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134 lines
5.8 KiB
134 lines
5.8 KiB
7 years ago
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<?xml version="1.0" encoding="utf-8" ?>
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<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd">
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<html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en" lang="en">
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<head>
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<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
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<meta name="generator" content="Docutils 0.14: http://docutils.sourceforge.net/" />
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<title>Mathematics</title>
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<script type="text/javascript" src="/usr/share/javascript/mathjax/MathJax.js?config=TeX-AMS_CHTML"></script>
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<link rel="stylesheet" href="../input/data/html4css1.css" type="text/css" />
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</head>
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<body>
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<div class="document" id="mathematics">
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<h1 class="title">Mathematics</h1>
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<p>Docutils supports inline math with the prefix or postfix <tt class="docutils literal">:math:</tt>
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role specificator, <span class="math">\(n! + \sin(x_n^2)\)</span> and <span class="math">\(A_\text{c} =
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\frac{\pi}{4} d^2\)</span>, as well as displayed math via the
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<cite>math</cite> directive:</p>
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<div class="math">
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\begin{equation*}
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f(\epsilon) = \frac{1}{1 + \exp\left(\frac{\varepsilon}{k_\text{B}T}\right)}
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\end{equation*}
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</div>
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<p>Content may start on the first line of the directive, e.g.</p>
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<div class="math">
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\begin{equation*}
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N = \frac{\text{number of apples}}{7}
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\end{equation*}
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</div>
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<p>Equations can be labeled with a reference name using the <tt class="docutils literal">:name:</tt> option.
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See <a class="reference internal" href="#eq-m">eq:M</a> and <a class="reference internal" href="#eq-schrodinger">eq:schrödinger</a> below.</p>
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<p>The determinant of the matrix</p>
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<div class="math" id="eq-m">
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\begin{equation*}
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\mathbf{M} = \left(\begin{matrix}a&b\\c&d\end{matrix}\right)
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\end{equation*}
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</div>
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<p>is <span class="math">\(|\mathbf{M}| = ad - bc\)</span>.</p>
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<p>More than one display math block can be put in one math directive.
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For example, the following sum and integral with limits:</p>
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<div class="math">
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\begin{equation*}
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\int_0^1 x^n dx = \frac{1}{n + 1}
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\end{equation*}
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</div>
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<div class="math">
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\begin{equation*}
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\sum_{n=1}^m n = \frac{m(m+1)}{2}
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\end{equation*}
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</div>
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<p>LaTeX-supported Unicode math symbols can be used in math roles and
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directives:</p>
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<p>The Schrödinger equation</p>
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<div class="math" id="eq-schrodinger">
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\begin{equation*}
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i\hbar \frac{\partial }{\partial t}\Psi = \hat{H}\Psi ,
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\end{equation*}
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</div>
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<p>with the <em>wave function</em> <span class="math">\(\Psi \)</span>, describes how the quantum state of a
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physical system changes in time.</p>
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<dl class="docutils">
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<dt>Math-Accents:</dt>
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<dd><table border="1" class="colwidths-given borderless first last docutils">
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<colgroup>
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<col width="33%" />
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<col width="33%" />
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<col width="33%" />
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</colgroup>
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<tbody valign="top">
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<tr><td><span class="math">\(\acute{a}\)</span> <tt class="docutils literal">\acute{a}</tt></td>
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<td><span class="math">\(\dot{t}\)</span> <tt class="docutils literal">\dot{t}</tt></td>
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<td><span class="math">\(\hat{\gamma}\)</span> <tt class="docutils literal"><span class="pre">\hat{\gamma}</span></tt></td>
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</tr>
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<tr><td><span class="math">\(\grave{a}\)</span> <tt class="docutils literal">\grave{a}</tt></td>
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<td><span class="math">\(\ddot{t}\)</span> <tt class="docutils literal">\ddot{t}</tt></td>
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<td><span class="math">\(\tilde{\alpha}\)</span> <tt class="docutils literal"><span class="pre">\tilde{\alpha}</span></tt></td>
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</tr>
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<tr><td><span class="math">\(\breve{x}\)</span> <tt class="docutils literal">\breve{x}</tt></td>
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<td><span class="math">\(\dddot{t}\)</span> <tt class="docutils literal">\dddot{t}</tt></td>
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<td><span class="math">\(\vec{\imath}\)</span> <tt class="docutils literal"><span class="pre">\vec{\imath}</span></tt></td>
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</tr>
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<tr><td><span class="math">\(\check{a}\)</span> <tt class="docutils literal">\check{a}</tt></td>
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<td><span class="math">\(\bar{a}\)</span> <tt class="docutils literal">\bar{a}</tt></td>
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<td><span class="math">\(\vec{R}\)</span> <tt class="docutils literal">\vec{R}</tt></td>
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</tr>
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</tbody>
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</table>
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</dd>
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</dl>
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<!-- \widetilde{xxx}
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\widehat{xxx} -->
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<p>Modulation Transfer Function:</p>
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<div class="math">
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\begin{equation*}
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\text{MTF} = \left|\frac{\mathcal{F}\{s(x)\}}
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{\mathcal{F}\{ s(x)\} |_{\omega _{x}=0}}\right|
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= \mathrm{abs}\left(\frac
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{\int _{-\infty }^{\infty }s(x) \mathrm{e}^{\mathrm{i}\omega _{x}x}\mathrm{d}{x}}
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{\int _{-\infty }^{\infty }s(x)\mathrm{d}{x}}
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\right).
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\end{equation*}
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</div>
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<p>Math split over two lines: If a double backslash is detected outside a
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<tt class="docutils literal"><span class="pre">\begin{...}</span> <span class="pre">\end{...}</span></tt> pair, the math code is wrapped in an <a class="reference external" href="ftp://ftp.ams.org/ams/doc/amsmath/short-math-guide.pdf">AMSmath</a>
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<tt class="docutils literal">align</tt> environment:</p>
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<div class="math">
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\begin{align*}
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s_{\mathrm{out}}(x) & = s_{\mathrm{in}}(x') * s_\delta (x-x') \\
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& = \int s_{\mathrm{in}}(x')s_\delta (x-x')\mathrm{d}x'
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\end{align*}
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</div>
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<p>Cases ("manually", with <tt class="docutils literal">matrix</tt> environment):</p>
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<div class="math">
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\begin{equation*}
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\mathrm{sgn}(x) = \left\{\begin{matrix}
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-1 & x<0\\
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1 & x>0
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\end{matrix}\right.
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\end{equation*}
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</div>
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<p>Cases with the <a class="reference external" href="ftp://ftp.ams.org/ams/doc/amsmath/short-math-guide.pdf">AMSmath</a> <tt class="docutils literal">cases</tt> environment (not (yet) supported by
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HTML writers with <tt class="docutils literal"><span class="pre">--math-output=MathML</span></tt>):</p>
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<div class="math">
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\begin{equation*}
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\mathrm{sgn}(x) = \begin{cases}
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-1 & x<0\\
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1 & x>0
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\end{cases}
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\end{equation*}
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</div>
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</div>
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</body>
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</html>
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