{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# LaTeX Mathematics in Jupyter Notebook\n", "\n", "[LaTex](http://www.latex-project.org/) : A document preparation system\n", "\n", "LaTeX is a high-quality typesetting system; it includes features designed for the production of `technical and scientific documentation`.\n", "\n", "**MathJax**\n", "\n", "Jupyter notebook recognizes **LaTeX** code written in **markdown** cells and renders the symbols in the browser using the [MathJax](https://www.mathjax.org/) JavaScript library.\n", "\n", "## 1 插入公式\n", "\n", "Jupyter Notebook中的[LaTex](http://www.latex-project.org/)数学公式有两种形式：行中公式和独立行公式。\n", "\n", "* **行中公式(inline)** 放在行文本中与其它文字混编。 Inline (within text) formulas\n", "\n", "* **独立行公式(separate)** 单独成行。Displayed equations\n", "\n", "### 1.1 行中公式\n", "\n", "`$数学公式$`\n", "\n", "例如：\n", "\n", "```bash\n", " 我们定义:$f(x) = \\sum_{i=0}^{N}\\int_{a}^{b} g(t,i) \\text{ d}t$.\n", "```\n", "\n", "我们定义:$f(x) = \\sum_{i=0}^{N}\\int_{a}^{b} g(t,i) \\text{ d}t$.\n", "\n", "\n", "### 1.2 独立行公式\n", "\n", "`$$数学公式$$`\n", "\n", "例如：\n", "\n", "```\n", "$$J_\\alpha(x) = \\sum_{m=0}^\\infty \\frac{(-1)^m}{m! \\Gamma (m + \\alpha + 1)} {\\left({ \\frac{x}{2} }\\right)}^{2m + \\alpha}$$\n", "```\n", "公式行中显示：\n", "\n", "\n", "$$J_\\alpha(x) = \\sum_{m=0}^\\infty \\frac{(-1)^m}{m! \\Gamma (m + \\alpha + 1)} {\\left({ \\frac{x}{2} }\\right)}^{2m + \\alpha}$$\n", "\n", "\n", "如果： \n", "```\n", "$J_\\alpha(x) = \\sum_{m=0}^\\infty \\frac{(-1)^m}{m! \\Gamma (m + \\alpha + 1)} {\\left({ \\frac{x}{2} }\\right)}^{2m + \\alpha}$\n", "```\n", "是只有公式的行，但仍是行内公式，因此，公式行左显示：\n", "\n", "$J_\\alpha(x) = \\sum_{m=0}^\\infty \\frac{(-1)^m}{m! \\Gamma (m + \\alpha + 1)} {\\left({ \\frac{x}{2} }\\right)}^{2m + \\alpha}$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 1.3 多行公式对齐\n", "\n", "```txt\n", "$$\\begin{align*}\n", " &\\frac{dS}{dt}=-\\frac{\\beta SI}{N}\\\\\n", " &\\frac{dI}{dt}=\\frac{\\beta SI}{N}-\\gamma \\\\\n", " &\\frac{dR}{dt}=\\frac{\\gamma}{I} \n", "\\end{align*}\n", "$$\n", "```\n", "\n", "$$\\begin{align*}\n", " &\\frac{dS}{dt}=-\\frac{\\beta SI}{N}\\\\\n", " &\\frac{dI}{dt}=\\frac{\\beta SI}{N}-\\gamma \\\\\n", " &\\frac{dR}{dt}=\\frac{\\gamma}{I} \n", "\\end{align*}\n", "$$\n", "\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 2．上下标\n", "\n", "^表示上标, _表示下标。如果上下标的内容多于一个字符，要用{}把这些内容括起来当成一个整体。上下标是可以嵌套的，也可以同时使用。\n", "\n", "例如：\n", "```\n", "$x^{y^z}=(1+{\\rm e}^x)^{-2xy^w}$\n", "```\n", "显示:\n", "\n", "$x^{y^z}=(1+{\\rm e}^x)^{-2xy^w}$\n", "\n", "另外，如果要在左右两边都有上下标，可以用\\sideset命令。\n", "\n", "例如：\n", "```\n", "$\\sideset{^1_2}{^3_4}\\bigotimes$\n", "```\n", "显示：\n", "\n", "$\\sideset{^1_2}{^3_4}\\bigotimes$ \n", "\n", "## 3．括号和分隔符\n", "\n", "()、[]和|表示自己，`\\{\\}`表示{}。当要显示大号的括号或分隔符时，要用\\left和\\right命令。\n", "\n", "例如：\n", "```\n", "$f(x,y,z) = 3y^2z \\left( 3+\\frac{7x+5}{1+y^2} \\right)$\n", "```\n", "\n", "显示：\n", "\n", "$f(x,y,z) = 3y^2z \\left( 3+\\frac{7x+5}{1+y^2} \\right)$\n", "\n", "有时,需要 \\left. 或 \\right. 进行匹配而不显示本身。\n", "\n", "例如：\n", "```\n", "$\\left. \\frac{{\\rm d}u}{{\\rm d}x} \\right| _{x=0}$\n", "```\n", "\n", "显示：\n", "$\\left. \\frac{{\\rm d}u}{{\\rm d}x} \\right| _{x=0}$\n", "\n", "## 4．分数\n", "\n", "例如：\n", "\n", "```$\\frac{1}{3}$``` 显示：$\\frac{1}{3}$\n", "\n", "\n", "```$1 \\over 3$``` 显示： $1 \\over 3$\n", "\n", "## 5． 开方 Roots\n", "\n", "例如：\n", "```\n", "$\\sqrt{2}$\n", "\n", "$\\sqrt[n]{3}$\n", "```\n", "\n", "显示：\n", "\n", "$\\sqrt{2}$\n", "\n", "$\\sqrt[n]{3}$\n", "\n", "## 6．省略号\n", "\n", "数学公式中常见的省略号有两种:\n", "\n", " \\ldots 表示与文本底线对齐的省略号;\n", "\n", " \\cdots 表示与文本中线对齐的省略号。\n", "\n", "例如：\n", "```\n", "$f(x_1,x_2,\\ldots,x_n) = x_1^2 + x_2^2 + \\cdots + x_n^2$\n", "\n", "```\n", "显示：\n", "\n", "$f(x_1,x_2,\\ldots,x_n) = x_1^2 + x_2^2 + \\cdots + x_n^2$\n", "\n", "## 7．矢量\n", "\n", "例如：\n", "```\n", "$\\vec{a} \\cdot \\vec{b}=0$\n", "```\n", "显示：\n", "\n", "$\\vec{a} \\cdot \\vec{b}=0$\n", "\n", "## 8．积分 integrals\n", "\n", "例如：\n", "```\n", "$\\int_0^1 x^2 {\\rm d}x$\n", "```\n", "显示：\n", "\n", "$\\int_0^1 x^2 {\\rm d}x$\n", "\n", "## 9．极限运算\n", "\n", "例如：\n", "```\n", "$\\lim_{n \\rightarrow +\\infty} \\frac{1}{n(n+1)}$\n", "```\n", "\n", "显示：\n", "\n", "$\\lim_{n \\rightarrow +\\infty} \\frac{1}{n(n+1)}$\n", "\n", "## 10．累加、累乘运算\n", "\n", "例如：\n", "```\n", "$\\sum_{i=0}^n \\frac{1}{i^2}　和　\\prod_{i=0}^n \\frac{1}{i^2}$\n", "```\n", "显示：\n", "\n", "$\\sum_{i=0}^n \\frac{1}{i^2}　和　\\prod_{i=0}^n \\frac{1}{i^2}$\n", "\n", "## 11．希腊字母 Greek letters\n", "\n", "例如：\n", "```\n", " \\alpha　\\beta　\\gamma　\\Gamma　\\delta　\\Delta　\\epsilon　\n", " \\varepsilon　　\\zeta　\\eta　\\theta　\\Theta　\\vartheta\n", " \\iota　\\kappa　\\lambda　\\Lambda　\\mu　\\nu　\n", " \\xi　\\Xi　\\pi　\\Pi　\\varpi　\\rho　\n", " \\varrho　\\sigma　\\Sigma　\\varsigma　\\tau　\\upsilon　\\Upsilon\n", " \\phi　\\Phi　\\varphi　　\\chi　X　\\psi　\\Psi　\\omega　\\Omega\n", "```\n", "\n", "显示：\n", "\n", "$\\alpha　\\beta　\\gamma　\\Gamma　\\delta　\\Delta　\\epsilon$\n", "\n", "$\\varepsilon　\\zeta　\\eta　\\theta　\\Theta　\\vartheta$\n", "\n", "$\\iota　\\kappa　\\lambda　\\Lambda　\\mu　\\nu$\n", "\n", "$\\xi　\\Xi　\\pi　\\Pi　\\varpi　　\\rho$\n", "\n", "$\\varrho　\\sigma　\\Sigma　\\varsigma　\\tau　\\upsilon　\\Upsilon$\n", "\n", "$\\phi　\\Phi　\\varphi　\\chi　\\psi　\\Psi　\\omega　\\Omega$\n", "\n", "## 12．其它特殊字符\n", "\n", "#### 关系运算符 Relation Symbols：\n", "\n", "```$\\pm$``` $\\pm$\n", "\n", "```$\\times$``` $\\times$\n", "\n", "```$\\div$``` $\\div$\n", "\n", "```$\\mid$``` $\\mid$\n", "\n", "```$\\nmid$``` $\\nmid$\n", "\n", "```$\\cdot$``` $\\cdot$\n", "\n", "```$\\circ$``` $\\circ$\n", "\n", "```$\\ast$``` $\\ast$\n", "\n", "```$\\bigodot$``` $\\bigodot$\n", "\n", "```$\\bigoplus$``` $\\bigoplus$\n", "\n", "```$\\leq$``` $\\leq$\n", "\n", "```$\\geq$``` $\\geq$\n", "\n", "```$\\neq$``` $\\neq$\n", "\n", "```$\\approx$``` $\\approx$\n", "\n", "```$\\equiv$``` $\\equiv$\n", "\n", "```$\\sum$``` $\\sum$\n", "\n", "```$\\prod$``` $\\prod$\n", "\n", "```$\\coprod$``` $\\coprod$\n", "\n", "#### 集合运算符 Set Notation\n", "\n", "```$\\emptyset$``` $\\emptyset$\n", "\n", "```$\\in$``` $\\in$\n", "\n", "```$\\notin$``` $\\notin$\n", "\n", "```$\\subset$``` $\\subset$\n", "\n", "```$\\supset$``` $\\supset$\n", "\n", "```$\\subseteq$``` $\\subseteq$\n", "\n", "```$\\supseteq$``` $\\supseteq$\n", "\n", "```$\\bigcap$``` $\\bigcap$\n", "\n", "```$\\bigcup$``` $\\bigcup$\n", "\n", "```$\\bigvee$``` $\\bigvee$\n", "\n", "```$\\bigwedge$``` $\\bigwedge$\n", "\n", "```$\\biguplus$``` $\\biguplus$\n", "\n", "```$\\bigsqcup$``` $\\bigsqcup$\n", "\n", "#### 对数运算符：\n", "\n", "```$\\log$``` $\\log$\n", "\n", "```$\\lg$``` $\\lg$\n", "\n", "```$\\ln$``` $\\ln$\n", "\n", "#### 三角运算符：\n", "\n", "```$\\bot$``` $\\bot$\n", "\n", "```$\\angle$``` $\\angle$\n", "\n", "```$30^\\circ$``` $30^\\circ$\n", "\n", "```$\\sin$``` $\\sin$\n", "\n", "```$\\cos$``` $\\cos$\n", "\n", "```$\\tan$``` $\\tan$\n", "\n", "```$\\cot$``` $\\cot$\n", "\n", "```$\\sec$``` $\\sec$\n", "\n", "```$\\csc$``` $\\csc$\n", "\n", "#### 微积分运算符：\n", "\n", "```$\\prime$``` $\\prime$\n", "\n", "```$\\int$``` $\\int$\n", "\n", "```$\\iint$``` $\\iint$ \n", "\n", "```$\\iiint$``` $\\iiint$\n", "\n", "```$\\iiiint$``` $\\iiiint$\n", "\n", "```$\\oint$``` $\\oint$\n", "\n", "```$\\lim$``` $\\lim$\n", "\n", "```$\\infty$``` $\\infty$\n", "\n", "```$\\nabla$``` $\\nabla$\n", "\n", "#### 逻辑运算符：\n", "\n", "```$\\because$``` $\\because$\n", "\n", "```$\\therefore$``` $\\therefore$\n", "\n", "```$\\forall$``` $\\forall$\n", "\n", "```$\\exists$``` $\\exists$\n", "\n", "```$\\not=$``` $\\not=$\n", "\n", "```$\\not>$``` $\\not>$\n", "\n", "```$\\not\\subset>$``` $\\not\\subset$\n", "\n", "#### 戴帽符号：\n", "\n", "```$\\hat{y}$``` $\\hat{y}$\n", "\n", "```$\\check{y}$``` $\\check{y}$\n", "\n", "```$\\breve{y}$``` $\\breve{y}$\n", "\n", "```$\\dot{y}$``` $\\dot{y}$\n", "\n", "#### 连线符号：\n", "\n", "```$\\overline{a+b+c+d}$``` $\\overline{a+b+c+d}$\n", "\n", "```$\\underline{a+b+c+d}$``` $\\underline{a+b+c+d}$\n", "\n", "```$\\overbrace{a+\\underbrace{b+c}_{1.0}+d}^{2.0}$``` $\\overbrace{a+\\underbrace{b+c}_{1.0}+d}^{2.0}$\n", "\n", "#### 箭头符号：\n", "\n", "```$\\uparrow$``` $\\uparrow$\n", "\n", "```$\\downarrow$``` $\\downarrow$\n", "\n", "```$\\Uparrow$``` $\\Uparrow$\n", "\n", "```$\\Downarrow$``` $\\Downarrow$\n", "\n", "```$\\rightarrow$``` $\\rightarrow$\n", "\n", "```$\\leftarrow$``` $\\leftarrow$\n", "\n", "```$\\Rightarrow$``` $\\Rightarrow$\n", "\n", "```$\\Leftarrow$``` $\\Leftarrow$\n", "\n", "```$\\longrightarrow$``` $\\longrightarrow$\n", "\n", "```$\\longleftarrow$``` $\\longleftarrow$\n", "\n", "```$\\Longrightarrow$``` $\\Longrightarrow$\n", "\n", "```$\\Longleftarrow$``` $\\Longleftarrow$\n", "\n", "#### 输出字符\n", "\n", "空格　#　$　%　&　_　{　}　\n", "\n", "用命令：　\n", "\n", "``` ``` 空 格\n", "\n", "```$\\#``` $\\#$\n", "\n", "```$\\$$``` $\\$$\n", "\n", "```$\\%$``` $\\%$\n", "\n", "```$\\&$``` $\\&$\n", "\n", "```$\\_$``` $\\_$\n", "\n", "```$\\{$``` $\\{$\n", "\n", "```$\\}$``` $\\}$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 13．字体转换\n", "\n", "要对公式的某一部分字符进行字体转换，可以用```{\\rm ```需转换的部分字符}命令，其中,```\\rm```可以参照下表选择合适的字体。一般情况下，公式默认为意大利体。\n", "\n", "罗马体: ```$\\rm ABC$``` $\\rm ABC$\n", "\n", "意大利体: ```$\\it ABC$``` $\\it ABC$\n", "\n", "黑体: ```$\\bf 123$``` $\\bf 123$\n", "\n", "花体: ```$\\cal ABC$``` $\\cal ABC$\n", "\n", "等线体: ```$\\sf ABC$``` $\\sf ABC$\n", "\n", "数学斜体: ```$\\mit ABC$``` $\\mit ABC$\n", "\n", "打字机字体: ```$\\tt ABC$``` $\\tt ABC$　\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 14. 表格\n", "\n", "使用```$$\\begin{array}{col style…\\end{array}$$```这样的形式来创建表格.\n", " \n", "* 列样式： \n", " \n", " * c： 居中\n", " * l ：左对齐\n", " * r : 右对齐\n", "\n", "* \\hline: 在本行前加入一条直线\n", "\n", "* | :表示一条竖线\n", "\n", "* 表格中各行分隔: \\\\\n", "\n", "* 表格中各列分隔: & \n", "\n", "```\n", "$$\n", "\\begin{array}{c|lcr}\n", "n & \\text{Left} & \\text{Center} & \\text{Right} \\\\\n", "\\hline\n", "1 & 0.24 & 1 & 125 \\\\\n", "2 & -1 & 189 & -8 \\\\\n", "3 & -20 & 2000 & 1+10i \\\\\n", "\\end{array}\n", "$$\n", "```\n", "$$\n", "\\begin{array}{c|lcr}\n", "n & \\text{Left} & \\text{Center} & \\text{Right} \\\\\n", "\\hline\n", "1 & 0.24 & 1 & 125 \\\\\n", "2 & -1 & 189 & -8 \\\\\n", "3 & -20 & 2000 & 1+10i \\\\\n", "\\end{array}\n", "$$\n", "```\n", "$$\n", "% outer vertical array of arrays\n", "\\begin{array}{c}\n", "% inner horizontal array of arrays\n", "\\begin{array}{cc}\n", "% inner array of minimum values\n", "\\begin{array}{c|cccc}\n", "\\text{min} & 0 & 1 & 2 & 3\\\\\n", "\\hline\n", "0 & 0 & 0 & 0 & 0\\\\\n", "1 & 0 & 1 & 1 & 1\\\\\n", "2 & 0 & 1 & 2 & 2\\\\\n", "3 & 0 & 1 & 2 & 3\n", "\\end{array}\n", "&\n", "% inner array of maximum values\n", "\\begin{array}{c|cccc}\n", "\\text{max}&0&1&2&3\\\\\n", "\\hline\n", "0 & 0 & 1 & 2 & 3\\\\\n", "1 & 1 & 1 & 2 & 3\\\\\n", "2 & 2 & 2 & 2 & 3\\\\\n", "3 & 3 & 3 & 3 & 3\n", "\\end{array}\n", "\\end{array}\n", "\\\\\n", "% inner array of delta values\n", "\\begin{array}{c|cccc}\n", "\\Delta&0&1&2&3\\\\\n", "\\hline\n", "0 & 0 & 1 & 2 & 3\\\\\n", "1 & 1 & 0 & 1 & 2\\\\\n", "2 & 2 & 1 & 0 & 1\\\\\n", "3 & 3 & 2 & 1 & 0\n", "\\end{array}\n", "\\end{array}\n", "$$\n", "```\n", "$$\n", "% outer vertical array of arrays\n", "\\begin{array}{c}\n", "% inner horizontal array of arrays\n", "\\begin{array}{cc}\n", "% inner array of minimum values\n", "\\begin{array}{c|cccc}\n", "\\text{min} & 0 & 1 & 2 & 3\\\\\n", "\\hline\n", "0 & 0 & 0 & 0 & 0\\\\\n", "1 & 0 & 1 & 1 & 1\\\\\n", "2 & 0 & 1 & 2 & 2\\\\\n", "3 & 0 & 1 & 2 & 3\n", "\\end{array}\n", "&\n", "% inner array of maximum values\n", "\\begin{array}{c|cccc}\n", "\\text{max}&0&1&2&3\\\\\n", "\\hline\n", "0 & 0 & 1 & 2 & 3\\\\\n", "1 & 1 & 1 & 2 & 3\\\\\n", "2 & 2 & 2 & 2 & 3\\\\\n", "3 & 3 & 3 & 3 & 3\n", "\\end{array}\n", "\\end{array}\n", "\\\\\n", "% inner array of delta values\n", "\\begin{array}{c|cccc}\n", "\\Delta&0&1&2&3\\\\\n", "\\hline\n", "0 & 0 & 1 & 2 & 3\\\\\n", "1 & 1 & 0 & 1 & 2\\\\\n", "2 & 2 & 1 & 0 & 1\\\\\n", "3 & 3 & 2 & 1 & 0\n", "\\end{array}\n", "\\end{array}\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 15. 矩阵\n", "\n", "使用 ```$$\\begin{matrix}…\\end{matrix}$$```这样的形式来表示矩阵，在```\\begin```与```\\end```之间加入矩阵中的元素即可。\n", "\n", "* 行之间分隔: ``\\\\``\n", "* 列之间分隔: ```&```\n", "\n", "```\n", "$$ At=b \\tag{3}$$\n", "\n", "$$ \n", "\\left[\n", " \\begin{matrix}\n", " 4 & -1 & -1 & 0 \\\\\n", " -1 & 4 & 0 & -1 \\\\\n", " -1 & 0 & 4 & -1 \\\\\n", " 0 & -1 & -1 & 4 \n", " \\end{matrix}\n", "\\right] \n", "\\left[\n", " \\begin{matrix}\n", " node[1,1] \\\\\n", " node[1,2] \\\\\n", " node[2,1] \\\\\n", " node[2,2] \n", " \\end{matrix}\n", "\\right]\n", "=\n", "\\left[\n", " \\begin{matrix}\n", " node[0,1]+node[1,0] \\\\\n", " node[0,2]+node[1,3] \\\\\n", " node[2,0]+node[3,1] \\\\\n", " node[2,3]+node[3,2] \n", " \\end{matrix}\n", "\\right]\n", "\\tag{4}\n", "$$\n", "```\n", "$$ At=b \\tag{3}$$\n", "\n", "$$ \n", "\\left[\n", " \\begin{matrix}\n", " 4 & -1 & -1 & 0 \\\\\n", " -1 & 4 & 0 & -1 \\\\\n", " -1 & 0 & 4 & -1 \\\\\n", " 0 & -1 & -1 & 4 \n", " \\end{matrix}\n", "\\right] \n", "\\left[\n", " \\begin{matrix}\n", " node[1,1] \\\\\n", " node[1,2] \\\\\n", " node[2,1] \\\\\n", " node[2,2] \n", " \\end{matrix}\n", "\\right]\n", "=\n", "\\left[\n", " \\begin{matrix}\n", " node[0,1]+node[1,0] \\\\\n", " node[0,2]+node[1,3] \\\\\n", " node[2,0]+node[3,1] \\\\\n", " node[2,3]+node[3,2] \n", " \\end{matrix}\n", "\\right]\n", "\\tag{4}\n", "$$\n", "\n", "### 增广矩阵\n", "\n", "增广矩阵需要使用array来实现\n", "```\n", "$$ \\left[\n", " \\begin{array}{cc|c}\n", " 1&2&3\\\\\n", " 4&5&6\n", " \\end{array}\n", " \\right]\n", "$$\n", "```\n", "$$ \\left[\n", " \\begin{array}{cc|c}\n", " 1&2&3\\\\\n", " 4&5&6\n", " \\end{array}\n", " \\right]\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 16. Crossing things out\n", "\n", "Use `\\require{cancel}` in the first formula in your post that requires cancelling; you need it only once per page. Then use:\n", "\n", "`$\\require{cancel}$` $\\require{cancel}$\n", "\n", "y+\\cancel{x} $y+\\cancel{x}$\n", "\n", "\\cancel{y+x} $\\cancel{y+x}$ \n", "\n", "y+\\bcancel{x} $y+\\bcancel{x}$\n", "\n", "y+\\xcancel{x} $y+\\xcancel{x}$\n", "\n", "y+\\cancelto{0}{x} $y+\\cancelto{0}{x}$ \n", "\n", "\n", "\\frac{1\\cancel9}{\\cancel95} = \\frac15 $\\frac{1\\cancel9}{\\cancel95} = \\frac15$\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 17 Color\n", "\n", "The package xcolor, described in Colors, allows us to add color to our equations. For example, \n", "\n", " $k = {\\color{red}x} \\mathbin{\\color{blue}-} 2$\n", " \n", " \n", " $k = {\\color{red}x} \\mathbin{\\color{blue}-} 2 $ \n", " \n", "The only problem is that this disrupts the default LaTeX formatting around the `-` operator. To fix this, we enclose it in a `\\mathbin` environment, since `-` is a binary operator" ] }, { "cell_type": "markdown", "metadata": { "collapsed": true }, "source": [ "\n", "## 参考：\n", "\n", "1. [LaTeX](http://www.latex-project.org/)\n", "\n", "\n", "2. [MathJax](https://www.mathjax.org/)\n", "\n", "\n", "3. [LaTex mathematics for equations: LaTeX/Mathematics](https://en.wikibooks.org/wiki/LaTeX/Mathematics)\n", "\n", "\n", "4. https://math.meta.stackexchange.com/questions/5020/mathjax-basic-tutorial-and-quick-reference/5044\n" ] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.8.7" }, "latex_envs": { "LaTeX_envs_menu_present": true, "autoclose": false, "autocomplete": true, "bibliofile": "biblio.bib", "cite_by": "apalike", "current_citInitial": 1, "eqLabelWithNumbers": true, "eqNumInitial": 1, "hotkeys": { "equation": "Ctrl-E", "itemize": "Ctrl-I" }, "labels_anchors": false, "latex_user_defs": false, "report_style_numbering": false, "user_envs_cfg": false }, "toc": { "base_numbering": 1, "nav_menu": {}, "number_sections": false, "sideBar": true, "skip_h1_title": false, "title_cell": "Table of Contents", "title_sidebar": "Contents", "toc_cell": false, "toc_position": { "height": "calc(100% - 180px)", "left": "10px", "top": "150px", "width": "218.663px" }, "toc_section_display": true, "toc_window_display": true }, "widgets": { "state": {}, "version": "1.1.2" } }, "nbformat": 4, "nbformat_minor": 1 }