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<?xml version="1.0" encoding="utf-8"?>
<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd">
<html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en" lang="en">
<head>
<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
<meta name="generator" content="Docutils 0.20.1b.dev: https://docutils.sourceforge.io/" />
<title>Mathematics</title>
<script type="text/javascript" src="/usr/share/javascript/mathjax/MathJax.js?config=TeX-AMS_CHTML"></script>
<link rel="stylesheet" href="../input/data/html4css1.css" type="text/css" />
</head>
<body>
<div class="document" id="mathematics">
<h1 class="title">Mathematics</h1>
<p>Docutils supports inline math with the prefix or postfix <tt class="docutils literal">:math:</tt>
role specificator, <span class="math">\(n! + \sin(x_n^2)\)</span> and <span class="math">\(A_\text{c} =
\frac{\pi}{4} d^2\)</span>, as well as displayed math via the
<cite>math</cite> directive:</p>
<div class="math">
\begin{equation*}
f(\epsilon) = \frac{1}{1 + \exp\left(\frac{\varepsilon}{k_\text{B}T}\right)}
\end{equation*}
</div>
<p>Content may start on the first line of the directive, e.g.</p>
<div class="math">
\begin{equation*}
N = \frac{\text{number of apples}}{7}
\end{equation*}
</div>
<p>Equations can be labeled with a reference name using the <tt class="docutils literal">:name:</tt> option.
See <a class="reference internal" href="#eq-m">eq:M</a> and <a class="reference internal" href="#eq-schrodinger">eq:schrödinger</a> below.</p>
<p>The determinant of the matrix</p>
<div class="math" id="eq-m">
\begin{equation*}
\mathbf{M} = \left(\begin{matrix}a&b\\c&d\end{matrix}\right)
\end{equation*}
</div>
<p>is <span class="math">\(|\mathbf{M}| = ad - bc\)</span>.</p>
<p>More than one display math block can be put in one math directive.
For example, the following sum and integral with limits:</p>
<div class="math">
\begin{equation*}
\int_0^1 x^n dx = \frac{1}{n + 1}
\end{equation*}
</div>
<div class="math">
\begin{equation*}
\sum_{n=1}^m n = \frac{m(m+1)}{2}
\end{equation*}
</div>
<p>LaTeX-supported Unicode math symbols can be used in math roles and
directives:</p>
<p>The Schrödinger equation</p>
<div class="math" id="eq-schrodinger">
\begin{equation*}
i\hbar \frac{\partial }{\partial t}\Psi = \hat{H}\Psi ,
\end{equation*}
</div>
<p>with the <em>wave function</em> <span class="math">\(\Psi \)</span>, describes how the quantum state of a
physical system changes in time.</p>
<dl class="docutils">
<dt>Math-Accents:</dt>
<dd><table border="1" class="borderless first last docutils" style="width: 100%">
<colgroup>
<col width="33%" />
<col width="33%" />
<col width="33%" />
</colgroup>
<tbody valign="top">
<tr><td><span class="math">\(\acute{a}\)</span> <tt class="docutils literal">\acute{a}</tt></td>
<td><span class="math">\(\dot{t}\)</span> <tt class="docutils literal">\dot{t}</tt></td>
<td><span class="math">\(\hat{\gamma}\)</span> <tt class="docutils literal"><span class="pre">\hat{\gamma}</span></tt></td>
</tr>
<tr><td><span class="math">\(\grave{a}\)</span> <tt class="docutils literal">\grave{a}</tt></td>
<td><span class="math">\(\ddot{t}\)</span> <tt class="docutils literal">\ddot{t}</tt></td>
<td><span class="math">\(\tilde{\alpha}\)</span> <tt class="docutils literal"><span class="pre">\tilde{\alpha}</span></tt></td>
</tr>
<tr><td><span class="math">\(\breve{x}\)</span> <tt class="docutils literal">\breve{x}</tt></td>
<td><span class="math">\(\dddot{t}\)</span> <tt class="docutils literal">\dddot{t}</tt></td>
<td><span class="math">\(\vec{\imath}\)</span> <tt class="docutils literal"><span class="pre">\vec{\imath}</span></tt></td>
</tr>
<tr><td><span class="math">\(\check{a}\)</span> <tt class="docutils literal">\check{a}</tt></td>
<td><span class="math">\(\bar{a}\)</span> <tt class="docutils literal">\bar{a}</tt></td>
<td><span class="math">\(\vec{R}\)</span> <tt class="docutils literal">\vec{R}</tt></td>
</tr>
</tbody>
</table>
</dd>
</dl>
<!-- \widetilde{xxx}
\widehat{xxx} -->
<p>Modulation Transfer Function:</p>
<div class="math">
\begin{equation*}
\text{MTF} = \left|\frac{\mathcal{F}\{s(x)\}}
{\mathcal{F}\{ s(x)\} |_{\omega _{x}=0}}\right|
= \mathrm{abs}\left(\frac
{\int _{-\infty }^{\infty }s(x) \mathrm{e}^{\mathrm{i}\omega _{x}x}\mathrm{d}{x}}
{\int _{-\infty }^{\infty }s(x)\mathrm{d}{x}}
\right).
\end{equation*}
</div>
<p>Math split over two lines: If a double backslash is detected outside a
<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>
<tt class="docutils literal">align</tt> environment:</p>
<div class="math">
\begin{align*}
s_{\mathrm{out}}(x) & = s_{\mathrm{in}}(x') * s_\delta (x-x') \\
& = \int s_{\mathrm{in}}(x')s_\delta (x-x')\mathrm{d}x'
\end{align*}
</div>
<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:</p>
<div class="math">
\begin{equation*}
\mathrm{sgn}(x) = \begin{cases}
-1 & x<0\\
\phantom{-} 1 & x>0
\end{cases}
\end{equation*}
</div>
</div>
</body>
</html>
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