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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 0.22: https://docutils.sourceforge.io/" />
<title>Mathematics</title>
<link rel="stylesheet" href="../input/data/html4css1.css" type="text/css" />
<link rel="stylesheet" href="../input/data/math.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="formula"><i>n</i>! + sin(<i>x</i><span class="scripts"><sup class="script">2</sup><sub class="script"><i>n</i></sub></span>)</span> and <span class="formula"><i>A</i><sub><span class="text">c</span></sub> = <span class="fraction"><span class="ignored">(</span><span class="numerator"><i>π</i></span><span class="ignored">)/(</span><span class="denominator">4</span><span class="ignored">)</span></span><i>d</i><sup>2</sup></span>, as well as displayed math via the
<cite>math</cite> directive:</p>
<div class="formula">
<i>f</i>(<i>ϵ</i>) = <span class="fraction"><span class="ignored">(</span><span class="numerator">1</span><span class="ignored">)/(</span><span class="denominator">1 + exp<span class="array"><span class="arrayrow"><span class="bracket align-left">⎛</span></span><span class="arrayrow"><span class="bracket align-left">⎝</span></span></span><span class="fraction"><span class="ignored">(</span><span class="numerator"><i>ε</i></span><span class="ignored">)/(</span><span class="denominator"><i>k</i><sub><span class="text">B</span></sub><i>T</i></span><span class="ignored">)</span></span><span class="array"><span class="arrayrow"><span class="bracket align-right">⎞</span></span><span class="arrayrow"><span class="bracket align-right">⎠</span></span></span></span><span class="ignored">)</span></span>
</div>
<p>Content may start on the first line of the directive, e.g.</p>
<div class="formula">
<i>N</i> = <span class="fraction"><span class="ignored">(</span><span class="numerator"><span class="text">number of apples</span></span><span class="ignored">)/(</span><span class="denominator">7</span><span class="ignored">)</span></span>
</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="formula" id="eq-m">
<b>M</b> = <span class="array"><span class="arrayrow"><span class="bracket align-left">⎛</span></span><span class="arrayrow"><span class="bracket align-left">⎜</span></span><span class="arrayrow"><span class="bracket align-left">⎝</span></span></span><span class="array"><span class="arrayrow">
<span class="arraycell align-c">
<i>a</i>
</span>
<span class="arraycell align-c">
<i>b</i>
</span>
</span>
<span class="arrayrow">
<span class="arraycell align-c">
</span>
<span class="arraycell align-c">
</span>
</span>
<span class="arrayrow">
<span class="arraycell align-c">
<i>c</i>
</span>
<span class="arraycell align-c">
<i>d</i>
</span>
</span>
</span><span class="array"><span class="arrayrow"><span class="bracket align-right">⎞</span></span><span class="arrayrow"><span class="bracket align-right">⎟</span></span><span class="arrayrow"><span class="bracket align-right">⎠</span></span></span>
</div>
<p>is <span class="formula">|<b>M</b>| = <i>ad</i> − <i>bc</i></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="formula">
<span class="limits"><sup class="limit">1</sup><span class="limit"><span class="bigoperator integral">∫</span></span><sub class="limit">0</sub></span><i>x</i><sup><i>n</i></sup><i>dx</i> = <span class="fraction"><span class="ignored">(</span><span class="numerator">1</span><span class="ignored">)/(</span><span class="denominator"><i>n</i> + 1</span><span class="ignored">)</span></span>
</div>
<div class="formula">
<span class="limits"><sup class="limit"><i>m</i></sup><span class="limit"><span class="bigoperator">∑</span></span><sub class="limit"><i>n</i> = 1</sub></span><i>n</i> = <span class="fraction"><span class="ignored">(</span><span class="numerator"><i>m</i>(<i>m</i> + 1)</span><span class="ignored">)/(</span><span class="denominator">2</span><span class="ignored">)</span></span>
</div>
<p>LaTeX-supported Unicode math symbols can be used in math roles and
directives:</p>
<p>The Schrödinger equation</p>
<div class="formula" id="eq-schrodinger">
<i>i</i><i>ħ</i><span class="fraction"><span class="ignored">(</span><span class="numerator">∂</span><span class="ignored">)/(</span><span class="denominator">∂<i>t</i></span><span class="ignored">)</span></span>Ψ = <i>Ĥ</i>Ψ,
</div>
<p>with the <em>wave function</em> <span class="formula">Ψ</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="formula"><i>á</i></span> <tt class="docutils literal">\acute{a}</tt></td>
<td><span class="formula"><i>ṫ</i></span> <tt class="docutils literal">\dot{t}</tt></td>
<td><span class="formula"><i>γ̂</i></span> <tt class="docutils literal"><span class="pre">\hat{\gamma}</span></tt></td>
</tr>
<tr><td><span class="formula"><i>à</i></span> <tt class="docutils literal">\grave{a}</tt></td>
<td><span class="formula"><i>ẗ</i></span> <tt class="docutils literal">\ddot{t}</tt></td>
<td><span class="formula"><i>α̃</i></span> <tt class="docutils literal"><span class="pre">\tilde{\alpha}</span></tt></td>
</tr>
<tr><td><span class="formula"><i>x̆</i></span> <tt class="docutils literal">\breve{x}</tt></td>
<td><span class="formula"><span class="embellished"><sup><span class="smallsymbol">⋯</span></sup><span class="base"><i>t</i></span></span></span> <tt class="docutils literal">\dddot{t}</tt></td>
<td><span class="formula"><span class="embellished"><sup><span class="smallsymbol">→</span></sup><span class="base"><i>ı</i></span></span></span> <tt class="docutils literal"><span class="pre">\vec{\imath}</span></tt></td>
</tr>
<tr><td><span class="formula"><i>ǎ</i></span> <tt class="docutils literal">\check{a}</tt></td>
<td><span class="formula"><span class="embellished"><sup>‒</sup><span class="base"><i>a</i></span></span></span> <tt class="docutils literal">\bar{a}</tt></td>
<td><span class="formula"><span class="embellished"><sup><span class="smallsymbol">→</span></sup><span class="base"><i>R</i></span></span></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="formula">
<span class="text">MTF</span> = <span class="array"><span class="arrayrow"><span class="bracket align-left">|</span></span><span class="arrayrow"><span class="bracket align-left">|</span></span></span><span class="fraction"><span class="ignored">(</span><span class="numerator">ℱ{<i>s</i>(<i>x</i>)}</span><span class="ignored">)/(</span><span class="denominator">ℱ{<i>s</i>(<i>x</i>)}|<sub><i>ω</i><sub><i>x</i></sub> = 0</sub></span><span class="ignored">)</span></span><span class="array"><span class="arrayrow"><span class="bracket align-right">|</span></span><span class="arrayrow"><span class="bracket align-right">|</span></span></span> = <span class="mathrm"> abs</span><span class="array"><span class="arrayrow"><span class="bracket align-left">⎛</span></span><span class="arrayrow"><span class="bracket align-left">⎝</span></span></span><span class="fraction"><span class="ignored">(</span><span class="numerator"><span class="limits"><span class="limit"><span class="bigoperator integral">∫</span></span></span><span class="scripts"><sup class="script">∞</sup><sub class="script"> − ∞</sub></span><i>s</i>(<i>x</i>)<span class="mathrm">e</span><sup><span class="mathrm">i</span><i>ω</i><sub><i>x</i></sub><i>x</i></sup><span class="mathrm">d</span><i>x</i></span><span class="ignored">)/(</span><span class="denominator"><span class="limits"><span class="limit"><span class="bigoperator integral">∫</span></span></span><span class="scripts"><sup class="script">∞</sup><sub class="script"> − ∞</sub></span><i>s</i>(<i>x</i>)<span class="mathrm">d</span><i>x</i></span><span class="ignored">)</span></span><span class="array"><span class="arrayrow"><span class="bracket align-right">⎞</span></span><span class="arrayrow"><span class="bracket align-right">⎠</span></span></span>.
</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="formula">
<span class="environment align"><span class="arrayrow">
<span class="arraycell align-r">
<i>s</i><sub><span class="mathrm">out</span></sub>(<i>x</i>)
</span>
<span class="arraycell align-l">
= <i>s</i><sub><span class="mathrm">in</span></sub>(<i>x</i>’)*<i>s</i><sub><i>δ</i></sub>(<i>x</i> − <i>x</i>’)
</span>
</span>
<span class="arrayrow">
<span class="arraycell align-r">
</span>
<span class="arraycell align-l">
</span>
</span>
<span class="arrayrow">
<span class="arraycell align-r">
</span>
<span class="arraycell align-l">
= <span class="limits"><span class="limit"><span class="bigoperator integral">∫</span></span></span><i>s</i><sub><span class="mathrm">in</span></sub>(<i>x</i>’)<i>s</i><sub><i>δ</i></sub>(<i>x</i> − <i>x</i>’)<span class="mathrm">d</span><i>x</i>’
</span>
</span>
</span>
</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="formula">
<span class="mathrm">sgn</span>(<i>x</i>) = <span class="array"><span class="arrayrow"><span class="bracket align-l">⎧</span></span><span class="arrayrow"><span class="bracket align-l">⎨</span></span><span class="arrayrow"><span class="bracket align-l">⎩</span></span></span><span class="bracketcases">
<span class="arrayrow">
<span class="case align-l">
− 1
</span>
<span class="case align-l">
<i>x</i> < 0
</span>
</span>
<span class="arrayrow">
<span class="case align-l">
</span>
<span class="case align-l">
</span>
</span>
<span class="arrayrow">
<span class="case align-l">
<span class="phantom"> − </span>1
</span>
<span class="case align-l">
<i>x</i> > 0
</span>
</span>
</span>
</div>
</div>
</body>
</html>
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