劉維爾公式

劉維爾公式（Liouville's Formula）是一個關於多重積分和歐拉積分（$\Gamma$函數）的公式，其形式如下：

$\int ...\iint _{x_{1},x_{2},...,x_{n}\geqslant 0;x_{1}+x_{2}+...+x_{n}\leqslant 1}f\left(x_{1}+x_{2}+...+x_{n}\right)x_{1}^{p_{1}-1}x_{2}^{p_{2}-1}...x_{n}^{p_{n}-1}\mathrm {d} x_{1}\mathrm {d} x_{2}...\mathrm {d} x_{n}$

$={\frac {\Gamma \left(p_{1}\right)\Gamma \left(p_{2}\right)...\Gamma \left(p_{n}\right)}{\Gamma \left(p_{1}+p_{2}+...+p_{n}\right)}}\int _{0}^{1}f\left(u\right)u^{p_{1}+p_{2}+...+p_{n}-1}\mathrm {d} u$

其中$p_{1},p_{2},...,p_{n}>0$，$f\left(u\right)$為連續函數。

證明

用數學歸納法。 當n=1時，公式顯然成立。

當n=2時，公式也成立，即

$\iint _{x_{1},x_{2}\geqslant 0;x_{1}+x_{2}\leqslant 1}f\left(x_{1}+x_{2}\right)x_{1}^{p_{1}-1}x_{2}^{p_{2}-1}\mathrm {d} x_{1}\mathrm {d} x_{2}={\frac {\Gamma \left(p_{1}\right)\Gamma \left(p_{2}\right)}{\Gamma \left(p_{1}+p_{2}\right)}}\int _{0}^{1}f\left(u\right)u^{p_{1}+p_{2}-1}\mathrm {d} u$

事實上，令$\Omega$表示區域：$x_{1}\geqslant 0,x_{2}\geqslant 0,x_{1}+x_{2}\leqslant 1$，作代換$x_{1}=\xi _{1},x_{1}+x_{2}=\xi _{2}$，以及$t={\frac {\xi _{1}}{\xi _{2}}}$，則有

$\iint _{x_{1},x_{2}\geqslant 0;x_{1}+x_{2}\leqslant 1}f\left(x_{1}+x_{2}\right)x_{1}^{p_{1}-1}x_{2}^{p_{2}-1}\mathrm {d} x_{1}\mathrm {d} x_{2}=\int _{0}^{1}f\left(\xi _{2}\right)\mathrm {d} \xi _{2}\int _{0}^{\xi _{2}}\xi _{1}^{p_{1}-1}\left(\xi _{2}-\xi {1}\right)^{p_{2}-1}\mathrm {d} \xi _{1}$

$\int _{0}^{1}f\left(\xi _{2}\right)\mathrm {d} \xi _{2}\int _{0}^{1}t^{p_{1}-1}\left(1-t\right)^{p_{2}-1}\xi _{2}^{p_{1}+p_{2}-1}\mathrm {d} t={\frac {\Gamma \left(p_{1}\right)\Gamma \left(p_{2}\right)}{\Gamma \left(p_{1}+p_{2}\right)}}\int _{0}^{1}f\left(\xi _{2}\right)\xi _{2}^{p_{1}+p_{2}-1}\mathrm {d} \xi _{2}={\frac {\Gamma \left(p_{1}\right)\Gamma \left(p_{2}\right)}{\Gamma \left(p_{1}+p_{2}\right)}}\int _{0}^{1}f\left(u\right)u^{p_{1}+p_{2}-1}\mathrm {d} u$

設公式對於n-1成立，今證對於n公式也成立。為此，將公式左端寫為

$\int ...\iint _{x_{1},x_{2},...,x_{n-1}\geqslant 0;x_{1}+x_{2}+...+x_{n-1}\leqslant 1}x_{1}^{p_{1}-1}x_{2}^{p_{2}-1}...x_{n-1}^{p_{n-1}-1}\mathrm {d} x_{1}\mathrm {d} x_{2}...\mathrm {d} x_{n-1}\int _{0}^{1-\left(x_{1}+x_{2}+...+x_{n-1}\right)}f\left(x_{1}+x_{2}+...+x_{n}\right)x_{n}^{p_{n}-1}\mathrm {d} x_{n}$

令$\psi \left(s\right)=\int _{0}^{1-s}f\left(s+x_{n}\right)x_{n}^{p_{n}-1}\mathrm {d} x_{n}$

代入上式，並利用公式對n-1成立的假定，得知上式為

${\frac {\Gamma \left(p_{1}\right)\Gamma \left(p_{2}\right)...\Gamma \left(p_{n-1}\right)}{\Gamma \left(p_{1}+p_{2}+...+p_{n-1}\right)}}\int _{0}^{1}\psi \left(s\right)s^{p_{1}+p_{2}+...+p_{n-1}-1}\mathrm {d} s$

利用上面已證的n=2時的公式，於是即得

$\int ...\iint _{x_{1},x_{2},...,x_{n}\geqslant 0;x_{1}+x_{2}+...+x_{n}\leqslant 1}f\left(x_{1}+x_{2}+...+x_{n}\right)x_{1}^{p_{1}-1}x_{2}^{p_{2}-1}...x_{n}^{p_{n}-1}\mathrm {d} x_{1}\mathrm {d} x_{2}...\mathrm {d} x_{n}$

$={\frac {\Gamma \left(p_{1}\right)\Gamma \left(p_{2}\right)...\Gamma \left(p_{n-1}\right)}{\Gamma \left(p_{1}+p_{2}+...+p_{n-1}\right)}}\int _{0}^{1}\mathrm {d} s\int _{0}^{1-s}f\left(s+x_{n}\right)s^{p_{1}+p_{2}+...+p_{n-1}-1}x_{n}^{p_{n}-1}\mathrm {d} x_{n}$

$={\frac {\Gamma \left(p_{1}\right)\Gamma \left(p_{2}\right)...\Gamma \left(p_{n-1}\right)}{\Gamma \left(p_{1}+p_{2}+...+p_{n-1}\right)}}\iint _{s,x_{n}\geqslant 0;s+x_{n}\leqslant 1}f\left(s+x_{n}\right)s^{p_{1}+p_{2}+...+p_{n-1}-1}x_{n}^{p_{n}-1}\mathrm {d} x_{n}$

$={\frac {\Gamma \left(p_{1}\right)\Gamma \left(p_{2}\right)...\Gamma \left(p_{n-1}\right)}{\Gamma \left(p_{1}+p_{2}+...+p_{n-1}\right)}}\cdot {\frac {\Gamma \left(p_{1}+p_{2}+...+p_{n-1}\right)\Gamma \left(p_{n}\right)}{\Gamma \left(p_{1}+p_{2}+...+p_{n}\right)}}\int _{0}^{1}f\left(u\right)u^{p_{1}+p_{2}+...+p_{n}-1}\mathrm {d} u$

$={\frac {\Gamma \left(p_{1}\right)\Gamma \left(p_{2}\right)...\Gamma \left(p_{n}\right)}{\Gamma \left(p_{1}+p_{2}+...+p_{n}\right)}}\int _{0}^{1}f\left(u\right)u^{p_{1}+p_{2}+...+p_{n}-1}\mathrm {d} u$

證明完畢。