# Β函数

（重定向自B函数

Β函数，又称为贝塔函数第一类欧拉积分，是一个特殊函数，由下式定义：

${\displaystyle \mathrm {\mathrm {B} } (x,y)=\int _{0}^{1}t^{x-1}(1-t)^{y-1}\,dt\!}$

## 性质

Β函数具有以下對稱性質：

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

${\displaystyle \mathrm {B} (x,y)={\dfrac {(x-1)!\,(y-1)!}{(x+y-1)!}}\!}$

${\displaystyle \mathrm {B} (x,y)={\dfrac {\Gamma (x)\,\Gamma (y)}{\Gamma (x+y)}}\!}$
${\displaystyle \mathrm {B} (x,y)=2\int _{0}^{\frac {\pi }{2}}(\sin \theta )^{2x-1}(\cos \theta )^{2y-1}\,d\theta ,\qquad {\textrm {Re}}(x)>0,\ {\textrm {Re}}(y)>0\!}$
${\displaystyle \mathrm {B} (x,y)=\int _{0}^{\infty }{\dfrac {t^{x-1}}{(1+t)^{x+y}}}\,dt,\qquad {\textrm {Re}}(x)>0,\ {\textrm {Re}}(y)>0\!}$
${\displaystyle \mathrm {B} (x,y)=\sum _{n=0}^{\infty }{\dfrac {n-y \choose n}{x+n}},\!}$
${\displaystyle \mathrm {B} (x,y)=\prod _{n=0}^{\infty }\left(1+{\dfrac {xy}{n(x+y+n)}}\right)^{-1},\!}$
${\displaystyle \mathrm {B} (x,y)\cdot \mathrm {B} (x+y,1-y)={\dfrac {\pi }{x\sin(\pi y)}},\!}$
${\displaystyle \mathrm {B} (x,y)={\dfrac {1}{y}}\sum _{n=0}^{\infty }(-1)^{n}{\dfrac {y^{n+1}}{n!(x+n)}}\!}$

${\displaystyle {n \choose k}={\frac {1}{(n+1)\mathrm {B} (n-k+1,k+1)}}}$

## 伽玛函数与贝塔函数之间的关系

${\displaystyle \Gamma (x)\Gamma (y)=\int _{0}^{\infty }\ e^{-u}u^{x-1}\,du\int _{0}^{\infty }\ e^{-v}v^{y-1}\,dv.\!}$

{\displaystyle {\begin{aligned}\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{aligned}}\!}

{\displaystyle {\begin{aligned}\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)\mathrm {B} (x,y).\end{aligned}}}

${\displaystyle \mathrm {B} (x,y)={\frac {\Gamma (x)\,\Gamma (y)}{\Gamma (x+y)}}.}$

## 导数

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

## 估计

${\displaystyle \mathrm {B} (x,y)\approx {\sqrt {2\pi }}{\frac {x^{x-{\frac {1}{2}}}y^{y-{\frac {1}{2}}}}{\left({x+y}\right)^{x+y-{\frac {1}{2}}}}}.}$

## 不完全贝塔函数

${\displaystyle \mathrm {B} (x;\,a,b)=\int _{0}^{x}t^{a-1}\,(1-t)^{b-1}\,dt.\!}$

x = 1，上式即化为贝塔函数。

${\displaystyle I_{x}(a,b)={\dfrac {\mathrm {B} (x;\,a,b)}{\mathrm {B} (a,b)}}.\!}$

ab是整数时，计算以上的积分（可以用分部积分法），可得：

${\displaystyle I_{x}(a,b)=\sum _{j=a}^{a+b-1}{(a+b-1)! \over j!(a+b-1-j)!}x^{j}(1-x)^{a+b-1-j}.}$

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

### 性质

${\displaystyle I_{0}(a,b)=0\,}$
${\displaystyle I_{1}(a,b)=1\,}$
${\displaystyle I_{x}(a,b)=1-I_{1-x}(b,a)\,}$