一些亟待解决的问题

今天回家，11:00的车，临走前写几个要解决的问题，虽然第三个积分在潘承洞的素数定理的初等证明里看到过.

$$\begin{align*}\gamma&=- \int_0^1 {\ln \ln \frac{1}{x}dx}  = \int_0^1 {\left( {\frac{1}{{1 - x}} + \frac{1}{{\ln x}}} \right)dx}  = \int_0^\infty  {{e^{ - x}}\ln xdx} \\\beta \left( 3 \right) &= \frac{{{{\left( { - 1} \right)}^{n - 1}}}}{{{{\left( {2n - 1} \right)}^3}}} = \frac{{{\pi ^3}}}{{32}}.\end{align*}$$

接下来是一个恒等式的证明：

$$\begin{align*}&\frac{1}{{\left( {{x_1} - {x_2}} \right)\left( {{x_1} - {x_3}} \right) \cdots \left( {{x_1} - {x_n}} \right)}} = \frac{1}{{{x_1} - {x_2}}}\left[ {\frac{1}{{\left( {{x_2} - {x_3}} \right)\left( {{x_2} - {x_4}} \right) \cdots \left( {{x_2} - {x_n}} \right)}}} \right] \\&+ \frac{1}{{{x_1} - {x_3}}}\left[ {\frac{1}{{\left( {{x_3} - {x_2}} \right)\left( {{x_3} - {x_4}} \right) \cdots \left( {{x_3} - {x_n}} \right)}}} \right] +  \cdots \frac{1}{{{x_1} - {x_n}}}\left[ {\frac{1}{{\left( {{x_n} - {x_2}} \right)\left( {{x_n} - {x_3}} \right) \cdots \left( {{x_n} - {x_{n - 1}}} \right)}}} \right].\end{align*}$$

也祝愿自己和亲们假期愉快！