カテゴリー: ベータ関数

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

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

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

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

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

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

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