\(\Gamma\)与\(B\)函数

· \(\Gamma\)函数：\[\Gamma(s) = \int_0 ^{+\infty}x^{s-1}e^{-x}dx.\]定义域：\((0,+\infty).\)
性质.
(1) \(\Gamma(s+1) = s\cdot \Gamma(s).\) 特别地，对于正整数k有\(\Gamma(k+1) = k!.\)
(2) \[\Gamma(s) = 2 \int_0^{+\infty} x^{2s-1}e^{-x^2}dx.\]
(3) \(\Gamma(s) \in C^\infty (0,+\infty).\)
(4) \(\Gamma(s),\ln \Gamma(x)\)都是严格凸函数。

·\(B\)函数：\[B(p,q) = \int_0^1 x^{p-1}(1-x)^{q-1}dx.\]
定义域：\((p,q)\in R^2_+.\)
性质.
(1) \(B(p,q)=B(q,p).\)
(2) \[B(p,q) = \frac{p-1}{p+q-1}B(p-1,q).\]
(3) \[B(p,q) = 2\int_0^{\frac{\pi}{2}}\cos^{2p-1}\theta\sin^{2q-1}\theta d\theta.\]
(4) \[B(p,q) = \int_0^{+\infty}\frac{x^{q-1}}{(1+x)^{p+q}}dx.\]

·\(\Gamma\)函数与\(B\)函数的关系
(1)\[B(p,q) = \frac{\Gamma(p)\Gamma(q)}{\Gamma(p+q)}.\]推论：\(B(p,q)\)在定义域内存在任意次偏导数。

(2) 余元公式：\[B(p,1-p) = \Gamma(p)\Gamma(1-p) = \frac{\pi}{\sin\ p\pi}.\]

    所属分类：数学分析     发表于2022-01-28