized incomplete beta function (or regularized beta function for short) is defined in terms of the incomplete beta function and the complete beta function:


 * $$ I_x(a,b) = \frac{\mathrm{B}_x(a,b)}{\mathrm{B}(a,b)} \!$$

Properties

 * $$ I_0(a,b) = 0 \, $$
 * $$ I_1(a,b) = 1 \, $$
 * $$ I_x(a,b) = 1 - I_{1-x}(b,a) \, $$

(Many other properties could be listed here.)