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：

$$\int ...\iint_{x_{1},x_{2},...,x_{n}\geqslant0;x_{1}+x_{2}+...+x_{n}\leqslant1}f\left(x_{1}+x_{2}+...+x_{n}\right)x_{1}^{p_{1}-1}x_{2}^{p_{2}-1}...x_{n}^{p_{n}-1}\mathrm{d}x_{1}\mathrm{d}x_{2}...\mathrm{d}x_{n}$$

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

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

$$\iint_{x_{1},x_{2}\geqslant0;x_{1}+x_{2}\leqslant1}f\left(x_{1}+x_{2}\right)x_{1}^{p_{1}-1}x_{2}^{p_{2}-1}\mathrm{d}x_{1}\mathrm{d}x_{2}=\frac{\Gamma\left(p_{1}\right)\Gamma\left(p_{2}\right)}{\Gamma\left(p_{1}+p_{2}\right)}\int_{0}^{1}f\left(u\right)u^{p_{1}+p_{2}-1}\mathrm{d}u$$

事實上，令$\Omega$表示區域：$x_{1}\geqslant0,x_{2}\geqslant0,x_{1}+x_{2}\leqslant1$，作代換$x_{1}=\xi_{1},x_{1}+x_{2}=\xi_{2}$，以及$t=\frac{\xi_{1}}{\xi_{2}}$，則有

$$\iint_{x_{1},x_{2}\geqslant0;x_{1}+x_{2}\leqslant1}f\left(x_{1}+x_{2}\right)x_{1}^{p_{1}-1}x_{2}^{p_{2}-1}\mathrm{d}x_{1}\mathrm{d}x_{2}=\int_{0}^{1}f\left(\xi_{2}\right)\mathrm{d}\xi_{2}\int_{0}^{\xi_{2}}\xi_{1}^{p_{1}-1}\left(\xi_{2}-\xi{1}\right)^{p_{2}-1}\mathrm{d}\xi_{1}$$

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

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

$$\int ...\iint_{x_{1},x_{2},...,x_{n-1}\geqslant0;x_{1}+x_{2}+...+x_{n-1}\leqslant1}x_{1}^{p_{1}-1}x_{2}^{p_{2}-1}...x_{n-1}^{p_{n-1}-1}\mathrm{d}x_{1}\mathrm{d}x_{2}...\mathrm{d}x_{n-1}\int_{0}^{1-\left(x_{1}+x_{2}+...+x_{n-1}\right)}f\left(x_{1}+x_{2}+...+x_{n}\right)x_{n}^{p_{n}-1}\mathrm{d}x_{n}$$

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

$$\int ...\iint_{x_{1},x_{2},...,x_{n}\geqslant0;x_{1}+x_{2}+...+x_{n}\leqslant1}f\left(x_{1}+x_{2}+...+x_{n}\right)x_{1}^{p_{1}-1}x_{2}^{p_{2}-1}...x_{n}^{p_{n}-1}\mathrm{d}x_{1}\mathrm{d}x_{2}...\mathrm{d}x_{n}$$

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

$$=\frac{\Gamma\left(p_{1}\right)\Gamma\left(p_{2}\right)...\Gamma\left(p_{n-1}\right)}{\Gamma\left(p_{1}+p_{2}+...+p_{n-1}\right)}\iint_{s,x_{n}\geqslant0;s+x_{n}\leqslant1}f\left(s+x_{n}\right)s^{p_{1}+p_{2}+...+p_{n-1}-1}x_{n}^{p_{n}-1}\mathrm{d}x_{n}$$

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

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

證明完畢。