ベータ関数

\[B\left(z;\alpha,\beta\right)=z^{\alpha}\sum_{k=0}^{\infty}\frac{C\left(k-\beta,k\right)}{\alpha+k}z^{k}\]

\[B\left(z;\alpha,\beta\right)=\frac{z^{\alpha}}{\alpha}F\left(\alpha,1-\beta;\alpha+1;z\right)\]

\[B\left(z;\alpha,1\right)=\frac{z^{\alpha}}{\alpha}\]

\[B\left(\alpha,1\right)=\frac{1}{\alpha}\]

\[B\left(z;\alpha+1,\beta\right)=\frac{1}{\alpha+\beta}\left(\alpha B\left(z;\alpha,\beta\right)-z^{\alpha}\left(1-z\right)^{\beta}\right)\]

\[B\left(z;\alpha,\beta\right)+B\left(1-z;\beta,\alpha\right)=B\left(\alpha,\beta\right)\]

\[B\left(\alpha,\beta\right)=B\left(\beta,\alpha\right)\]

\[B\left(\alpha,\beta\right)=\int_{0}^{1}t^{\alpha-1}\left(1-t\right)^{\beta-1}dt\]

ベータ関数$B\left(p,q\right)$は$\Re\left(p\right)>0\;\land\;\Re\left(q\right)>0$で絶対収束

\[B(x,y)=\frac{C(y-1,-x)\pi}{\sin(\pi x)}\]

\[\frac{\partial}{\partial x}B(x,y)=B(x,y)\left\{ \psi(x)-\psi(x+y)\right\} \]

\[B(x,y)=\sum_{k=0}^{\infty}\frac{C(k-y,k)}{x+k}\]

\[B(x,y)=\frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}\]

\[xB(x,y+1)=yB(x+1,y)\]

\[\int_{0}^{\frac{\pi}{2}}\sin^{x}t\cos^{y}tdt=\frac{1}{2}B\left(\frac{x+1}{2},\frac{y+1}{2}\right)\]