Edit File: cmath.html
<!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"> <head> <meta http-equiv="Content-Type" content="text/html; charset=utf-8" /> <title>9.3. cmath — Mathematical functions for complex numbers — Python 2.7.5 documentation</title> <link rel="stylesheet" href="../_static/default.css" type="text/css" /> <link rel="stylesheet" href="../_static/pygments.css" type="text/css" /> <script type="text/javascript"> var DOCUMENTATION_OPTIONS = { URL_ROOT: '../', VERSION: '2.7.5', COLLAPSE_INDEX: false, FILE_SUFFIX: '.html', HAS_SOURCE: true }; </script> <script type="text/javascript" src="../_static/jquery.js"></script> <script type="text/javascript" src="../_static/underscore.js"></script> <script type="text/javascript" src="../_static/doctools.js"></script> <script type="text/javascript" src="../_static/sidebar.js"></script> <link rel="search" type="application/opensearchdescription+xml" title="Search within Python 2.7.5 documentation" href="../_static/opensearch.xml"/> <link rel="author" title="About these documents" href="../about.html" /> <link rel="copyright" title="Copyright" href="../copyright.html" /> <link rel="top" title="Python 2.7.5 documentation" href="../index.html" /> <link rel="up" title="9. Numeric and Mathematical Modules" href="numeric.html" /> <link rel="next" title="9.4. decimal — Decimal fixed point and floating point arithmetic" href="decimal.html" /> <link rel="prev" title="9.2. math — Mathematical functions" href="math.html" /> <link rel="shortcut icon" type="image/png" href="../_static/py.png" /> <script type="text/javascript" src="../_static/copybutton.js"></script> </head> <body> <div class="related"> <h3>Navigation</h3> <ul> <li class="right" style="margin-right: 10px"> <a href="../genindex.html" title="General Index" accesskey="I">index</a></li> <li class="right" > <a href="../py-modindex.html" title="Python Module Index" >modules</a> |</li> <li class="right" > <a href="decimal.html" title="9.4. decimal — Decimal fixed point and floating point arithmetic" accesskey="N">next</a> |</li> <li class="right" > <a href="math.html" title="9.2. math — Mathematical functions" accesskey="P">previous</a> |</li> <li><img src="../_static/py.png" alt="" style="vertical-align: middle; margin-top: -1px"/></li> <li><a href="http://www.python.org/">Python</a> »</li> <li> <a href="../index.html">Python 2.7.5 documentation</a> » </li> <li><a href="index.html" >The Python Standard Library</a> »</li> <li><a href="numeric.html" accesskey="U">9. Numeric and Mathematical Modules</a> »</li> </ul> </div> <div class="document"> <div class="documentwrapper"> <div class="bodywrapper"> <div class="body"> <div class="section" id="module-cmath"> <span id="cmath-mathematical-functions-for-complex-numbers"></span><h1>9.3. <a class="reference internal" href="#module-cmath" title="cmath: Mathematical functions for complex numbers."><tt class="xref py py-mod docutils literal"><span class="pre">cmath</span></tt></a> — Mathematical functions for complex numbers<a class="headerlink" href="#module-cmath" title="Permalink to this headline">¶</a></h1> <p>This module is always available. It provides access to mathematical functions for complex numbers. The functions in this module accept integers, floating-point numbers or complex numbers as arguments. They will also accept any Python object that has either a <a class="reference internal" href="../reference/datamodel.html#object.__complex__" title="object.__complex__"><tt class="xref py py-meth docutils literal"><span class="pre">__complex__()</span></tt></a> or a <a class="reference internal" href="../reference/datamodel.html#object.__float__" title="object.__float__"><tt class="xref py py-meth docutils literal"><span class="pre">__float__()</span></tt></a> method: these methods are used to convert the object to a complex or floating-point number, respectively, and the function is then applied to the result of the conversion.</p> <div class="admonition note"> <p class="first admonition-title">Note</p> <p class="last">On platforms with hardware and system-level support for signed zeros, functions involving branch cuts are continuous on <em>both</em> sides of the branch cut: the sign of the zero distinguishes one side of the branch cut from the other. On platforms that do not support signed zeros the continuity is as specified below.</p> </div> <div class="section" id="conversions-to-and-from-polar-coordinates"> <h2>9.3.1. Conversions to and from polar coordinates<a class="headerlink" href="#conversions-to-and-from-polar-coordinates" title="Permalink to this headline">¶</a></h2> <p>A Python complex number <tt class="docutils literal"><span class="pre">z</span></tt> is stored internally using <em>rectangular</em> or <em>Cartesian</em> coordinates. It is completely determined by its <em>real part</em> <tt class="docutils literal"><span class="pre">z.real</span></tt> and its <em>imaginary part</em> <tt class="docutils literal"><span class="pre">z.imag</span></tt>. In other words:</p> <div class="highlight-python"><div class="highlight"><pre><span class="n">z</span> <span class="o">==</span> <span class="n">z</span><span class="o">.</span><span class="n">real</span> <span class="o">+</span> <span class="n">z</span><span class="o">.</span><span class="n">imag</span><span class="o">*</span><span class="mi">1</span><span class="n">j</span> </pre></div> </div> <p><em>Polar coordinates</em> give an alternative way to represent a complex number. In polar coordinates, a complex number <em>z</em> is defined by the modulus <em>r</em> and the phase angle <em>phi</em>. The modulus <em>r</em> is the distance from <em>z</em> to the origin, while the phase <em>phi</em> is the counterclockwise angle, measured in radians, from the positive x-axis to the line segment that joins the origin to <em>z</em>.</p> <p>The following functions can be used to convert from the native rectangular coordinates to polar coordinates and back.</p> <dl class="function"> <dt id="cmath.phase"> <tt class="descclassname">cmath.</tt><tt class="descname">phase</tt><big>(</big><em>x</em><big>)</big><a class="headerlink" href="#cmath.phase" title="Permalink to this definition">¶</a></dt> <dd><p>Return the phase of <em>x</em> (also known as the <em>argument</em> of <em>x</em>), as a float. <tt class="docutils literal"><span class="pre">phase(x)</span></tt> is equivalent to <tt class="docutils literal"><span class="pre">math.atan2(x.imag,</span> <span class="pre">x.real)</span></tt>. The result lies in the range [-π, π], and the branch cut for this operation lies along the negative real axis, continuous from above. On systems with support for signed zeros (which includes most systems in current use), this means that the sign of the result is the same as the sign of <tt class="docutils literal"><span class="pre">x.imag</span></tt>, even when <tt class="docutils literal"><span class="pre">x.imag</span></tt> is zero:</p> <div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="n">phase</span><span class="p">(</span><span class="nb">complex</span><span class="p">(</span><span class="o">-</span><span class="mf">1.0</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">))</span> <span class="go">3.1415926535897931</span> <span class="gp">>>> </span><span class="n">phase</span><span class="p">(</span><span class="nb">complex</span><span class="p">(</span><span class="o">-</span><span class="mf">1.0</span><span class="p">,</span> <span class="o">-</span><span class="mf">0.0</span><span class="p">))</span> <span class="go">-3.1415926535897931</span> </pre></div> </div> <p class="versionadded"> <span class="versionmodified">New in version 2.6.</span></p> </dd></dl> <div class="admonition note"> <p class="first admonition-title">Note</p> <p class="last">The modulus (absolute value) of a complex number <em>x</em> can be computed using the built-in <a class="reference internal" href="functions.html#abs" title="abs"><tt class="xref py py-func docutils literal"><span class="pre">abs()</span></tt></a> function. There is no separate <a class="reference internal" href="#module-cmath" title="cmath: Mathematical functions for complex numbers."><tt class="xref py py-mod docutils literal"><span class="pre">cmath</span></tt></a> module function for this operation.</p> </div> <dl class="function"> <dt id="cmath.polar"> <tt class="descclassname">cmath.</tt><tt class="descname">polar</tt><big>(</big><em>x</em><big>)</big><a class="headerlink" href="#cmath.polar" title="Permalink to this definition">¶</a></dt> <dd><p>Return the representation of <em>x</em> in polar coordinates. Returns a pair <tt class="docutils literal"><span class="pre">(r,</span> <span class="pre">phi)</span></tt> where <em>r</em> is the modulus of <em>x</em> and phi is the phase of <em>x</em>. <tt class="docutils literal"><span class="pre">polar(x)</span></tt> is equivalent to <tt class="docutils literal"><span class="pre">(abs(x),</span> <span class="pre">phase(x))</span></tt>.</p> <p class="versionadded"> <span class="versionmodified">New in version 2.6.</span></p> </dd></dl> <dl class="function"> <dt id="cmath.rect"> <tt class="descclassname">cmath.</tt><tt class="descname">rect</tt><big>(</big><em>r</em>, <em>phi</em><big>)</big><a class="headerlink" href="#cmath.rect" title="Permalink to this definition">¶</a></dt> <dd><p>Return the complex number <em>x</em> with polar coordinates <em>r</em> and <em>phi</em>. Equivalent to <tt class="docutils literal"><span class="pre">r</span> <span class="pre">*</span> <span class="pre">(math.cos(phi)</span> <span class="pre">+</span> <span class="pre">math.sin(phi)*1j)</span></tt>.</p> <p class="versionadded"> <span class="versionmodified">New in version 2.6.</span></p> </dd></dl> </div> <div class="section" id="power-and-logarithmic-functions"> <h2>9.3.2. Power and logarithmic functions<a class="headerlink" href="#power-and-logarithmic-functions" title="Permalink to this headline">¶</a></h2> <dl class="function"> <dt id="cmath.exp"> <tt class="descclassname">cmath.</tt><tt class="descname">exp</tt><big>(</big><em>x</em><big>)</big><a class="headerlink" href="#cmath.exp" title="Permalink to this definition">¶</a></dt> <dd><p>Return the exponential value <tt class="docutils literal"><span class="pre">e**x</span></tt>.</p> </dd></dl> <dl class="function"> <dt id="cmath.log"> <tt class="descclassname">cmath.</tt><tt class="descname">log</tt><big>(</big><em>x</em><span class="optional">[</span>, <em>base</em><span class="optional">]</span><big>)</big><a class="headerlink" href="#cmath.log" title="Permalink to this definition">¶</a></dt> <dd><p>Returns the logarithm of <em>x</em> to the given <em>base</em>. If the <em>base</em> is not specified, returns the natural logarithm of <em>x</em>. There is one branch cut, from 0 along the negative real axis to -∞, continuous from above.</p> <p class="versionchanged"> <span class="versionmodified">Changed in version 2.4: </span><em>base</em> argument added.</p> </dd></dl> <dl class="function"> <dt id="cmath.log10"> <tt class="descclassname">cmath.</tt><tt class="descname">log10</tt><big>(</big><em>x</em><big>)</big><a class="headerlink" href="#cmath.log10" title="Permalink to this definition">¶</a></dt> <dd><p>Return the base-10 logarithm of <em>x</em>. This has the same branch cut as <a class="reference internal" href="#cmath.log" title="cmath.log"><tt class="xref py py-func docutils literal"><span class="pre">log()</span></tt></a>.</p> </dd></dl> <dl class="function"> <dt id="cmath.sqrt"> <tt class="descclassname">cmath.</tt><tt class="descname">sqrt</tt><big>(</big><em>x</em><big>)</big><a class="headerlink" href="#cmath.sqrt" title="Permalink to this definition">¶</a></dt> <dd><p>Return the square root of <em>x</em>. This has the same branch cut as <a class="reference internal" href="#cmath.log" title="cmath.log"><tt class="xref py py-func docutils literal"><span class="pre">log()</span></tt></a>.</p> </dd></dl> </div> <div class="section" id="trigonometric-functions"> <h2>9.3.3. Trigonometric functions<a class="headerlink" href="#trigonometric-functions" title="Permalink to this headline">¶</a></h2> <dl class="function"> <dt id="cmath.acos"> <tt class="descclassname">cmath.</tt><tt class="descname">acos</tt><big>(</big><em>x</em><big>)</big><a class="headerlink" href="#cmath.acos" title="Permalink to this definition">¶</a></dt> <dd><p>Return the arc cosine of <em>x</em>. There are two branch cuts: One extends right from 1 along the real axis to ∞, continuous from below. The other extends left from -1 along the real axis to -∞, continuous from above.</p> </dd></dl> <dl class="function"> <dt id="cmath.asin"> <tt class="descclassname">cmath.</tt><tt class="descname">asin</tt><big>(</big><em>x</em><big>)</big><a class="headerlink" href="#cmath.asin" title="Permalink to this definition">¶</a></dt> <dd><p>Return the arc sine of <em>x</em>. This has the same branch cuts as <a class="reference internal" href="#cmath.acos" title="cmath.acos"><tt class="xref py py-func docutils literal"><span class="pre">acos()</span></tt></a>.</p> </dd></dl> <dl class="function"> <dt id="cmath.atan"> <tt class="descclassname">cmath.</tt><tt class="descname">atan</tt><big>(</big><em>x</em><big>)</big><a class="headerlink" href="#cmath.atan" title="Permalink to this definition">¶</a></dt> <dd><p>Return the arc tangent of <em>x</em>. There are two branch cuts: One extends from <tt class="docutils literal"><span class="pre">1j</span></tt> along the imaginary axis to <tt class="docutils literal"><span class="pre">∞j</span></tt>, continuous from the right. The other extends from <tt class="docutils literal"><span class="pre">-1j</span></tt> along the imaginary axis to <tt class="docutils literal"><span class="pre">-∞j</span></tt>, continuous from the left.</p> <p class="versionchanged"> <span class="versionmodified">Changed in version 2.6: </span>direction of continuity of upper cut reversed</p> </dd></dl> <dl class="function"> <dt id="cmath.cos"> <tt class="descclassname">cmath.</tt><tt class="descname">cos</tt><big>(</big><em>x</em><big>)</big><a class="headerlink" href="#cmath.cos" title="Permalink to this definition">¶</a></dt> <dd><p>Return the cosine of <em>x</em>.</p> </dd></dl> <dl class="function"> <dt id="cmath.sin"> <tt class="descclassname">cmath.</tt><tt class="descname">sin</tt><big>(</big><em>x</em><big>)</big><a class="headerlink" href="#cmath.sin" title="Permalink to this definition">¶</a></dt> <dd><p>Return the sine of <em>x</em>.</p> </dd></dl> <dl class="function"> <dt id="cmath.tan"> <tt class="descclassname">cmath.</tt><tt class="descname">tan</tt><big>(</big><em>x</em><big>)</big><a class="headerlink" href="#cmath.tan" title="Permalink to this definition">¶</a></dt> <dd><p>Return the tangent of <em>x</em>.</p> </dd></dl> </div> <div class="section" id="hyperbolic-functions"> <h2>9.3.4. Hyperbolic functions<a class="headerlink" href="#hyperbolic-functions" title="Permalink to this headline">¶</a></h2> <dl class="function"> <dt id="cmath.acosh"> <tt class="descclassname">cmath.</tt><tt class="descname">acosh</tt><big>(</big><em>x</em><big>)</big><a class="headerlink" href="#cmath.acosh" title="Permalink to this definition">¶</a></dt> <dd><p>Return the hyperbolic arc cosine of <em>x</em>. There is one branch cut, extending left from 1 along the real axis to -∞, continuous from above.</p> </dd></dl> <dl class="function"> <dt id="cmath.asinh"> <tt class="descclassname">cmath.</tt><tt class="descname">asinh</tt><big>(</big><em>x</em><big>)</big><a class="headerlink" href="#cmath.asinh" title="Permalink to this definition">¶</a></dt> <dd><p>Return the hyperbolic arc sine of <em>x</em>. There are two branch cuts: One extends from <tt class="docutils literal"><span class="pre">1j</span></tt> along the imaginary axis to <tt class="docutils literal"><span class="pre">∞j</span></tt>, continuous from the right. The other extends from <tt class="docutils literal"><span class="pre">-1j</span></tt> along the imaginary axis to <tt class="docutils literal"><span class="pre">-∞j</span></tt>, continuous from the left.</p> <p class="versionchanged"> <span class="versionmodified">Changed in version 2.6: </span>branch cuts moved to match those recommended by the C99 standard</p> </dd></dl> <dl class="function"> <dt id="cmath.atanh"> <tt class="descclassname">cmath.</tt><tt class="descname">atanh</tt><big>(</big><em>x</em><big>)</big><a class="headerlink" href="#cmath.atanh" title="Permalink to this definition">¶</a></dt> <dd><p>Return the hyperbolic arc tangent of <em>x</em>. There are two branch cuts: One extends from <tt class="docutils literal"><span class="pre">1</span></tt> along the real axis to <tt class="docutils literal"><span class="pre">∞</span></tt>, continuous from below. The other extends from <tt class="docutils literal"><span class="pre">-1</span></tt> along the real axis to <tt class="docutils literal"><span class="pre">-∞</span></tt>, continuous from above.</p> <p class="versionchanged"> <span class="versionmodified">Changed in version 2.6: </span>direction of continuity of right cut reversed</p> </dd></dl> <dl class="function"> <dt id="cmath.cosh"> <tt class="descclassname">cmath.</tt><tt class="descname">cosh</tt><big>(</big><em>x</em><big>)</big><a class="headerlink" href="#cmath.cosh" title="Permalink to this definition">¶</a></dt> <dd><p>Return the hyperbolic cosine of <em>x</em>.</p> </dd></dl> <dl class="function"> <dt id="cmath.sinh"> <tt class="descclassname">cmath.</tt><tt class="descname">sinh</tt><big>(</big><em>x</em><big>)</big><a class="headerlink" href="#cmath.sinh" title="Permalink to this definition">¶</a></dt> <dd><p>Return the hyperbolic sine of <em>x</em>.</p> </dd></dl> <dl class="function"> <dt id="cmath.tanh"> <tt class="descclassname">cmath.</tt><tt class="descname">tanh</tt><big>(</big><em>x</em><big>)</big><a class="headerlink" href="#cmath.tanh" title="Permalink to this definition">¶</a></dt> <dd><p>Return the hyperbolic tangent of <em>x</em>.</p> </dd></dl> </div> <div class="section" id="classification-functions"> <h2>9.3.5. Classification functions<a class="headerlink" href="#classification-functions" title="Permalink to this headline">¶</a></h2> <dl class="function"> <dt id="cmath.isinf"> <tt class="descclassname">cmath.</tt><tt class="descname">isinf</tt><big>(</big><em>x</em><big>)</big><a class="headerlink" href="#cmath.isinf" title="Permalink to this definition">¶</a></dt> <dd><p>Return <em>True</em> if the real or the imaginary part of x is positive or negative infinity.</p> <p class="versionadded"> <span class="versionmodified">New in version 2.6.</span></p> </dd></dl> <dl class="function"> <dt id="cmath.isnan"> <tt class="descclassname">cmath.</tt><tt class="descname">isnan</tt><big>(</big><em>x</em><big>)</big><a class="headerlink" href="#cmath.isnan" title="Permalink to this definition">¶</a></dt> <dd><p>Return <em>True</em> if the real or imaginary part of x is not a number (NaN).</p> <p class="versionadded"> <span class="versionmodified">New in version 2.6.</span></p> </dd></dl> </div> <div class="section" id="constants"> <h2>9.3.6. Constants<a class="headerlink" href="#constants" title="Permalink to this headline">¶</a></h2> <dl class="data"> <dt id="cmath.pi"> <tt class="descclassname">cmath.</tt><tt class="descname">pi</tt><a class="headerlink" href="#cmath.pi" title="Permalink to this definition">¶</a></dt> <dd><p>The mathematical constant <em>π</em>, as a float.</p> </dd></dl> <dl class="data"> <dt id="cmath.e"> <tt class="descclassname">cmath.</tt><tt class="descname">e</tt><a class="headerlink" href="#cmath.e" title="Permalink to this definition">¶</a></dt> <dd><p>The mathematical constant <em>e</em>, as a float.</p> </dd></dl> <p id="index-0">Note that the selection of functions is similar, but not identical, to that in module <a class="reference internal" href="math.html#module-math" title="math: Mathematical functions (sin() etc.)."><tt class="xref py py-mod docutils literal"><span class="pre">math</span></tt></a>. The reason for having two modules is that some users aren’t interested in complex numbers, and perhaps don’t even know what they are. They would rather have <tt class="docutils literal"><span class="pre">math.sqrt(-1)</span></tt> raise an exception than return a complex number. Also note that the functions defined in <a class="reference internal" href="#module-cmath" title="cmath: Mathematical functions for complex numbers."><tt class="xref py py-mod docutils literal"><span class="pre">cmath</span></tt></a> always return a complex number, even if the answer can be expressed as a real number (in which case the complex number has an imaginary part of zero).</p> <p>A note on branch cuts: They are curves along which the given function fails to be continuous. They are a necessary feature of many complex functions. It is assumed that if you need to compute with complex functions, you will understand about branch cuts. Consult almost any (not too elementary) book on complex variables for enlightenment. For information of the proper choice of branch cuts for numerical purposes, a good reference should be the following:</p> <div class="admonition-see-also admonition seealso"> <p class="first admonition-title">See also</p> <p class="last">Kahan, W: Branch cuts for complex elementary functions; or, Much ado about nothing’s sign bit. In Iserles, A., and Powell, M. (eds.), The state of the art in numerical analysis. Clarendon Press (1987) pp165-211.</p> </div> </div> </div> </div> </div> </div> <div class="sphinxsidebar"> <div class="sphinxsidebarwrapper"> <h3><a href="../contents.html">Table Of Contents</a></h3> <ul> <li><a class="reference internal" href="#">9.3. <tt class="docutils literal"><span class="pre">cmath</span></tt> — Mathematical functions for complex numbers</a><ul> <li><a class="reference internal" href="#conversions-to-and-from-polar-coordinates">9.3.1. Conversions to and from polar coordinates</a></li> <li><a class="reference internal" href="#power-and-logarithmic-functions">9.3.2. Power and logarithmic functions</a></li> <li><a class="reference internal" href="#trigonometric-functions">9.3.3. Trigonometric functions</a></li> <li><a class="reference internal" href="#hyperbolic-functions">9.3.4. Hyperbolic functions</a></li> <li><a class="reference internal" href="#classification-functions">9.3.5. Classification functions</a></li> <li><a class="reference internal" href="#constants">9.3.6. Constants</a></li> </ul> </li> </ul> <h4>Previous topic</h4> <p class="topless"><a href="math.html" title="previous chapter">9.2. <tt class="docutils literal"><span class="pre">math</span></tt> — Mathematical functions</a></p> <h4>Next topic</h4> <p class="topless"><a href="decimal.html" title="next chapter">9.4. <tt class="docutils literal"><span class="pre">decimal</span></tt> — Decimal fixed point and floating point arithmetic</a></p> <h3>This Page</h3> <ul class="this-page-menu"> <li><a href="../bugs.html">Report a Bug</a></li> <li><a href="../_sources/library/cmath.txt" rel="nofollow">Show Source</a></li> </ul> <div id="searchbox" style="display: none"> <h3>Quick search</h3> <form class="search" action="../search.html" method="get"> <input type="text" name="q" /> <input type="submit" value="Go" /> <input type="hidden" name="check_keywords" value="yes" /> <input type="hidden" name="area" value="default" /> </form> <p class="searchtip" style="font-size: 90%"> Enter search terms or a module, class or function name. </p> </div> <script type="text/javascript">$('#searchbox').show(0);</script> </div> </div> <div class="clearer"></div> </div> <div class="related"> <h3>Navigation</h3> <ul> <li class="right" style="margin-right: 10px"> <a href="../genindex.html" title="General Index" >index</a></li> <li class="right" > <a href="../py-modindex.html" title="Python Module Index" >modules</a> |</li> <li class="right" > <a href="decimal.html" title="9.4. decimal — Decimal fixed point and floating point arithmetic" >next</a> |</li> <li class="right" > <a href="math.html" title="9.2. math — Mathematical functions" >previous</a> |</li> <li><img src="../_static/py.png" alt="" style="vertical-align: middle; margin-top: -1px"/></li> <li><a href="http://www.python.org/">Python</a> »</li> <li> <a href="../index.html">Python 2.7.5 documentation</a> » </li> <li><a href="index.html" >The Python Standard Library</a> »</li> <li><a href="numeric.html" >9. Numeric and Mathematical Modules</a> »</li> </ul> </div> <div class="footer"> © <a href="../copyright.html">Copyright</a> 1990-2020, Python Software Foundation. <br /> The Python Software Foundation is a non-profit corporation. <a href="http://www.python.org/psf/donations/">Please donate.</a> <br /> Last updated on Oct 13, 2020. <a href="../bugs.html">Found a bug</a>? <br /> Created using <a href="http://sphinx.pocoo.org/">Sphinx</a> 1.1.3. </div> </body> </html>