【流水】2022.11.19

随便掺点东西罢

大家没事也可以打打

基本上不熟练的半个小时也就行了

ps：看不到的多刷新几遍

实在不行粘源码

$$\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}$$

<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.24999999999999992em" columnalign="center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mtext mathvariant="bold">KaTeX入门</mtext><mspace width="20em"></mspace><mtext>fixed&nbsp;by&nbsp;离散小波变换</mtext></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><menclose notation="box"><mstyle scriptlevel="0" displaystyle="false"><mstyle scriptlevel="0" displaystyle="false"><mstyle scriptlevel="0" displaystyle="true"><mtable rowspacing="0.24999999999999992em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mspace width="1.5em"></mspace><mtable rowspacing="0.24999999999999992em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mtext>先假设你有一个简单的公式。</mtext></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><mo fence="true">{</mo><mtable rowspacing="0.3599999999999999em" columnalign="left left" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo>+</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mn>2</mn><mo stretchy="false">)</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi>x</mi><mo>≥</mo><mn>3</mn></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>1</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mtext>Otherwise.</mtext></mstyle></mtd></mtr></mtable></mrow></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mtext>假设现在有人又给了一个公式。</mtext></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><mo fence="true">{</mo><mtable rowspacing="0.3599999999999999em" columnalign="left left" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo>+</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi>x</mi><mo>≥</mo><mn>2</mn></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi>f</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mtext>Otherwise.</mtext></mstyle></mtd></mtr></mtable></mrow></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mtext>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。</mtext></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mtext>也就是说，只要多写写公式你的水平自然会提升。</mtext></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mtext>这就是公式的基本写法了。</mtext></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mtext>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</mtext></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mtext>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。</mtext><mspace width="1.5em"></mspace></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><munder accentunder="true"><mspace width="31em"></mspace><mo stretchy="true">‾</mo></munder></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><munder accentunder="true"><mspace width="31em"></mspace><mo stretchy="true">‾</mo></munder></mrow></mstyle></mtd></mtr></mtable></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mspace width="1em"></mspace><menclose notation="box"><mstyle scriptlevel="0" displaystyle="false"><mstyle scriptlevel="0" displaystyle="false"><mstyle scriptlevel="0" displaystyle="true"><mo lspace="0em" rspace="0em"><mover><mo><mstyle mathsize="1em"><mspace width="1em"><mtext>例题&nbsp;1.8</mtext><mspace width="1em"></mspace></mspace></mstyle></mo><mstyle mathsize="1em"><mspace width="1em"><mtext mathvariant="bold">试试看!</mtext><mspace width="1em"></mspace></mspace></mstyle></mover></mo></mstyle></mstyle></mstyle></menclose></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mtable rowspacing="0.24999999999999992em" columnalign="center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mspace width="0.5em"></mspace><mi>log</mi><mo>⁡</mo><mi mathvariant="normal">Π</mi><mo stretchy="false">(</mo><mi>N</mi><mo stretchy="false">)</mo><mo>=</mo><mo fence="false">(</mo><mi>N</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo fence="false">)</mo><mi>log</mi><mo>⁡</mo><mi>N</mi><mo>−</mo><mi>N</mi><mo>+</mo><mi>A</mi><mo>−</mo><msubsup><mo>∫</mo><mi>N</mi><mi mathvariant="normal">∞</mi></msubsup><mfrac><mrow><msub><mover accent="true"><mi>B</mi><mo stretchy="true">‾</mo></mover><mn>1</mn></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><mi>x</mi></mrow><mi>x</mi></mfrac><mo separator="true">,</mo><mi>A</mi><mo>=</mo><mn>1</mn><mo>+</mo><msubsup><mo>∫</mo><mn>1</mn><mi mathvariant="normal">∞</mi></msubsup><mfrac><mrow><msub><mover accent="true"><mi>B</mi><mo stretchy="true">‾</mo></mover><mn>1</mn></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><mi>x</mi></mrow><mi>x</mi></mfrac></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>log</mi><mo>⁡</mo><mi mathvariant="normal">Π</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>=</mo><mo fence="false">(</mo><mi>s</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo fence="false">)</mo><mi>log</mi><mo>⁡</mo><mi>s</mi><mo>−</mo><mi>s</mi><mo>+</mo><mi>A</mi><mo>−</mo><msubsup><mo>∫</mo><mn>0</mn><mi mathvariant="normal">∞</mi></msubsup><mfrac><mrow><msub><mover accent="true"><mi>B</mi><mo stretchy="true">‾</mo></mover><mn>1</mn></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><mi>t</mi></mrow><mrow><mi>t</mi><mo>+</mo><mi>s</mi></mrow></mfrac></mrow></mstyle></mtd></mtr></mtable></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mspace width="0.5em"></mspace><mtable rowspacing="0.24999999999999992em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>log</mi><mo>⁡</mo><mi mathvariant="normal">Π</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>=</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><munder><mo><mi>lim</mi><mo>⁡</mo></mo><mrow><mi>n</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></munder><mo fence="false">[</mo><mi>s</mi><mi>log</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>N</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mo>+</mo><munderover><mo>∑</mo><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mi>N</mi></munderover><mi>log</mi><mo>⁡</mo><mi>n</mi><mo>−</mo><munderover><mo>∑</mo><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mi>N</mi></munderover><mi>log</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>s</mi><mo>+</mo><mi>n</mi><mo stretchy="false">)</mo><mo fence="false">]</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mo lspace="0em" rspace="0em">=</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><munder><mo><mi>lim</mi><mo>⁡</mo></mo><mrow><mi>n</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></munder><mo fence="false">[</mo><mi>s</mi><mi>log</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>N</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mo>+</mo><msubsup><mo>∫</mo><mn>1</mn><mi>N</mi></msubsup><mi>log</mi><mo>⁡</mo><mi>x</mi><mi mathvariant="normal">d</mi><mi>x</mi><mo>−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>log</mi><mo>⁡</mo><mi>N</mi><mo>+</mo><msubsup><mo>∫</mo><mn>1</mn><mi>N</mi></msubsup><mfrac><mrow><msub><mover accent="true"><mi>B</mi><mo stretchy="true">‾</mo></mover><mn>1</mn></msub><mi mathvariant="normal">d</mi><mi>x</mi></mrow><mi>x</mi></mfrac></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>−</mo><msubsup><mo>∫</mo><mn>1</mn><mi>N</mi></msubsup><mi>log</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>s</mi><mo>+</mo><mi>x</mi><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><mi>x</mi><mo>−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo stretchy="false">[</mo><mi>log</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>s</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mo>+</mo><mi>log</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>s</mi><mo>+</mo><mi>N</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>−</mo><msubsup><mo>∫</mo><mn>1</mn><mi>N</mi></msubsup><mfrac><mrow><msub><mover accent="true"><mi>B</mi><mo stretchy="true">‾</mo></mover><mn>1</mn></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><mi>x</mi></mrow><mrow><mi>s</mi><mo>+</mo><mi>x</mi></mrow></mfrac><mo fence="false">]</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mo lspace="0em" rspace="0em">=</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><munder><mo><mi>lim</mi><mo>⁡</mo></mo><mrow><mi>n</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></munder><mo fence="false">[</mo><mi>s</mi><mi>log</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>N</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mo>+</mo><mi>N</mi><mi>log</mi><mo>⁡</mo><mi>N</mi><mo>−</mo><mi>N</mi><mo>+</mo><mn>1</mn><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>log</mi><mo>⁡</mo><mi>N</mi><mo>+</mo><msubsup><mo>∫</mo><mn>1</mn><mi>N</mi></msubsup><mfrac><mrow><msub><mover accent="true"><mi>B</mi><mo stretchy="true">‾</mo></mover><mn>1</mn></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><mi>x</mi></mrow><mi>x</mi></mfrac></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>−</mo><mo stretchy="false">(</mo><mi>s</mi><mo>+</mo><mi>N</mi><mo stretchy="false">)</mo><mi>log</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>s</mi><mo>+</mo><mi>N</mi><mo stretchy="false">)</mo><mo>+</mo><mo stretchy="false">(</mo><mi>s</mi><mo>+</mo><mi>N</mi><mo stretchy="false">)</mo><mo>+</mo><mo stretchy="false">(</mo><mi>s</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mi>log</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>s</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>−</mo><mo stretchy="false">(</mo><mi>s</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mo>−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>log</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>s</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mo>−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>log</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>s</mi><mo>+</mo><mi>N</mi><mo stretchy="false">)</mo><mo>−</mo><msubsup><mo>∫</mo><mn>1</mn><mi>N</mi></msubsup><mfrac><mrow><msub><mover accent="true"><mi>B</mi><mo stretchy="true">‾</mo></mover><mn>1</mn></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><mi>x</mi></mrow><mrow><mi>s</mi><mo>+</mo><mi>x</mi></mrow></mfrac><mo fence="false">]</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mo lspace="0em" rspace="0em">=</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo fence="false">(</mo><mi>s</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo fence="false">)</mo><mi>log</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>s</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mo>+</mo><msubsup><mo>∫</mo><mn>1</mn><mi mathvariant="normal">∞</mi></msubsup><mfrac><mrow><msub><mover accent="true"><mi>B</mi><mo stretchy="true">‾</mo></mover><mn>1</mn></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><mi>x</mi></mrow><mi>x</mi></mfrac><mo>−</mo><msubsup><mo>∫</mo><mn>1</mn><mi>N</mi></msubsup><mfrac><mrow><msub><mover accent="true"><mi>B</mi><mo stretchy="true">‾</mo></mover><mn>1</mn></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><mi>x</mi></mrow><mrow><mi>s</mi><mo>+</mo><mi>x</mi></mrow></mfrac></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>+</mo><munder><mo><mi>lim</mi><mo>⁡</mo></mo><mrow><mi>n</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></munder><mo fence="false">[</mo><mi>s</mi><mi>log</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>N</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mo>+</mo><mo fence="false">(</mo><mi>N</mi><mfrac><mn>1</mn><mn>2</mn></mfrac><mo fence="false">)</mo><mi>log</mi><mo>⁡</mo><mi>N</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>−</mo><mo fence="false">(</mo><mi>s</mi><mo>+</mo><mi>N</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo fence="false">)</mo><mi>log</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>s</mi><mo>+</mo><mi>N</mi><mo stretchy="false">)</mo><mo fence="false">]</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mo lspace="0em" rspace="0em">=</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo fence="false">(</mo><mi>s</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo fence="false">)</mo><mi>log</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>s</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mo>+</mo><mo stretchy="false">(</mo><mi>A</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo>−</mo><msubsup><mo>∫</mo><mn>1</mn><mi mathvariant="normal">∞</mi></msubsup><mfrac><mrow><msub><mover accent="true"><mi>B</mi><mo stretchy="true">‾</mo></mover><mn>1</mn></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><mi>x</mi></mrow><mrow><mi>s</mi><mo>+</mo><mi>x</mi></mrow></mfrac></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>+</mo><mi>lim</mi><mo>⁡</mo><mo fence="false">[</mo><mi>s</mi><mi>log</mi><mo>⁡</mo><mfrac><mrow><mi>N</mi><mo>+</mo><mn>1</mn></mrow><mn>2</mn></mfrac><mo>−</mo><mo fence="false">(</mo><mi>N</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo fence="false">)</mo><mi>log</mi><mo>⁡</mo><mo fence="false">(</mo><mn>1</mn><mo>+</mo><mfrac><mi>s</mi><mn>2</mn></mfrac><mo fence="false">)</mo><mo fence="false">]</mo></mrow></mstyle></mtd></mtr></mtable></mrow></mstyle></mtd></mtr></mtable></mstyle></mstyle></mstyle></menclose></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mover accent="true"><mspace width="34.7em"></mspace><mo stretchy="true">‾</mo></mover></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{gathered}
\textbf{\KaTeX 入门}\kern{200pt}\text{fixed by 离散小波变换} \cr
\boxed{\begin{aligned}
&amp;\kern{15pt}
    \begin{aligned} \cr
    &amp;\text{先假设你有一个简单的公式。}\cr
    &amp;f(x)=\begin{cases}
        f(x-1)+f(x-2) &amp; x\geq 3\cr
        1 &amp; \text{Otherwise.}
    \end{cases}\cr
    &amp;\text{假设现在有人又给了一个公式。}\cr
    &amp;g(x)=\begin{cases}
        g(x-1)+f(x) &amp; x\geq 2\cr
        f(1) &amp; \text{Otherwise.}
    \end{cases}\cr
    &amp;\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
    &amp;\text{也就是说，只要多写写公式你的水平自然会提升。}\cr
    \cr
    &amp;\text{这就是公式的基本写法了。}\cr \cr
    &amp;\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
    &amp;\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
    &amp;\underline{\kern{310pt}}\cr[-13pt]
    &amp;\underline{\kern{310pt}}\cr
    \end{aligned}\cr
    &amp;\kern{10pt}\boxed{\stackrel{\normalsize\quad\textbf{试试看!}\quad}{\normalsize\quad\text{例题 1.8}\quad}}\cr
    &amp;\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
    &amp;\kern{5pt}\begin{aligned}
        \log \Pi(s)=&amp;\lim_{n\to \infty}\Big[s \log(N+1)+\sum_{n=1}^{N}\log n-\sum_{n=1}^{N}\log(s+n)\Big]\cr
        =&amp;\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
        &amp;-\int_{1}^{N}\log(s+x){\rm d} x-\dfrac{1}{2}[\log(s+1)+\log(s+N)]\cr
        &amp;-\int_{1}^{N}\dfrac{\overline{B}_1(x){\rm d} x}{s+x}\Big]\cr
        =&amp;\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
        &amp;-(s+N)\log(s+N)+(s+N)+(s+1)\log(s+1)\cr
        &amp;-(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
        =&amp;\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
        &amp;+\lim_{n \to \infty}\Big[s\log(N+1)+\Big(N\dfrac{1}{2}\Big)\log N\cr
        &amp;-\Big(s+N+\dfrac{1}{2}\Big)\log(s+N)\Big]\cr
        =&amp;\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
        &amp;+\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}
</annotation></semantics></math></span></span></span></p>

今天考了场试

寄了，寄大方了

还有他妈6天NOIP我好慌啊，连个省一都拿不到我是不是个寄吧啊qwq

退役倒寄时是吧