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<!DOCTYPE html>
<html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en" lang="en">
<head>
<meta charset="utf-8"/>
<meta name="generator" content="Docutils 0.14: http://docutils.sourceforge.net/" />
<title>Mathematics</title>
<link rel="stylesheet" href="../input/data/minimal.css" type="text/css" />
<link rel="stylesheet" href="../input/data/plain.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 <span class="docutils literal">:math:</span>
role specificator, <math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow><mi>n</mi><mo>!</mo><mo>+</mo><mo>sin</mo><mo>(</mo><msubsup><mi>x</mi><mi>n</mi><mn>2</mn></msubsup><mo>)</mo></mrow></math> and <math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow><msub><mi>A</mi><mtext>c</mtext></msub><mo>=</mo><mfrac>
<mrow><mi>π</mi></mrow>
<mrow><mn>4</mn></mrow></mfrac><msup><mi>d</mi><mn>2</mn></msup></mrow></math>, as well as displayed math via the
<cite>math</cite> directive:</p>
<div>
<math xmlns="http://www.w3.org/1998/Math/MathML" mode="display">
<mtable>
<mtr>
<mtd><mi>f</mi><mo>(</mo><mi>ϵ</mi><mo>)</mo><mo>=</mo><mfrac>
<mrow><mn>1</mn></mrow>
<mrow><mn>1</mn><mo>+</mo><mo>exp</mo><mfenced open="(" close=")">
<mrow><mfrac>
<mrow><mi>ε</mi></mrow>
<mrow><msub><mi>k</mi><mtext>B</mtext></msub><mi>T</mi></mrow></mfrac></mrow></mfenced></mrow></mfrac></mtd></mtr></mtable></math>
</div>
<p>Content may start on the first line of the directive, e.g.</p>
<div>
<math xmlns="http://www.w3.org/1998/Math/MathML" mode="display">
<mtable>
<mtr>
<mtd><mi>N</mi><mo>=</mo><mfrac>
<mrow><mtext>number of apples</mtext></mrow>
<mrow><mn>7</mn></mrow></mfrac></mtd></mtr></mtable></math>
</div>
<p>Equations can be labeled with a reference name using the <span class="docutils literal">:name:</span> 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 id="eq-m">
<math xmlns="http://www.w3.org/1998/Math/MathML" mode="display">
<mtable>
<mtr>
<mtd><mstyle fontweight="bold">
<mrow><mi>M</mi></mrow></mstyle><mo>=</mo><mfenced open="(" close=")">
<mrow>
<mtable>
<mtr>
<mtd><mi>a</mi></mtd>
<mtd><mi>b</mi></mtd></mtr>
<mtr>
<mtd><mi>c</mi></mtd>
<mtd><mi>d</mi></mtd></mtr></mtable></mrow></mfenced></mtd></mtr></mtable></math>
</div>
<p>is <math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow><mo>|</mo><mstyle fontweight="bold">
<mrow><mi>M</mi></mrow></mstyle><mo>|</mo><mo>=</mo><mi>a</mi><mi>d</mi><mo>-</mo><mi>b</mi><mi>c</mi></mrow></math>.</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>
<math xmlns="http://www.w3.org/1998/Math/MathML" mode="display">
<mtable>
<mtr>
<mtd><munderover><mo>∫</mo><mn>0</mn><mn>1</mn></munderover><msup><mi>x</mi><mi>n</mi></msup><mi>d</mi><mi>x</mi><mo>=</mo><mfrac>
<mrow><mn>1</mn></mrow>
<mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mtd></mtr></mtable></math>
</div>
<div>
<math xmlns="http://www.w3.org/1998/Math/MathML" mode="display">
<mtable>
<mtr>
<mtd><munderover><mo>∑</mo>
<mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mi>m</mi></munderover><mi>n</mi><mo>=</mo><mfrac>
<mrow><mi>m</mi><mo>(</mo><mi>m</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow>
<mrow><mn>2</mn></mrow></mfrac></mtd></mtr></mtable></math>
</div>
<p>LaTeX-supported Unicode math symbols can be used in math roles and
directives:</p>
<p>The Schrödinger equation</p>
<div id="eq-schrodinger">
<math xmlns="http://www.w3.org/1998/Math/MathML" mode="display">
<mtable>
<mtr>
<mtd><mi>i</mi><mi>ℏ</mi><mfrac>
<mrow><mo>∂</mo></mrow>
<mrow><mo>∂</mo><mi>t</mi></mrow></mfrac><mo>Ψ</mo><mo>=</mo><mover>
<mrow><mi>H</mi></mrow><mo>^</mo></mover><mo>Ψ</mo><mo>,</mo></mtd></mtr></mtable></math>
</div>
<p>with the <em>wave function</em> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow><mo>Ψ</mo></mrow></math>, describes how the quantum state of a
physical system changes in time.</p>
<dl>
<dt>Math-Accents:</dt>
<dd><table class="colwidths-given borderless">
<colgroup>
<col style="width: 33%" />
<col style="width: 33%" />
<col style="width: 33%" />
</colgroup>
<tbody>
<tr><td><p><math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow><mover>
<mrow><mi>a</mi></mrow><mo>´</mo></mover></mrow></math>      <span class="docutils literal">\acute{a}</span></p></td>
<td><p><math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow><mover>
<mrow><mi>t</mi></mrow><mo>˙</mo></mover></mrow></math>        <span class="docutils literal">\dot{t}</span></p></td>
<td><p><math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow><mover>
<mrow><mi>γ</mi></mrow><mo>^</mo></mover></mrow></math>   <span class="docutils literal"><span class="pre">\hat{\gamma}</span></span></p></td>
</tr>
<tr><td><p><math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow><mover>
<mrow><mi>a</mi></mrow><mo>`</mo></mover></mrow></math>      <span class="docutils literal">\grave{a}</span></p></td>
<td><p><math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow><mover>
<mrow><mi>t</mi></mrow><mo>¨</mo></mover></mrow></math>       <span class="docutils literal">\ddot{t}</span></p></td>
<td><p><math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow><mover>
<mrow><mi>α</mi></mrow><mo>˜</mo></mover></mrow></math> <span class="docutils literal"><span class="pre">\tilde{\alpha}</span></span></p></td>
</tr>
<tr><td><p><math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow><mover>
<mrow><mi>x</mi></mrow><mo>˘</mo></mover></mrow></math>      <span class="docutils literal">\breve{x}</span></p></td>
<td><p><math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow><mover>
<mrow><mi>t</mi></mrow><mo>⃛</mo></mover></mrow></math>      <span class="docutils literal">\dddot{t}</span></p></td>
<td><p><math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow><mover>
<mrow><mi>ı</mi></mrow><mo>⃗</mo></mover></mrow></math>   <span class="docutils literal"><span class="pre">\vec{\imath}</span></span></p></td>
</tr>
<tr><td><p><math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow><mover>
<mrow><mi>a</mi></mrow><mo>ˇ</mo></mover></mrow></math>      <span class="docutils literal">\check{a}</span></p></td>
<td><p><math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow><mover>
<mrow><mi>a</mi></mrow><mo>¯</mo></mover></mrow></math>        <span class="docutils literal">\bar{a}</span></p></td>
<td><p><math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow><mover>
<mrow><mi>R</mi></mrow><mo>⃗</mo></mover></mrow></math>        <span class="docutils literal">\vec{R}</span></p></td>
</tr>
</tbody>
</table>
</dd>
</dl>
<!-- \widetilde{xxx}
\widehat{xxx} -->
<p>Modulation Transfer Function:</p>
<div>
<math xmlns="http://www.w3.org/1998/Math/MathML" mode="display">
<mtable>
<mtr>
<mtd><mtext>MTF</mtext><mo>=</mo><mfenced open="|" close="|">
<mrow><mfrac>
<mrow><mi>ℱ</mi><mo>{</mo><mi>s</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>}</mo></mrow>
<mrow><mi>ℱ</mi><mo>{</mo><mi>s</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>}</mo><msub><mo>|</mo>
<mrow><msub><mi>ω</mi>
<mrow><mi>x</mi></mrow></msub><mo>=</mo><mn>0</mn></mrow></msub></mrow></mfrac></mrow></mfenced><mo>=</mo><mtext>abs</mtext><mfenced open="(" close=")">
<mrow><mfrac>
<mrow><munderover><mo>∫</mo>
<mrow><mo>-</mo><mo>∞</mo></mrow>
<mrow><mo>∞</mo></mrow></munderover><mi>s</mi><mo>(</mo><mi>x</mi><mo>)</mo><msup><mtext>e</mtext>
<mrow><mtext>i</mtext><msub><mi>ω</mi>
<mrow><mi>x</mi></mrow></msub><mi>x</mi></mrow></msup><mtext>d</mtext>
<mrow><mi>x</mi></mrow></mrow>
<mrow><munderover><mo>∫</mo>
<mrow><mo>-</mo><mo>∞</mo></mrow>
<mrow><mo>∞</mo></mrow></munderover><mi>s</mi><mo>(</mo><mi>x</mi><mo>)</mo><mtext>d</mtext>
<mrow><mi>x</mi></mrow></mrow></mfrac></mrow></mfenced><mo>.</mo></mtd></mtr></mtable></math>
</div>
<p>Math split over two lines: If a double backslash is detected outside a
<span class="docutils literal"><span class="pre">\begin{...}</span> <span class="pre">\end{...}</span></span> 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>
<span class="docutils literal">align</span> environment:</p>
<div>
<math xmlns="http://www.w3.org/1998/Math/MathML" mode="display">
<mtable>
<mtr>
<mtd><msub><mi>s</mi>
<mrow><mtext>out</mtext></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo></mtd>
<mtd><mo>=</mo><msub><mi>s</mi>
<mrow><mtext>in</mtext></mrow></msub><mo>(</mo><mi>x</mi><mo>'</mo><mo>)</mo><mo>*</mo><msub><mi>s</mi><mi>δ</mi></msub><mo>(</mo><mi>x</mi><mo>-</mo><mi>x</mi><mo>'</mo><mo>)</mo></mtd></mtr>
<mtr>
<mtd></mtd>
<mtd><mo>=</mo><mo>∫</mo><msub><mi>s</mi>
<mrow><mtext>in</mtext></mrow></msub><mo>(</mo><mi>x</mi><mo>'</mo><mo>)</mo><msub><mi>s</mi><mi>δ</mi></msub><mo>(</mo><mi>x</mi><mo>-</mo><mi>x</mi><mo>'</mo><mo>)</mo><mtext>d</mtext><mi>x</mi><mo>'</mo></mtd></mtr></mtable></math>
</div>
<p>Cases (&quot;manually&quot;, with <span class="docutils literal">matrix</span> environment):</p>
<div>
<math xmlns="http://www.w3.org/1998/Math/MathML" mode="display">
<mtable>
<mtr>
<mtd><mtext>sgn</mtext><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mfenced open="{" close="">
<mrow>
<mtable>
<mtr>
<mtd><mo>-</mo><mn>1</mn></mtd>
<mtd><mi>x</mi><mo>&lt;</mo><mn>0</mn></mtd></mtr>
<mtr>
<mtd><mn>1</mn></mtd>
<mtd><mi>x</mi><mo>&gt;</mo><mn>0</mn></mtd></mtr></mtable></mrow></mfenced></mtd></mtr></mtable></math>
</div>
<p>Cases with the <a class="reference external" href="ftp://ftp.ams.org/ams/doc/amsmath/short-math-guide.pdf">AMSmath</a> <span class="docutils literal">cases</span> environment (not (yet) supported by
HTML writers with <span class="docutils literal"><span class="pre">--math-output=MathML</span></span>):</p>
<div class="system-message">
<p class="system-message-title">System Message: ERROR/3 (<span class="docutils literal">functional/input/data/math.txt</span>, line 108)</p>
<p>
Environment not supported! Supported environment: "matrix".</p>
<pre class="literal-block">
\mathrm{sgn}(x) = \begin{cases}
                    -1 &amp; x&lt;0\\
                     1 &amp; x&gt;0
                  \end{cases}
</pre>
</div>
</div>
</body>
</html>