刘维尔公式

刘维尔公式（Liouville's Formula）是一个关于多重积分和欧拉积分（ Γ {\displaystyle \Gamma } 函数）的公式，其形式如下： ∫ . . . ∬ x 1 , x 2 , . . . , x n ⩾ 0 ; x 1 + x 2 + . . . + x n ⩽ 1 f ( x 1 + x 2 + . . . + x n ) x 1 p 1 − 1 x 2 p 2 − 1 . . . x n p n − 1 d x 1 d x 2 . . . d x n {\displaystyle \int ...\iint _{x_{1},x_{2},...,x_{n}\geqslant 0;x_{1}+x_{2}+...+x_{n}\leqslant 1}f\left(x_{1}+x_{2}+...+x_{n}\right)x_{1}^{p_{1}-1}x_{2}^{p_{2}-1}...x_{n}^{p_{n}-1}\mathrm {d} x_{1}\mathrm {d} x_{2}...\mathrm {d} x_{n}} = Γ ( p 1 ) Γ ( p 2 ) . . . Γ ( p n ) Γ ( p 1 + p 2 + . . . + p n ) ∫ 0 1 f ( u ) u p 1 + p 2 + . . . + p n − 1 d u {\displaystyle ={\frac {\Gamma \left(p_{1}\right)\Gamma \left(p_{2}\right)...\Gamma \left(p_{n}\right)}{\Gamma \left(p_{1}+p_{2}+...+p_{n}\right)}}\int _{0}^{1}f\left(u\right)u^{p_{1}+p_{2}+...+p_{n}-1}\mathrm {d} u} 其中 p 1 , p 2 , . . . , p n > 0 {\displaystyle p_{1},p_{2},...,p_{n}>0} ， f ( u ) {\displaystyle f\left(u\right)} 为连续函数。 用数学归纳法。 当n=1时，公式显然成立。 当n=2时，公式也成立，即 ∬ x 1 , x 2 ⩾ 0 ; x 1 + x 2 ⩽ 1 f ( x 1 + x 2 ) x 1 p 1 − 1 x 2 p 2 − 1 d x 1 d x 2 = Γ ( p 1 ) Γ ( p 2 ) Γ ( p 1 + p 2 ) ∫ 0 1 f ( u ) u p 1 + p 2 − 1 d u {\displaystyle \iint _{x_{1},x_{2}\geqslant 0;x_{1}+x_{2}\leqslant 1}f\left(x_{1}+x_{2}\right)x_{1}^{p_{1}-1}x_{2}^{p_{2}-1}\mathrm {d} x_{1}\mathrm {d} x_{2}={\frac {\Gamma \left(p_{1}\right)\Gamma \left(p_{2}\right)}{\Gamma \left(p_{1}+p_{2}\right)}}\int _{0}^{1}f\left(u\right)u^{p_{1}+p_{2}-1}\mathrm {d} u} 事实上，令 Ω {\displaystyle \Omega } 表示区域： x 1 ⩾ 0 , x 2 ⩾ 0 , x 1 + x 2 ⩽ 1 {\displaystyle x_{1}\geqslant 0,x_{2}\geqslant 0,x_{1}+x_{2}\leqslant 1} ，作代换 x 1 = ξ 1 , x 1 + x 2 = ξ 2 {\displaystyle x_{1}=\xi _{1},x_{1}+x_{2}=\xi _{2}} ，以及 t = ξ 1 ξ 2 {\displaystyle t={\frac {\xi _{1}}{\xi _{2}}}} ，则有 ∬ x 1 , x 2 ⩾ 0 ; x 1 + x 2 ⩽ 1 f ( x 1 + x 2 ) x 1 p 1 − 1 x 2 p 2 − 1 d x 1 d x 2 = ∫ 0 1 f ( ξ 2 ) d ξ 2 ∫ 0 ξ 2 ξ 1 p 1 − 1 ( ξ 2 − ξ 1 ) p 2 − 1 d ξ 1 {\displaystyle \iint _{x_{1},x_{2}\geqslant 0;x_{1}+x_{2}\leqslant 1}f\left(x_{1}+x_{2}\right)x_{1}^{p_{1}-1}x_{2}^{p_{2}-1}\mathrm {d} x_{1}\mathrm {d} x_{2}=\int _{0}^{1}f\left(\xi _{2}\right)\mathrm {d} \xi _{2}\int _{0}^{\xi _{2}}\xi _{1}^{p_{1}-1}\left(\xi _{2}-\xi {1}\right)^{p_{2}-1}\mathrm {d} \xi _{1}} ∫ 0 1 f ( ξ 2 ) d ξ 2 ∫ 0 1 t p 1 − 1 ( 1 − t ) p 2 − 1 ξ 2 p 1 + p 2 − 1 d t = Γ ( p 1 ) Γ ( p 2 ) Γ ( p 1 + p 2 ) ∫ 0 1 f ( ξ 2 ) ξ 2 p 1 + p 2 − 1 d ξ 2 = Γ ( p 1 ) Γ ( p 2 ) Γ ( p 1 + p 2 ) ∫ 0 1 f ( u ) u p 1 + p 2 − 1 d u {\displaystyle \int _{0}^{1}f\left(\xi _{2}\right)\mathrm {d} \xi _{2}\int _{0}^{1}t^{p_{1}-1}\left(1-t\right)^{p_{2}-1}\xi _{2}^{p_{1}+p_{2}-1}\mathrm {d} t={\frac {\Gamma \left(p_{1}\right)\Gamma \left(p_{2}\right)}{\Gamma \left(p_{1}+p_{2}\right)}}\int _{0}^{1}f\left(\xi _{2}\right)\xi _{2}^{p_{1}+p_{2}-1}\mathrm {d} \xi _{2}={\frac {\Gamma \left(p_{1}\right)\Gamma \left(p_{2}\right)}{\Gamma \left(p_{1}+p_{2}\right)}}\int _{0}^{1}f\left(u\right)u^{p_{1}+p_{2}-1}\mathrm {d} u} 设公式对于n-1成立，今证对于n公式也成立。为此，将公式左端写为 ∫ . . . ∬ x 1 , x 2 , . . . , x n − 1 ⩾ 0 ; x 1 + x 2 + . . . + x n − 1 ⩽ 1 x 1 p 1 − 1 x 2 p 2 − 1 . . . x n − 1 p n − 1 − 1 d x 1 d x 2 . . . d x n − 1 ∫ 0 1 − ( x 1 + x 2 + . . . + x n − 1 ) f ( x 1 + x 2 + . . . + x n ) x n p n − 1 d x n {\displaystyle \int ...\iint _{x_{1},x_{2},...,x_{n-1}\geqslant 0;x_{1}+x_{2}+...+x_{n-1}\leqslant 1}x_{1}^{p_{1}-1}x_{2}^{p_{2}-1}...x_{n-1}^{p_{n-1}-1}\mathrm {d} x_{1}\mathrm {d} x_{2}...\mathrm {d} x_{n-1}\int _{0}^{1-\left(x_{1}+x_{2}+...+x_{n-1}\right)}f\left(x_{1}+x_{2}+...+x_{n}\right)x_{n}^{p_{n}-1}\mathrm {d} x_{n}} 令 ψ ( s ) = ∫ 0 1 − s f ( s + x n ) x n p n − 1 d x n {\displaystyle \psi \left(s\right)=\int _{0}^{1-s}f\left(s+x_{n}\right)x_{n}^{p_{n}-1}\mathrm {d} x_{n}} 代入上式，并利用公式对n-1成立的假定，得知上式为 Γ ( p 1 ) Γ ( p 2 ) . . . Γ ( p n − 1 ) Γ ( p 1 + p 2 + . . . + p n − 1 ) ∫ 0 1 ψ ( s ) s p 1 + p 2 + . . . + p n − 1 − 1 d s {\displaystyle {\frac {\Gamma \left(p_{1}\right)\Gamma \left(p_{2}\right)...\Gamma \left(p_{n-1}\right)}{\Gamma \left(p_{1}+p_{2}+...+p_{n-1}\right)}}\int _{0}^{1}\psi \left(s\right)s^{p_{1}+p_{2}+...+p_{n-1}-1}\mathrm {d} s} 利用上面已证的n=2时的公式，于是即得 ∫ . . . ∬ x 1 , x 2 , . . . , x n ⩾ 0 ; x 1 + x 2 + . . . + x n ⩽ 1 f ( x 1 + x 2 + . . . + x n ) x 1 p 1 − 1 x 2 p 2 − 1 . . . x n p n − 1 d x 1 d x 2 . . . d x n {\displaystyle \int ...\iint _{x_{1},x_{2},...,x_{n}\geqslant 0;x_{1}+x_{2}+...+x_{n}\leqslant 1}f\left(x_{1}+x_{2}+...+x_{n}\right)x_{1}^{p_{1}-1}x_{2}^{p_{2}-1}...x_{n}^{p_{n}-1}\mathrm {d} x_{1}\mathrm {d} x_{2}...\mathrm {d} x_{n}} = Γ ( p 1 ) Γ ( p 2 ) . . . Γ ( p n − 1 ) Γ ( p 1 + p 2 + . . . + p n − 1 ) ∫ 0 1 d s ∫ 0 1 − s f ( s + x n ) s p 1 + p 2 + . . . + p n − 1 − 1 x n p n − 1 d x n {\displaystyle ={\frac {\Gamma \left(p_{1}\right)\Gamma \left(p_{2}\right)...\Gamma \left(p_{n-1}\right)}{\Gamma \left(p_{1}+p_{2}+...+p_{n-1}\right)}}\int _{0}^{1}\mathrm {d} s\int _{0}^{1-s}f\left(s+x_{n}\right)s^{p_{1}+p_{2}+...+p_{n-1}-1}x_{n}^{p_{n}-1}\mathrm {d} x_{n}} = Γ ( p 1 ) Γ ( p 2 ) . . . Γ ( p n − 1 ) Γ ( p 1 + p 2 + . . . + p n − 1 ) ∬ s , x n ⩾ 0 ; s + x n ⩽ 1 f ( s + x n ) s p 1 + p 2 + . . . + p n − 1 − 1 x n p n − 1 d x n {\displaystyle ={\frac {\Gamma \left(p_{1}\right)\Gamma \left(p_{2}\right)...\Gamma \left(p_{n-1}\right)}{\Gamma \left(p_{1}+p_{2}+...+p_{n-1}\right)}}\iint _{s,x_{n}\geqslant 0;s+x_{n}\leqslant 1}f\left(s+x_{n}\right)s^{p_{1}+p_{2}+...+p_{n-1}-1}x_{n}^{p_{n}-1}\mathrm {d} x_{n}} = Γ ( p 1 ) Γ ( p 2 ) . . . Γ ( p n − 1 ) Γ ( p 1 + p 2 + . . . + p n − 1 ) ⋅ Γ ( p 1 + p 2 + . . . + p n − 1 ) Γ ( p n ) Γ ( p 1 + p 2 + . . . + p n ) ∫ 0 1 f ( u ) u p 1 + p 2 + . . . + p n − 1 d u {\displaystyle ={\frac {\Gamma \left(p_{1}\right)\Gamma \left(p_{2}\right)...\Gamma \left(p_{n-1}\right)}{\Gamma \left(p_{1}+p_{2}+...+p_{n-1}\right)}}\cdot {\frac {\Gamma \left(p_{1}+p_{2}+...+p_{n-1}\right)\Gamma \left(p_{n}\right)}{\Gamma \left(p_{1}+p_{2}+...+p_{n}\right)}}\int _{0}^{1}f\left(u\right)u^{p_{1}+p_{2}+...+p_{n}-1}\mathrm {d} u} = Γ ( p 1 ) Γ ( p 2 ) . . . Γ ( p n ) Γ ( p 1 + p 2 + . . . + p n ) ∫ 0 1 f ( u ) u p 1 + p 2 + . . . + p n − 1 d u {\displaystyle ={\frac {\Gamma \left(p_{1}\right)\Gamma \left(p_{2}\right)...\Gamma \left(p_{n}\right)}{\Gamma \left(p_{1}+p_{2}+...+p_{n}\right)}}\int _{0}^{1}f\left(u\right)u^{p_{1}+p_{2}+...+p_{n}-1}\mathrm {d} u} 证明完毕。 Б.П.吉米多维奇. 《吉米多维奇数学分析习题集题解》. 济南: 山东科学技术出版社. 2014. ISBN 978-7-5331-5895-8.