上次我展示了一个极为惊艳的积分:

\[\int_0^1 {{e^{i\pi x}}} \; {x^x}{(1 - x)^{1 - x}}\; dx = \frac{e}{2}\frac{\pi }{3}\frac{i}{4}\]

这个积分神奇的把$0,1,2,3,4,i,e,\pi,x^x$结合在了一起.

Well,可是有很多吃瓜群众仍不满足,执意要把欧拉常数$\gamma$塞进去...

这个积分很精密的,禁不住这么魔改.不过还真有能塞一起的式子,不但能把$e,\pi,\gamma$一网打尽还能塞个Glaisher常数$\rm A$进去.

\[ - \zeta'(2) = \sum\limits_{k = 1}^\infty  {\frac{{\ln k}}{{{k^2}}}}  = \ln \prod\limits_{k = 1}^\infty  {\sqrt[{{k^2}}]{k}}  = \zeta (2)\ln \left( {\frac{{{{\rm A}^{12}}}}{{2\pi {e^\gamma }}}} \right)\]

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\[\int_0^1 {x\;{\text{d}}x}  = \frac{1}{2}\]

这个式子不会算的...来,我们谈谈人生.

接下来我们推广下计算这个:

\[\int_0^1 {\int_0^1 {{x^y}\;{\text{d}}x} } \;{\text{d}}y = \ln 2\]

不会算的我觉得也得来谈谈人生.

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格拉瑟主定理(Glasser's Master Theorem)的标准表述如下:

恒等式$PV\int_{ - \infty }^{ + \infty } {F(\phi (x)){\rm{dx}}}  = PV\int_{ - \infty }^{ + \infty } {F(x){\rm{dx}}} $对任意可积函数$F(x)$和$\phi (x) = |a|x - \sum\limits_{n = 1}^N {\frac{{|{\alpha _n}|}}{{x - {\beta _n}}}} $成立.

其中$a,\left\{ {{\alpha _n}} \right\}_{n = 1}^N,\left\{ {{\beta _n}} \right\}_{n = 1}^N$可以是任意常数,PV是柯西主值(Cauchy Principal Value)不可以两边约掉的啊魂淡可我怎么总是有种两边划掉的强烈冲动怎么办才好啊好烦啊要控制不住了的说....

So....这玩意儿有什么用呢?

叫你证仨积分,汝敢答应否?

$$\begin{aligned}
{I_1} &= \int_{ - \infty }^{ + \infty } {\frac{{{x^8} + 12{x^7} + 58{x^6} + 144{x^5} + 193{x^4} + 132{x^3} + 36{x^2}}}{{{x^{10}} + 12{x^9} + 51{x^8} + 72{x^7} - 81{x^6} - 300{x^5} - 43{x^4} + 576{x^3} + 664{x^2} + 264x + 36}}{\rm{d}}x} = \pi \\
{I_2} &= \int_{ - \infty }^{ + \infty } {\frac{{{x^8} + 4{x^7} + 6{x^6} + 4{x^5} + {x^4}}}{{{x^{12}} + 4{x^{11}} - 2{x^{10}} - 24{x^9} - 10{x^8} + 56{x^7} + 48{x^6} - 40{x^5} - 49{x^4} + 4{x^3} + 20{x^2} + 8x + 1}}{\rm{d}}x = \frac{\pi }{{\sqrt 2 }}} \\
{I_3} &=\int_0^\infty \frac{x^{14}-15x^{12}+82x^{10}-190x^8+184x^6-60x^4+16x^2}{x^{16}-20x^{14}+156x^{12}-616x^{10}+1388x^8-1792x^6+1152x^4-224x^2+16}\; dx = \frac{\pi}{2}
\end{aligned}$$

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