# Β函数

Β函数，又称为贝塔函数或第一类欧拉积分，是一个特殊函数，由下式定义：

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

## 性质

Β函数具有以下對稱性質：

$\Beta(x,y) = \Beta(y,x). \!$

$\Beta(x,y)=\dfrac{(x-1)!\,(y-1)!}{(x+y-1)!} \!$

$\Beta(x,y)=\dfrac{\Gamma(x)\,\Gamma(y)}{\Gamma(x+y)} \!$
$\Beta(x,y) = 2\int_0^{\pi/2}(\sin\theta)^{2x-1}(\cos\theta)^{2y-1}\,d\theta, \qquad \textrm{Re}(x)>0,\ \textrm{Re}(y)>0 \!$
$\Beta(x,y) = \int_0^\infty\dfrac{t^{x-1}}{(1+t)^{x+y}}\,dt, \qquad \textrm{Re}(x)>0,\ \textrm{Re}(y)>0 \!$
$\Beta(x,y) = \sum_{n=0}^\infty \dfrac{{n-y \choose n}} {x+n}, \!$
$\Beta(x,y) = \prod_{n=0}^\infty \left( 1+ \dfrac{x y}{n (x+y+n)}\right)^{-1}, \!$
$\Beta(x,y) \cdot \Beta(x+y,1-y) = \dfrac{\pi}{x \sin(\pi y)}, \!$
$\Beta(x,y) = \dfrac{1}{y}\sum_{n=0}^\infty(-1)^n\dfrac{y^{n+1}}{n!(x+n)} \!$

${n \choose k} = \frac1{(n+1) \Beta(n-k+1, k+1)}$

## 伽玛函数与贝塔函数之间的关系

$\Gamma(x)\Gamma(y) = \int_0^\infty\ e^{-u} u^{x-1}\,du \int_0^\infty\ e^{-v} v^{y-1}\,dv. \!$

\begin{align} \Gamma(x)\Gamma(y) & {} = 4\int_0^\infty\ e^{-a^2} a^{2x-1}\mathrm{d}a \int_0^\infty\ e^{-b^2} b^{2y-1}\,db \\ & {} = \int_{-\infty}^\infty\ \int_{-\infty}^\infty\ e^{-(a^2+b^2)} |a|^{2x-1} |b|^{2y-1} \,da \,db. \end{align} \!

\begin{align} \Gamma(x)\Gamma(y) & {} = \int_0^{2\pi}\ \int_0^\infty\ e^{-r^2} |r\cos\theta|^{2x-1} |r\sin\theta|^{2y-1} r \, dr \,d\theta \\ & {} = \int_0^\infty\ e^{-r^2} r^{2x+2y-2} r\, dr \int_0^{2\pi}\ |(\cos\theta)^{2x-1} (\sin\theta)^{2y-1}| \, d\theta \\ & {} = \frac{1}{2}\int_0^\infty\ e^{-r^2} r^{2(x+y-1)} \, d(r^2) 4\int_0^{\frac{\pi}{2}}\ (\cos\theta)^{2x-1} (\sin\theta)^{2y-1} \,d\theta \\ & {} = \Gamma(x+y) 2\int_0^{\frac{\pi}{2}}\ (\cos\theta)^{2x-1} (\sin\theta)^{2y-1} \, d\theta \\ & {} = \Gamma(x+y) \Beta(x,y). \end{align}

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

## 导数

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

## 估计

$\Beta(x,y) \sim \sqrt {2\pi } \frac{{x^{x - \frac{1}{2}} y^{y - \frac{1}{2}} }}{{\left( {x + y} \right)^{x + y - \frac{1}{2}} }}.$

## 不完全贝塔函数

$\Beta(x;\,a,b) = \int_0^x t^{a-1}\,(1-t)^{b-1}\,dt. \!$

x = 1，上式即化为贝塔函数。

$I_x(a,b) = \dfrac{\Beta(x;\,a,b)}{\Beta(a,b)}. \!$

ab是整数时，计算以上的积分（可以用分部积分法），可得：

$I_x(a,b) = \sum_{j=a}^{a+b-1} {(a+b-1)! \over j!(a+b-1-j)!} x^j (1-x)^{a+b-1-j}.$

$F(k;n,p) = \Pr(X \le k) = I_{1-p}(n-k, k+1) = 1 - I_p(k+1,n-k)$

### 性质

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