\[
\begin{gathered}
\textbf{\KaTeX 入门}\kern{200pt}\text{fixed by 离散小波变换} \cr
\boxed{\begin{aligned}
&\kern{15pt}
\begin{aligned} \cr
&\text{先假设你有一个简单的公式。}\cr
&f(x)=\begin{cases}
f(x-1)+f(x-2) & x\geq 3\cr
1 & \text{Otherwise.}
\end{cases}\cr
&\text{假设现在有人又给了一个公式。}\cr
&g(x)=\begin{cases}
g(x-1)+f(x) & x\geq 2\cr
f(1) & \text{Otherwise.}
\end{cases}\cr
&\text{现在，看一看你所写的公式的码量。你会发现你的\KaTeX技能提升了。}\cr
&\text{也就是说，只要多写写公式你的水平自然会提升。}\cr
\cr
&\text{这就是公式的基本写法了。}\cr \cr
&\text{那么，现在你已经对\KaTeX的基本用法有了一定的了解，就让我们来看}\cr
&\text{一看下面这个简单的例子，来把我们刚刚学到的东西运用到实践中吧。}\kern{15pt}\cr
&\underline{\kern{310pt}}\cr[-13pt]
&\underline{\kern{310pt}}\cr
\end{aligned}\cr
&\kern{10pt}\boxed{\stackrel{\normalsize\quad\textbf{试试看!}\quad}{\normalsize\quad\text{例题 1.8}\quad}}\cr
&\begin{gathered}
\kern{5pt}\log \Pi(N)=\Big(N+\dfrac{1}{2}\Big)\log N -N+A-\int_{N}^{\infty}\dfrac{\overline{B}_1(x){\rm d} x}{x}, A=1+\int_{1}^{\infty}\dfrac{\overline{B}_1(x){\rm d} x}{x} \cr
\log \Pi(s)=\Big(s+\dfrac{1}{2}\Big)\log s-s+A-\int_{0}^{\infty}\dfrac{\overline{B}_1(t){\rm d} t}{t+s}
\end{gathered}\cr
&\kern{5pt}\begin{aligned}
\log \Pi(s)=&\lim_{n\to \infty}\Big[s \log(N+1)+\sum_{n=1}^{N}\log n-\sum_{n=1}^{N}\log(s+n)\Big]\cr
=&\lim_{n\to \infty}\Big[s \log (N+1)+\int_{1}^{N}\log x {\rm d} x-\dfrac{1}{2}\log N +\int_{1}^{N}\dfrac{\overline{B}_1{\rm d} x}{x}\cr
&-\int_{1}^{N}\log(s+x){\rm d} x-\dfrac{1}{2}[\log(s+1)+\log(s+N)]\cr
&-\int_{1}^{N}\dfrac{\overline{B}_1(x){\rm d} x}{s+x}\Big]\cr
=&\lim_{n\to \infty}\Big[s\log(N+1)+N\log N-N+1+\dfrac{1}{2}\log N+\int_{1}^{N}\dfrac{\overline{B}_1(x){\rm d} x}{x} \cr
&-(s+N)\log(s+N)+(s+N)+(s+1)\log(s+1)\cr
&-(s+1)-\dfrac{1}{2}\log(s+1)-\dfrac{1}{2}\log(s+N)-\int_{1}^{N}\dfrac{\overline{B}_1(x){\rm d} x}{s+x}\Big]\cr
=&\Big(s+\dfrac{1}{2} \Big)\log(s+1)+\int_{1}^{\infty}\dfrac{\overline{B}_1(x){\rm d} x}{x}-\int_{1}^{N}\dfrac{\overline{B}_1(x){\rm d} x}{s+x}\cr
&+\lim_{n \to \infty}\Big[s\log(N+1)+\Big(N\dfrac{1}{2}\Big)\log N\cr
&-\Big(s+N+\dfrac{1}{2}\Big)\log(s+N)\Big]\cr
=&\Big(s+\dfrac{1}{2}\Big)\log(s+1)+(A-1)-\int_{1}^{\infty}\dfrac{\overline{B}_1(x){\rm d} x}{s+x}\cr
&+\lim\Big[s\log\dfrac{N+1}{2}-\Big(N+\dfrac{1}{2}\Big)\log\Big(1+\dfrac{s}{2}\Big)\Big]
\end{aligned}
\end{aligned}}\cr
\cr[-44pt]\overline{\kern{347pt}}\cr[-7pt]\color{white}\rule{350pt}{18pt}\cr[-22pt]
\color{black}\textbf{假如让写\KaTeX的那些人来出教程}
\end{gathered}
\]

\(\Large\fcolorbox{e00000}{e0efaf}{\textbf{入口地址}}\)