Latex 的一大优点就是其可以写出优美的数学公式，本博客也使用 MathJax 来得到 Latex 中数学公式的效果。但是由于其有自己的语法，所以写了这篇文章以整理数学公式相关的内容。
由于本文中公式较多，所以可能需要些许渲染的时间。
常见公式
| 名称 | 代码 | 效果 |
| 上下标 | _1^2X_3^4 | \(_1^2X_3^4\) |
| 组合数 | \mathrm{C}_n^r | \(\mathrm{C}_n^r\) |
| 排列数 | \mathrm{P}_n^r | \(\mathrm{P}_n^r\) |
| 上划线 | \overline{abc} | \(\overline{abc}\) |
| 下划线 | \underline{abc} | \(\underline{abc}\) |
| 求和 | \sum_{k=1}^N k^2 | \(\sum\limits_{k=1}^N k^2\) |
| 求积 | \prod_{i=1}^N x_i | \(\prod\limits_{i=1}^N x_i\) |
| 极限 | \lim_{n \to \infty}n | \(\lim\limits_{n \to \infty}n\) |
| 根号 | \sqrt[n]{x}| | \(\sqrt[n]{x}\) |
| 上括号 | \overbrace{a+b+\cdots+z}^26 | \(\overbrace{a+b+\cdots+z}^{26}\) |
| 下括号 | \underbrace{a+b+\cdots+z}_26 | \(\underbrace{a+b+\cdots+z}_{26}\) |
| 向量 | \overrightarrow{AB} | \(\overrightarrow{AB}\) |
分数
分数最普通的形式为\frac{x}{y}，也可以用\tfrac{x}{y}来得到较小的分数。
\[\frac{x}{y}\ \ \tfrac{x}{y}\]
用\frac{x}{1+\frac{y}}可以得到连分数，但是使用\cfrac代替可以得到更大的分数。
\[\frac{2}{c + \frac{2}{d + \frac{2}{4}}} \ \ \cfrac{2}{c + \cfrac{2}{d + \cfrac{2}{4}}}\]
矩阵
矩阵首尾以\begin{matrix}和\end{matrix}标记，各元素以&分隔，以\\换行。
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| \begin{matrix}
a_{11} & \cdots & a_{1n} \\
\vdots & \ddots & \vdots \\
a_{n1} & \cdots & a_{nn}
\end{matrix}
|
如上则是一个矩阵的代码，可以得到如下的效果。
\[\begin{matrix} a_{11} & \cdots & a_{1n} \\ \vdots & \ddots & \vdots \\ a_{n1} & \cdots & a_{nn} \end{matrix}\]
此外对于{matrix}中 matrix 可以替换成如下内容，以得到不同的边框。
| 替换内容 | 效果 | 替换内容 | 效果 |
| vmatrix | \(\begin{vmatrix}a & b \\c & d\end{vmatrix}\) | Vmatrix | \(\begin{Vmatrix}a & b \\c & d\end{Vmatrix}\) |
| bmatrix | \(\begin{bmatrix}a & b \\c & d\end{bmatrix}\) | Bmatrix | \(\begin{Bmatrix}a & b \\c & d\end{Bmatrix}\) |
| pmatrix | \(\begin{pmatrix}a & b \\c & d\end{pmatrix}\) | smallmatrix | \(\begin{smallmatrix}a & b \\c & d\end{smallmatrix}\) |
分段函数 & 方程组
首尾以\begin{cases}和\end{cases}标记，各元素以&分隔，以\\换行。
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| f(x) =
\begin{cases}
x, & x\ge0 \\
-x, & x<0
\end{cases}
|
可以得到如下效果。
\[f(x) = \begin{cases} x, & x\ge0 \\ -x, & x<0 \end{cases}\]
如下所示，可以对上面的稍加修改，
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| \begin{cases}
x+y=1 \\
3x+4y=5 \\
\end{cases}
|
便可以得到方程组的形式。
\[\begin{cases} x+y=1 \\ 3x+4y=5 \\ \end{cases}\]
多行等式
首尾以\begin{align}和\end{align}标记，各元素以&分隔，以\\换行。
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| \begin{align}
(x+y)^3 & = (x+y)^2(x+y)
& = (x+y)(x+y)(x+y)
\end{align}
|
实际效果如下。
\[\begin{align} (x+y)^3 & = (x+y)^2(x+y)\\ & = (x+y)(x+y)(x+y)\\ \end{align}\]
空格
| 功能 | 代码 | 效果 | 宽度 |
| 2个quad空格 | \qquad | \(A\qquad B\) | \(2m\) |
| quad空格 | \quad | \(A\quad B\) | \(m\) |
| 大空格 | \ | \(A\ B\) | \(\frac{m}{3}\) |
| 中等空格 | \; | \(A\;B\) | \(\frac{2m}{7}\) |
| 小空格 | \, | \(A\,B\) | \(\frac{m}{6}\) |
| 没有空格 | ` ` | \(AB\) | \(0\) |
| 紧贴 | \! | \(A\!B\) | \(-\frac{m}{6}\) |
括号
用\left和\right来控制左右括号。
| 功能 | 语法 | 显示 |
| 小括号 | \left( \frac{a}{b} \right) | \(\left( \frac{a}{b} \right)\) |
| 中括号 | \left[ \frac{a}{b} \right] | \(\left[ \frac{a}{b} \right]\) |
| 大括号 | \left\{ \frac{a}{b} \right\} | \(\left\{ \frac{a}{b} \right\}\) |
| 角括号 | \left \langle \frac{a}{b} \right \rangle | \(\left\langle \frac{a}{b} \right \rangle\) |
| 双竖线 | \left\Vert \frac{a}{b} \right\Vert | \(\left\Vert \frac{a}{b} \right\Vert\) |
| 绝对值 | \left\| \frac{a}{b} \right \| | \(\left| \frac{a}{b} \right |\) |
| 向下取整 | \left \lfloor \frac{a}{b} \right \rfloor | \(\left \lfloor \frac{a}{b} \right \rfloor\) |
| 向上取整 | \left \lceil \frac{c}{d} \right \rceil | \(\left \lceil \frac{c}{d} \right \rceil\) |
| 斜线与反斜线 | \left / \frac{a}{b} \right \backslash | \(\left / \frac{a}{b} \right \backslash\) |
| 上下箭头 | \left \uparrow \frac{a}{b} \right \downarrow | \(\left \uparrow \frac{a}{b} \right \downarrow\) |
| 上下箭头 | \left \Uparrow \frac{a}{b} \right \Downarrow | \(\left \Uparrow \frac{a}{b} \right \Downarrow\) |
括号同时可以用\big \Big \bigg \Bigg修饰。
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| \Bigg( \bigg[ \Big\{ \big\langle\left| \frac{a}{b} \right| \big\rangle \Big\} \bigg] \Bigg)
|
如上代码可以得到如下效果。
\[\Bigg( \bigg[ \Big\{ \big\langle\left| \frac{a}{b} \right| \big\rangle \Big\} \bigg] \Bigg)\]
微积分
| 代码 | 效果 | 代码 | 效果 | 代码 | 效果 |
\nabla | \(\nabla\) | \partial{x} | \(\partial{x}\) | \mathrm{d}x | \(\mathrm{d}x\) |
\dot{x} | \(\dot{x}\) | \ddot{y} | \(\ddot{y}\) | \frac{\mathrm{d}y}{\mathrm{d}x} | \(\frac{\mathrm{d}y}{\mathrm{d}x}\) |
f\prime(x) | \(f\prime(x)\) | \int_{-N}^{N} | \(\int_{-N}^{N}\) | \iint_{-N}^{N} | \(\iint_{-N}^{N}\) |
\iiint_{-N}^{N} | \(\iiint_{-N}^{N}\) | \iiiint_{-N}^{N} | \(\iiint_{-N}^{N}\) | \oint_{C} | \(\oint_{C}\) |
特殊字体
| 名称 | 代码 | 效果 |
| 黑板粗体 | \mathbb{ONLY\ UPPERCASE} | \(\mathbb{ONLY\ UPPERCASE}\) |
| 正粗体 | \mathbf{abcDEFGHI012} | \(\mathbf{abcDEFGHI012}\) |
| 斜粗体 | \boldsymbol{abcDEFGHI012} | \(\boldsymbol{abcDEFGHI012}\) |
| 斜体数字 | \mathit{0123456789} | \(\mathit{0123456789}\) |
| 罗马体 | \mathrm{012} \mbox{abc} \operatorname{ABC} | \(\mathrm{012} \mbox{abc} \operatorname{ABC}\) |
| 哥特体 | \mathfrak{abcDEFGHI012} | \(\mathfrak{abcDEFGHI012}\) |
| 手写体 | \mathcal{abcDEFGHI012} | \(\mathcal{abcDEFGHI012}\) |
| 希伯来字母 | \aleph\beth\gimel\daleth | \(\aleph\beth\gimel\daleth\) |
字符标记
| 代码 | 效果 | 代码 | 效果 | 代码 | 效果 |
\acute{a} | \(\acute{a}\) | \grave{a} | \(\grave{a}\) | \hat{a} | \(\hat{a}\) |
\tilde{a} | \(\tilde{a}\) | \breve{a} | \(\breve{a}\) | \check{a} | \(\check{a}\) |
\bar{a} | \(\bar{a}\) | \ddot{a} | \(\ddot{a}\) | \dot{a} | \(\dot{a}\) |
常见函数
| 代码 | 效果 | 代码 | 效果 | 代码 | 效果 |
\sin | \(\sin\) | \cos | \(\cos\) | \tan | \(\tan\) |
\sec | \(\sec\) | \csc | \(\csc\) | \cot | \(\cot\) |
\arcsin | \(\arcsin\) | \arccos | \(\arccos\) | \arctan | \(\arctan\) |
\sinh | \(\sinh\) | \cosh | \(\cosh\) | \tanh | \(\tanh\) |
\coth | \(\coth\) | \lim | \(\lim\) | \limsup | \(\limsup\) |
\liminf | \(\liminf\) | \min | \(\min\) | \max | \(\max\) |
\inf | \(\inf\) | \sup | \(\sup\) | \exp | \(\exp\) |
\ln | \(\ln\) | \lg | \(\lg\) | \log | \(\log\) |
\log_{10} | \(\log_{10}\) | \ker | \(\ker\) | \deg | \(\deg\) |
\gcd | \(\gcd\) | \Pr | \(\Pr\) | \det | \(\det\) |
\hom | \(\hom\) | \arg | \(\arg\) | \dim | \(\dim\) |
\bmod | \(\bmod\) | | | | |
二元运算符
| 代码 | 效果 | 代码 | 效果 | 代码 | 效果 |
+ | \(+\) | - | \(-\) | \pm | \(\pm\) |
\mp | \(\mp\) | \dotplus | \(\dotplus\) | \times | \(\times\) |
\div | \(\div\) | \divideontimes | \(\divideontimes\) | \backslash | \(\backslash\) |
\cdot | \(\cdot\) | \ast | \(\ast\) | \star | \(\star\) |
\circ | \(\circ\) | \bullet | \(\bullet\) | \boxplus | \(\boxplus\) |
\boxminus | \(\boxminus\) | \boxtimes | \(\boxtimes\) | \boxdot | \(\boxdot\) |
\oplus | \(\oplus\) | \ominus | \(\ominus\) | \otimes | \(\otimes\) |
\oslash | \(\oslash\) | \odot | \(\odot\) | \circleddash | \(\circleddash\) |
\circledcirc | \(\circledcirc\) | \circledast | \(\circledast\) | \bigoplus | \(\bigoplus\) |
\bigotimes | \(\bigotimes\) | \bigodot | \(\bigodot\) | \equiv | \(\equiv\) |
集合
| 代码 | 效果 | 代码 | 效果 | 代码 | 效果 |
\forall | \(\forall\) | \exists | \(\exists\) | \emptyset | \(\emptyset\) |
\in | \(\in\) | \ni | \(\ni\) | \notin | \(\notin\) |
\subset | \(\subset\) | \subseteq | \(\subseteq\) | \supset | \(\supset\) |
\supseteq | \(\supseteq\) | \bigcap | \(\bigcap\) | \cup | \(\cup\) |
\bigcup | \(\bigcup\) | \biguplus | \(\biguplus\) | \setminus | \(\setminus\) |
\smallsetminus | \(\smallsetminus\) | \sqsubset | \(\sqsubset\) | \sqsubseteq | \(\sqsubseteq\) |
\sqsupset | \(\sqsupset\) | \sqsupseteq | \(\sqsupseteq\) | \sqcap | \(\sqcap\) |
\sqcup | \(\sqcup\) | \bigsqcup | \(\bigsqcup\) | \cap | \(\cap\) |
\varnothing | \(\varnothing\) | \cap | \(\cap\) | | |
\nsubseteq | \(\nsubseteq\) | \nsupseteq | \(\nsupseteq\) | \nsubseteqq | \(\nsubseteqq\) |
\nsupseteqq | \(\nsupseteqq\) | \subseteqq | \(\sqsubset\) | \supseteqq | \(\sqsubseteq\) |
\sqsupset | \(\sqsupset\) | \sqsupseteq | \(\sqsupseteq\) | \sqcap | \(\sqcap\) |
逻辑运算符
| 代码 | 效果 | 代码 | 效果 | 代码 | 效果 |
\land | \(\land\) | \wedge | \(\wedge\) | \bigwedge | \(\bigwedge\) |
\bar{q} | \(\bar{A}\) | \to | \(\to\) | \lor | \(\lor\) |
\vee | \(\vee\) | \bigvee | \(\bigvee\) | \lnot | \(\lnot\) |
\neg | \(\neg\) | \And | \(\And\) | | |
关系运算符
| 代码 | 效果 | 代码 | 效果 | 代码 | 效果 |
\sim | \(\sim\) | \approx | \(\approx\) | \simeq | \(\simeq\) |
\cong | \(\cong\) | \dot= | \(\dot=\) | \gtrapprox | \(\gtrapprox\) |
< | \(<\) | \le | \(\le\) | \ll | \(\ll\) |
\gg | \(\gg\) | \ge | \(\ge\) | > | \(>\) |
\equiv | \(\equiv\) | \not\equiv | \(\not\equiv\) | \ne | \(\ne\) |
\mbox{or} | \(\mbox{or}\) | \neq | \(\neq\) | \propto | \(\propto\) |
\lessapprox | \(\lessapprox\) | \lesssim | \(\lesssim\) | \eqslantless | \(\eqslantless\) |
\leqslant | \(\leqslant\) | \leqq | \(\leqq\) | \geqq | \(\geqq\) |
\geqslant | \(\geqslant\) | \eqslantgtr | \(\eqslantgtr\) | \gtrsim | \(\gtrsim\) |
几何符号
| 代码 | 效果 | 代码 | 效果 | 代码 | 效果 |
\Diamond | \(\Diamond\) | \Box | \(\Box\) | \triangle | \(\triangle\) |
\angle | \(\angle\) | \perp | \(\perp\) | \mid | \(\mid\) |
\nmid | \(\nmid\) | \| | \(|\) | 45^\circ | \(45^\circ\) |
\sphericalangle | \(\sphericalangle\) | \bigtriangledown | \(\bigtriangledown\) | | |
希腊字母
| 代码 | 效果 | 代码 | 效果 | 代码 | 效果 | 代码 | 效果 |
\alpha | \(\alpha\) | \theta | \(\theta\) | o | \(o\) | \tau | \(\tau\) |
\beta | \(\beta\) | \vartheta | \(\vartheta\) | \pi | \(\pi\) | \upsilon | \(\upsilon\) |
\gamma | \(\gamma\) | \iota | \(\iota\) | \varpi | \(\varpi\) | \phi | \(\phi\) |
\delta | \(\delta\) | \kappa | \(\kappa\) | \rho | \(\rho\) | \varphi | \(\varphi\) |
\epsilon | \(\epsilon\) | \lambda | \(\lambda\) | \varrho | \(\varrho\) | \chi | \(\chi\) |
\varepsilon | \(\varepsilon\) | \mu | \(\mu\) | \sigma | \(\sigma\) | \psi | \(\psi\) |
\zeta | \(\zeta\) | \nu | \(\nu\) | \varsigma | \(\varsigma\) | \omega | \(\omega\) |
\eta | \(\eta\) | \xi | \(\xi\) | | | | |
\Gamma | \(\Gamma\) | \Lambda | \(\Lambda\) | \Sigma | \(\Sigma\) | \Psi | \(\Psi\) |
\Delta | \(\Delta\) | \Xi | \(\Xi\) | \Upsilon | \(\Upsilon\) | \Omega | \(\Omega\) |
\Theta | \(\Theta\) | \Pi | \(\Pi\) | \Phi | \(\Phi\) | | |
箭头
| 代码 | 效果 | 代码 | 效果 | 代码 | 效果 |
\leftarrow | \(\leftarrow\) | \rightarrow | \(\rightarrow\) | \nleftarrow | \(\nleftarrow\) |
\nrightarrow | \(\nrightarrow\) | \leftrightarrow | \(\leftrightarrow\) | \nleftrightarrow | \(\nleftrightarrow\) |
\longleftarrow | \(\longleftarrow\) | \longrightarrow | \(\longrightarrow\) | \longleftrightarrow | \(\longleftrightarrow\) |
\leftarrow | \(\leftarrow\) | \rightarrow | \(\rightarrow\) | \nleftarrow | \(\nleftarrow\) |
\nrightarrow | \(\nrightarrow\) | \leftrightarrow | \(\leftrightarrow\) | \nleftrightarrow | \(\nleftrightarrow\) |
\longleftarrow | \(\longleftarrow\) | \longrightarrow | \(\longrightarrow\) | \longleftrightarrow | \(\longleftrightarrow\) |
\Leftarrow | \(\Leftarrow\) | \Rightarrow | \(\Rightarrow\) | \nLeftarrow | \(\nLeftarrow\) |
\nRightarrow | \(\nRightarrow\) | \Leftrightarrow | \(\Leftrightarrow\) | \nLeftrightarrow | \(\nLeftrightarrow\) |
\Longleftarrow | \(\Longleftarrow\) | \Longrightarrow | \(\Longrightarrow\) | \Longleftrightarrow | \(\Longleftrightarrow\) |
\Leftarrow | \(\Leftarrow\) | \Rightarrow | \(\Rightarrow\) | \nLeftarrow | \(\nLeftarrow\) |
\nRightarrow | \(\nRightarrow\) | \Leftrightarrow | \(\Leftrightarrow\) | \nLeftrightarrow | \(\nLeftrightarrow\) |
\Longleftarrow | \(\Longleftarrow\) | \Longrightarrow | \(\Longrightarrow\) | \Longleftrightarrow | \(\Longleftrightarrow\) |
\uparrow | \(\uparrow\) | \downarrow | \(\downarrow\) | \updownarrow | \(\updownarrow\) |
\Uparrow | \(\Uparrow\) | \Downarrow | \(\Downarrow\) | \Updownarrow | \(\Updownarrow\) |
\nearrow | \(\nearrow\) | \searrow | \(\searrow\) | \swarrow | \(\swarrow\) |
\nwarrow | \(\nwarrow\) | \uparrow | \(\uparrow\) | \downarrow | \(\downarrow\) |
\updownarrow | \(\updownarrow\) | \Uparrow | \(\Uparrow\) | \Downarrow | \(\Downarrow\) |
\Updownarrow | \(\Updownarrow\) | \nearrow | \(\nearrow\) | \searrow | \(\searrow\) |
\swarrow | \(\swarrow\) | \nwarrow | \(\nwarrow\) | \rightharpoonup | \(\rightharpoonup\) |
\rightharpoondown | \(\rightharpoondown\) | \leftharpoonup | \(\leftharpoonup\) | \leftharpoondown | \(\leftharpoondown\) |
\upharpoonleft | \(\upharpoonleft\) | \upharpoonright | \(\upharpoonright\) | \downharpoonleft | \(\downharpoonleft\) |
\downharpoonright | \(\downharpoonright\) | \rightleftharpoons | \(\rightleftharpoons\) | \leftrightharpoons | \(\leftrightharpoons\) |
\rightharpoonup | \(\rightharpoonup\) | \rightharpoondown | \(\rightharpoondown\) | \leftharpoonup | \(\leftharpoonup\) |
\leftharpoondown | \(\leftharpoondown\) | \upharpoonleft | \(\upharpoonleft\) | \upharpoonright | \(\upharpoonright\) |
\downharpoonleft | \(\downharpoonleft\) | \downharpoonright | \(\downharpoonright\) | \rightleftharpoons | \(\rightleftharpoons\) |
\leftrightharpoons | \(\leftrightharpoons\) | \curvearrowleft | \(\curvearrowleft\) | \circlearrowleft | \(\circlearrowleft\) |
\Lsh | \(\Lsh\) | \upuparrows | \(\upuparrows\) | \rightrightarrows | \(\rightrightarrows\) |
\rightleftarrows | \(\rightleftarrows\) | \Rrightarrow | \(\Rrightarrow\) | \rightarrowtail | \(\rightarrowtail\) |
\looparrowright | \(\looparrowright\) | \curvearrowleft | \(\curvearrowleft\) | \circlearrowleft | \(\circlearrowleft\) |
\Lsh | \(\Lsh\) | \upuparrows | \(\upuparrows\) | \rightrightarrows | \(\rightrightarrows\) |
\rightleftarrows | \(\rightleftarrows\) | \Rrightarrow | \(\Rrightarrow\) | \rightarrowtail | \(\rightarrowtail\) |
\looparrowright | \(\looparrowright\) | \curvearrowright | \(\curvearrowright\) | \circlearrowright | \(\circlearrowright\) |
\Rsh | \(\Rsh\) | \downdownarrows | \(\downdownarrows\) | \leftleftarrows | \(\leftleftarrows\) |
\leftrightarrows | \(\leftrightarrows\) | \Lleftarrow | \(\Lleftarrow\) | \leftarrowtail | \(\leftarrowtail\) |
\looparrowleft | \(\looparrowleft\) | \curvearrowright | \(\curvearrowright\) | \circlearrowright | \(\circlearrowright\) |
\Rsh | \(\Rsh\) | \downdownarrows | \(\downdownarrows\) | \leftleftarrows | \(\leftleftarrows\) |
\leftrightarrows | \(\leftrightarrows\) | \Lleftarrow | \(\Lleftarrow\) | \leftarrowtail | \(\leftarrowtail\) |
\looparrowleft | \(\looparrowleft\) | \mapsto | \(\mapsto\) | \longmapsto | \(\longmapsto\) |
\hookrightarrow | \(\hookrightarrow\) | \hookleftarrow | \(\hookleftarrow\) | \multimap | \(\multimap\) |
\leftrightsquigarrow | \(\leftrightsquigarrow\) | \rightsquigarrow | \(\rightsquigarrow\) | | |
此外还有几个可以在箭头上下方写字的箭头。
| 代码 | 效果 |
\xleftarrow[下方公式]{上方公式} | \(A\xleftarrow[下方公式]{上方公式}B\) |
\xrightarrow[下方公式]{上方公式} | \(A\xrightarrow[下方公式]{上方公式}B\) |
\underrightarrow{演示} | \(A\underrightarrow{演示}B\) |
\underleftarrow{演示} | \(A\underleftarrow{演示}B\) |
\underleftrightarrow{演示} | \(A\underleftrightarrow{演示}B\) |
其他各类符号
| 代码 | 效果 | 代码 | 效果 | 代码 | 效果 |
\ldots | \(\ldots\) | \cdots | \(\cdots\) | \vdots | \(\vdots\) |
\ddots | \(\ddots\) | \aleph | \(\aleph\) | \prime | \(\prime\) |
\infty | \(\infty\) | \hbar | \(\hbar\) | \Box$ | \(\Box\) |
\imath | \(\imath\) | \jmath | \(\jmath\) | \surd | \(\surd\) |
\flat | \(\flat\) | \triangle | \(\triangle\) | \ell | \(\ell\) |
\top | \(\top\) | \natural | \(\natural\) | \clubsuit | \(\clubsuit\) |
\wp | \(\wp\) | \bot | \(\bot\) | \sharp | \(\sharp\) |
\diamondsuit | \(\diamondsuit\) | \Re | \(\Re\) | \| | \(|\) |
\backslash | \(\backslash\) | \heartsuit | \(\heartsuit\) | \Im | \(\Im\) |
\spadesuit | \(\spadesuit\) | \mho | \(\mho\) | | |
主要根据 维基百科 制作