**刘维尔公式**（Liouville's Formula）是一个关于多重积分和欧拉积分（\Gamma 函数）的公式，其形式如下：

{\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}}
{\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{\displaystyle p\_{1},p\_{2},...p\_{n}>0}，{\displaystyle f\left(u\right)}为连续函数。[1]

证明
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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

{\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\Omega 表示区域：{\displaystyle x\_{1}\geqslant 0,x\_{2}\geqslant 0,x\_{1}+x\_{2}\leqslant 1}，作代换{\displaystyle x\_{1}=\xi \_{1},x\_{1}+x\_{2}=\xi \_{2}}，以及{\displaystyle t={\frac {\xi \_{1}}{\xi \_{2}}}}，则有

{\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}}
{\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

{\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{\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

{\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

{\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}}
{\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}}

{\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}}

{\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}

{\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。[1]

参考资料
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1. 1 2  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.