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	<li><a class="reference internal" href="#"><code class="xref py py-mod docutils literal notranslate"><span class="pre">cmath</span></code> — Mathematical functions for complex numbers</a><ul>
	<li><a class="reference internal" href="#conversions-to-and-from-polar-coordinates">Conversions to and from polar coordinates</a></li>
	<li><a class="reference internal" href="#power-and-logarithmic-functions">Power and logarithmic functions</a></li>
	<li><a class="reference internal" href="#trigonometric-functions">Trigonometric functions</a></li>
	<li><a class="reference internal" href="#hyperbolic-functions">Hyperbolic functions</a></li>
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	<section id="module-cmath">
	<span id="cmath-mathematical-functions-for-complex-numbers"></span><h1><code class="xref py py-mod docutils literal notranslate"><span class="pre">cmath</span></code> — Mathematical functions for complex numbers<a class="headerlink" href="#module-cmath" title="Link to this heading">¶</a></h1>
	<hr class="docutils" />
	<p>This module 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__"><code class="xref py py-meth docutils literal notranslate"><span class="pre">__complex__()</span></code></a> or a <a class="reference internal" href="../reference/datamodel.html#object.__float__" title="object.__float__"><code class="xref py py-meth docutils literal notranslate"><span class="pre">__float__()</span></code></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="admonition-title">Note</p>
	<p>For functions involving branch cuts, we have the problem of deciding how to
	define those functions on the cut itself. Following Kahan’s “Branch cuts for
	complex elementary functions” paper, as well as Annex G of C99 and later C
	standards, we use the sign of zero to distinguish one side of the branch cut
	from the other: for a branch cut along (a portion of) the real axis we look
	at the sign of the imaginary part, while for a branch cut along the
	imaginary axis we look at the sign of the real part.</p>
	<p>For example, the <a class="reference internal" href="#cmath.sqrt" title="cmath.sqrt"><code class="xref py py-func docutils literal notranslate"><span class="pre">cmath.sqrt()</span></code></a> function has a branch cut along the
	negative real axis. An argument of <code class="docutils literal notranslate"><span class="pre">-2-0j</span></code> is treated as
	though it lies <em>below</em> the branch cut, and so gives a result on the negative
	imaginary axis:</p>
	<div class="highlight-python3 notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">cmath</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="o">-</span><span class="mi">2</span><span class="o">-</span><span class="mi">0</span><span class="n">j</span><span class="p">)</span>
	<span class="go">-1.4142135623730951j</span>
	</pre></div>
	</div>
	<p>But an argument of <code class="docutils literal notranslate"><span class="pre">-2+0j</span></code> is treated as though it lies above
	the branch cut:</p>
	<div class="highlight-python3 notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">cmath</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="o">-</span><span class="mi">2</span><span class="o">+</span><span class="mi">0</span><span class="n">j</span><span class="p">)</span>
	<span class="go">1.4142135623730951j</span>
	</pre></div>
	</div>
	</div>
	<table class="docutils align-default">
	<tbody>
	<tr class="row-odd"><td colspan="2"><p><strong>Conversions to and from polar coordinates</strong></p></td>
	</tr>
	<tr class="row-even"><td><p><a class="reference internal" href="#cmath.phase" title="cmath.phase"><code class="xref py py-func docutils literal notranslate"><span class="pre">phase(z)</span></code></a></p></td>
	<td><p>Return the phase of <em>z</em></p></td>
	</tr>
	<tr class="row-odd"><td><p><a class="reference internal" href="#cmath.polar" title="cmath.polar"><code class="xref py py-func docutils literal notranslate"><span class="pre">polar(z)</span></code></a></p></td>
	<td><p>Return the representation of <em>z</em> in polar coordinates</p></td>
	</tr>
	<tr class="row-even"><td><p><a class="reference internal" href="#cmath.rect" title="cmath.rect"><code class="xref py py-func docutils literal notranslate"><span class="pre">rect(r,</span> <span class="pre">phi)</span></code></a></p></td>
	<td><p>Return the complex number <em>z</em> with polar coordinates <em>r</em> and <em>phi</em></p></td>
	</tr>
	<tr class="row-odd"><td colspan="2"><p><strong>Power and logarithmic functions</strong></p></td>
	</tr>
	<tr class="row-even"><td><p><a class="reference internal" href="#cmath.exp" title="cmath.exp"><code class="xref py py-func docutils literal notranslate"><span class="pre">exp(z)</span></code></a></p></td>
	<td><p>Return <em>e</em> raised to the power <em>z</em></p></td>
	</tr>
	<tr class="row-odd"><td><p><a class="reference internal" href="#cmath.log" title="cmath.log"><code class="xref py py-func docutils literal notranslate"><span class="pre">log(z[,</span> <span class="pre">base])</span></code></a></p></td>
	<td><p>Return the logarithm of <em>z</em> to the given <em>base</em> (<em>e</em> by default)</p></td>
	</tr>
	<tr class="row-even"><td><p><a class="reference internal" href="#cmath.log10" title="cmath.log10"><code class="xref py py-func docutils literal notranslate"><span class="pre">log10(z)</span></code></a></p></td>
	<td><p>Return the base-10 logarithm of <em>z</em></p></td>
	</tr>
	<tr class="row-odd"><td><p><a class="reference internal" href="#cmath.sqrt" title="cmath.sqrt"><code class="xref py py-func docutils literal notranslate"><span class="pre">sqrt(z)</span></code></a></p></td>
	<td><p>Return the square root of <em>z</em></p></td>
	</tr>
	<tr class="row-even"><td colspan="2"><p><strong>Trigonometric functions</strong></p></td>
	</tr>
	<tr class="row-odd"><td><p><a class="reference internal" href="#cmath.acos" title="cmath.acos"><code class="xref py py-func docutils literal notranslate"><span class="pre">acos(z)</span></code></a></p></td>
	<td><p>Return the arc cosine of <em>z</em></p></td>
	</tr>
	<tr class="row-even"><td><p><a class="reference internal" href="#cmath.asin" title="cmath.asin"><code class="xref py py-func docutils literal notranslate"><span class="pre">asin(z)</span></code></a></p></td>
	<td><p>Return the arc sine of <em>z</em></p></td>
	</tr>
	<tr class="row-odd"><td><p><a class="reference internal" href="#cmath.atan" title="cmath.atan"><code class="xref py py-func docutils literal notranslate"><span class="pre">atan(z)</span></code></a></p></td>
	<td><p>Return the arc tangent of <em>z</em></p></td>
	</tr>
	<tr class="row-even"><td><p><a class="reference internal" href="#cmath.cos" title="cmath.cos"><code class="xref py py-func docutils literal notranslate"><span class="pre">cos(z)</span></code></a></p></td>
	<td><p>Return the cosine of <em>z</em></p></td>
	</tr>
	<tr class="row-odd"><td><p><a class="reference internal" href="#cmath.sin" title="cmath.sin"><code class="xref py py-func docutils literal notranslate"><span class="pre">sin(z)</span></code></a></p></td>
	<td><p>Return the sine of <em>z</em></p></td>
	</tr>
	<tr class="row-even"><td><p><a class="reference internal" href="#cmath.tan" title="cmath.tan"><code class="xref py py-func docutils literal notranslate"><span class="pre">tan(z)</span></code></a></p></td>
	<td><p>Return the tangent of <em>z</em></p></td>
	</tr>
	<tr class="row-odd"><td colspan="2"><p><strong>Hyperbolic functions</strong></p></td>
	</tr>
	<tr class="row-even"><td><p><a class="reference internal" href="#cmath.acosh" title="cmath.acosh"><code class="xref py py-func docutils literal notranslate"><span class="pre">acosh(z)</span></code></a></p></td>
	<td><p>Return the inverse hyperbolic cosine of <em>z</em></p></td>
	</tr>
	<tr class="row-odd"><td><p><a class="reference internal" href="#cmath.asinh" title="cmath.asinh"><code class="xref py py-func docutils literal notranslate"><span class="pre">asinh(z)</span></code></a></p></td>
	<td><p>Return the inverse hyperbolic sine of <em>z</em></p></td>
	</tr>
	<tr class="row-even"><td><p><a class="reference internal" href="#cmath.atanh" title="cmath.atanh"><code class="xref py py-func docutils literal notranslate"><span class="pre">atanh(z)</span></code></a></p></td>
	<td><p>Return the inverse hyperbolic tangent of <em>z</em></p></td>
	</tr>
	<tr class="row-odd"><td><p><a class="reference internal" href="#cmath.cosh" title="cmath.cosh"><code class="xref py py-func docutils literal notranslate"><span class="pre">cosh(z)</span></code></a></p></td>
	<td><p>Return the hyperbolic cosine of <em>z</em></p></td>
	</tr>
	<tr class="row-even"><td><p><a class="reference internal" href="#cmath.sinh" title="cmath.sinh"><code class="xref py py-func docutils literal notranslate"><span class="pre">sinh(z)</span></code></a></p></td>
	<td><p>Return the hyperbolic sine of <em>z</em></p></td>
	</tr>
	<tr class="row-odd"><td><p><a class="reference internal" href="#cmath.tanh" title="cmath.tanh"><code class="xref py py-func docutils literal notranslate"><span class="pre">tanh(z)</span></code></a></p></td>
	<td><p>Return the hyperbolic tangent of <em>z</em></p></td>
	</tr>
	<tr class="row-even"><td colspan="2"><p><strong>Classification functions</strong></p></td>
	</tr>
	<tr class="row-odd"><td><p><a class="reference internal" href="#cmath.isfinite" title="cmath.isfinite"><code class="xref py py-func docutils literal notranslate"><span class="pre">isfinite(z)</span></code></a></p></td>
	<td><p>Check if all components of <em>z</em> are finite</p></td>
	</tr>
	<tr class="row-even"><td><p><a class="reference internal" href="#cmath.isinf" title="cmath.isinf"><code class="xref py py-func docutils literal notranslate"><span class="pre">isinf(z)</span></code></a></p></td>
	<td><p>Check if any component of <em>z</em> is infinite</p></td>
	</tr>
	<tr class="row-odd"><td><p><a class="reference internal" href="#cmath.isnan" title="cmath.isnan"><code class="xref py py-func docutils literal notranslate"><span class="pre">isnan(z)</span></code></a></p></td>
	<td><p>Check if any component of <em>z</em> is a NaN</p></td>
	</tr>
	<tr class="row-even"><td><p><a class="reference internal" href="#cmath.isclose" title="cmath.isclose"><code class="xref py py-func docutils literal notranslate"><span class="pre">isclose(a,</span> <span class="pre">b,</span> <span class="pre">*,</span> <span class="pre">rel_tol,</span> <span class="pre">abs_tol)</span></code></a></p></td>
	<td><p>Check if the values <em>a</em> and <em>b</em> are close to each other</p></td>
	</tr>
	<tr class="row-odd"><td colspan="2"><p><strong>Constants</strong></p></td>
	</tr>
	<tr class="row-even"><td><p><a class="reference internal" href="#cmath.pi" title="cmath.pi"><code class="xref py py-data docutils literal notranslate"><span class="pre">pi</span></code></a></p></td>
	<td><p><em>π</em> = 3.141592…</p></td>
	</tr>
	<tr class="row-odd"><td><p><a class="reference internal" href="#cmath.e" title="cmath.e"><code class="xref py py-data docutils literal notranslate"><span class="pre">e</span></code></a></p></td>
	<td><p><em>e</em> = 2.718281…</p></td>
	</tr>
	<tr class="row-even"><td><p><a class="reference internal" href="#cmath.tau" title="cmath.tau"><code class="xref py py-data docutils literal notranslate"><span class="pre">tau</span></code></a></p></td>
	<td><p><em>τ</em> = 2<em>π</em> = 6.283185…</p></td>
	</tr>
	<tr class="row-odd"><td><p><a class="reference internal" href="#cmath.inf" title="cmath.inf"><code class="xref py py-data docutils literal notranslate"><span class="pre">inf</span></code></a></p></td>
	<td><p>Positive infinity</p></td>
	</tr>
	<tr class="row-even"><td><p><a class="reference internal" href="#cmath.infj" title="cmath.infj"><code class="xref py py-data docutils literal notranslate"><span class="pre">infj</span></code></a></p></td>
	<td><p>Pure imaginary infinity</p></td>
	</tr>
	<tr class="row-odd"><td><p><a class="reference internal" href="#cmath.nan" title="cmath.nan"><code class="xref py py-data docutils literal notranslate"><span class="pre">nan</span></code></a></p></td>
	<td><p>“Not a number” (NaN)</p></td>
	</tr>
	<tr class="row-even"><td><p><a class="reference internal" href="#cmath.nanj" title="cmath.nanj"><code class="xref py py-data docutils literal notranslate"><span class="pre">nanj</span></code></a></p></td>
	<td><p>Pure imaginary NaN</p></td>
	</tr>
	</tbody>
	</table>
	<section id="conversions-to-and-from-polar-coordinates">
	<h2>Conversions to and from polar coordinates<a class="headerlink" href="#conversions-to-and-from-polar-coordinates" title="Link to this heading">¶</a></h2>
	<p>A Python complex number <code class="docutils literal notranslate"><span class="pre">z</span></code> is stored internally using <em>rectangular</em>
	or <em>Cartesian</em> coordinates. It is completely determined by its <em>real
	part</em> <code class="docutils literal notranslate"><span class="pre">z.real</span></code> and its <em>imaginary part</em> <code class="docutils literal notranslate"><span class="pre">z.imag</span></code>.</p>
	<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="py function">
	<dt class="sig sig-object py" id="cmath.phase">
	<span class="sig-prename descclassname"><span class="pre">cmath.</span></span><span class="sig-name descname"><span class="pre">phase</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">z</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#cmath.phase" title="Link to this definition">¶</a></dt>
	<dd><p>Return the phase of <em>z</em> (also known as the <em>argument</em> of <em>z</em>), as a float.
	<code class="docutils literal notranslate"><span class="pre">phase(z)</span></code> is equivalent to <code class="docutils literal notranslate"><span class="pre">math.atan2(z.imag,</span> <span class="pre">z.real)</span></code>. The result
	lies in the range [-<em>π</em>, <em>π</em>], and the branch cut for this operation lies
	along the negative real axis. The sign of the result is the same as the
	sign of <code class="docutils literal notranslate"><span class="pre">z.imag</span></code>, even when <code class="docutils literal notranslate"><span class="pre">z.imag</span></code> is zero:</p>
	<div class="highlight-python3 notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">phase</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="o">+</span><span class="mi">0</span><span class="n">j</span><span class="p">)</span>
	<span class="go">3.141592653589793</span>
	<span class="gp">>>> </span><span class="n">phase</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="o">-</span><span class="mi">0</span><span class="n">j</span><span class="p">)</span>
	<span class="go">-3.141592653589793</span>
	</pre></div>
	</div>
	</dd></dl>

	<div class="admonition note">
	<p class="admonition-title">Note</p>
	<p>The modulus (absolute value) of a complex number <em>z</em> can be
	computed using the built-in <a class="reference internal" href="functions.html#abs" title="abs"><code class="xref py py-func docutils literal notranslate"><span class="pre">abs()</span></code></a> function. There is no
	separate <code class="xref py py-mod docutils literal notranslate"><span class="pre">cmath</span></code> module function for this operation.</p>
	</div>
	<dl class="py function">
	<dt class="sig sig-object py" id="cmath.polar">
	<span class="sig-prename descclassname"><span class="pre">cmath.</span></span><span class="sig-name descname"><span class="pre">polar</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">z</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#cmath.polar" title="Link to this definition">¶</a></dt>
	<dd><p>Return the representation of <em>z</em> in polar coordinates. Returns a
	pair <code class="docutils literal notranslate"><span class="pre">(r,</span> <span class="pre">phi)</span></code> where <em>r</em> is the modulus of <em>z</em> and <em>phi</em> is the
	phase of <em>z</em>. <code class="docutils literal notranslate"><span class="pre">polar(z)</span></code> is equivalent to <code class="docutils literal notranslate"><span class="pre">(abs(z),</span>
	<span class="pre">phase(z))</span></code>.</p>
	</dd></dl>

	<dl class="py function">
	<dt class="sig sig-object py" id="cmath.rect">
	<span class="sig-prename descclassname"><span class="pre">cmath.</span></span><span class="sig-name descname"><span class="pre">rect</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">r</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">phi</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#cmath.rect" title="Link to this definition">¶</a></dt>
	<dd><p>Return the complex number <em>z</em> with polar coordinates <em>r</em> and <em>phi</em>.
	Equivalent to <code class="docutils literal notranslate"><span class="pre">complex(r</span> <span class="pre"></span> <span class="pre">math.cos(phi),</span> <span class="pre">r</span> <span class="pre"></span> <span class="pre">math.sin(phi))</span></code>.</p>
	</dd></dl>

	</section>
	<section id="power-and-logarithmic-functions">
	<h2>Power and logarithmic functions<a class="headerlink" href="#power-and-logarithmic-functions" title="Link to this heading">¶</a></h2>
	<dl class="py function">
	<dt class="sig sig-object py" id="cmath.exp">
	<span class="sig-prename descclassname"><span class="pre">cmath.</span></span><span class="sig-name descname"><span class="pre">exp</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">z</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#cmath.exp" title="Link to this definition">¶</a></dt>
	<dd><p>Return <em>e</em> raised to the power <em>z</em>, where <em>e</em> is the base of natural
	logarithms.</p>
	</dd></dl>

	<dl class="py function">
	<dt class="sig sig-object py" id="cmath.log">
	<span class="sig-prename descclassname"><span class="pre">cmath.</span></span><span class="sig-name descname"><span class="pre">log</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">z</span></span></em><span class="optional">[</span>, <em class="sig-param"><span class="n"><span class="pre">base</span></span></em><span class="optional">]</span><span class="sig-paren">)</span><a class="headerlink" href="#cmath.log" title="Link to this definition">¶</a></dt>
	<dd><p>Return the logarithm of <em>z</em> to the given <em>base</em>. If the <em>base</em> is not
	specified, returns the natural logarithm of <em>z</em>. There is one branch cut,
	from 0 along the negative real axis to -∞.</p>
	</dd></dl>

	<dl class="py function">
	<dt class="sig sig-object py" id="cmath.log10">
	<span class="sig-prename descclassname"><span class="pre">cmath.</span></span><span class="sig-name descname"><span class="pre">log10</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">z</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#cmath.log10" title="Link to this definition">¶</a></dt>
	<dd><p>Return the base-10 logarithm of <em>z</em>. This has the same branch cut as
	<a class="reference internal" href="#cmath.log" title="cmath.log"><code class="xref py py-func docutils literal notranslate"><span class="pre">log()</span></code></a>.</p>
	</dd></dl>

	<dl class="py function">
	<dt class="sig sig-object py" id="cmath.sqrt">
	<span class="sig-prename descclassname"><span class="pre">cmath.</span></span><span class="sig-name descname"><span class="pre">sqrt</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">z</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#cmath.sqrt" title="Link to this definition">¶</a></dt>
	<dd><p>Return the square root of <em>z</em>. This has the same branch cut as <a class="reference internal" href="#cmath.log" title="cmath.log"><code class="xref py py-func docutils literal notranslate"><span class="pre">log()</span></code></a>.</p>
	</dd></dl>

	</section>
	<section id="trigonometric-functions">
	<h2>Trigonometric functions<a class="headerlink" href="#trigonometric-functions" title="Link to this heading">¶</a></h2>
	<dl class="py function">
	<dt class="sig sig-object py" id="cmath.acos">
	<span class="sig-prename descclassname"><span class="pre">cmath.</span></span><span class="sig-name descname"><span class="pre">acos</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">z</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#cmath.acos" title="Link to this definition">¶</a></dt>
	<dd><p>Return the arc cosine of <em>z</em>. There are two branch cuts: One extends right
	from 1 along the real axis to ∞. The other extends left from -1 along the
	real axis to -∞.</p>
	</dd></dl>

	<dl class="py function">
	<dt class="sig sig-object py" id="cmath.asin">
	<span class="sig-prename descclassname"><span class="pre">cmath.</span></span><span class="sig-name descname"><span class="pre">asin</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">z</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#cmath.asin" title="Link to this definition">¶</a></dt>
	<dd><p>Return the arc sine of <em>z</em>. This has the same branch cuts as <a class="reference internal" href="#cmath.acos" title="cmath.acos"><code class="xref py py-func docutils literal notranslate"><span class="pre">acos()</span></code></a>.</p>
	</dd></dl>

	<dl class="py function">
	<dt class="sig sig-object py" id="cmath.atan">
	<span class="sig-prename descclassname"><span class="pre">cmath.</span></span><span class="sig-name descname"><span class="pre">atan</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">z</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#cmath.atan" title="Link to this definition">¶</a></dt>
	<dd><p>Return the arc tangent of <em>z</em>. There are two branch cuts: One extends from
	<code class="docutils literal notranslate"><span class="pre">1j</span></code> along the imaginary axis to <code class="docutils literal notranslate"><span class="pre">∞j</span></code>. The other extends from <code class="docutils literal notranslate"><span class="pre">-1j</span></code>
	along the imaginary axis to <code class="docutils literal notranslate"><span class="pre">-∞j</span></code>.</p>
	</dd></dl>

	<dl class="py function">
	<dt class="sig sig-object py" id="cmath.cos">
	<span class="sig-prename descclassname"><span class="pre">cmath.</span></span><span class="sig-name descname"><span class="pre">cos</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">z</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#cmath.cos" title="Link to this definition">¶</a></dt>
	<dd><p>Return the cosine of <em>z</em>.</p>
	</dd></dl>

	<dl class="py function">
	<dt class="sig sig-object py" id="cmath.sin">
	<span class="sig-prename descclassname"><span class="pre">cmath.</span></span><span class="sig-name descname"><span class="pre">sin</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">z</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#cmath.sin" title="Link to this definition">¶</a></dt>
	<dd><p>Return the sine of <em>z</em>.</p>
	</dd></dl>

	<dl class="py function">
	<dt class="sig sig-object py" id="cmath.tan">
	<span class="sig-prename descclassname"><span class="pre">cmath.</span></span><span class="sig-name descname"><span class="pre">tan</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">z</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#cmath.tan" title="Link to this definition">¶</a></dt>
	<dd><p>Return the tangent of <em>z</em>.</p>
	</dd></dl>

	</section>
	<section id="hyperbolic-functions">
	<h2>Hyperbolic functions<a class="headerlink" href="#hyperbolic-functions" title="Link to this heading">¶</a></h2>
	<dl class="py function">
	<dt class="sig sig-object py" id="cmath.acosh">
	<span class="sig-prename descclassname"><span class="pre">cmath.</span></span><span class="sig-name descname"><span class="pre">acosh</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">z</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#cmath.acosh" title="Link to this definition">¶</a></dt>
	<dd><p>Return the inverse hyperbolic cosine of <em>z</em>. There is one branch cut,
	extending left from 1 along the real axis to -∞.</p>
	</dd></dl>

	<dl class="py function">
	<dt class="sig sig-object py" id="cmath.asinh">
	<span class="sig-prename descclassname"><span class="pre">cmath.</span></span><span class="sig-name descname"><span class="pre">asinh</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">z</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#cmath.asinh" title="Link to this definition">¶</a></dt>
	<dd><p>Return the inverse hyperbolic sine of <em>z</em>. There are two branch cuts:
	One extends from <code class="docutils literal notranslate"><span class="pre">1j</span></code> along the imaginary axis to <code class="docutils literal notranslate"><span class="pre">∞j</span></code>. The other
	extends from <code class="docutils literal notranslate"><span class="pre">-1j</span></code> along the imaginary axis to <code class="docutils literal notranslate"><span class="pre">-∞j</span></code>.</p>
	</dd></dl>

	<dl class="py function">
	<dt class="sig sig-object py" id="cmath.atanh">
	<span class="sig-prename descclassname"><span class="pre">cmath.</span></span><span class="sig-name descname"><span class="pre">atanh</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">z</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#cmath.atanh" title="Link to this definition">¶</a></dt>
	<dd><p>Return the inverse hyperbolic tangent of <em>z</em>. There are two branch cuts: One
	extends from <code class="docutils literal notranslate"><span class="pre">1</span></code> along the real axis to <code class="docutils literal notranslate"><span class="pre">∞</span></code>. The other extends from
	<code class="docutils literal notranslate"><span class="pre">-1</span></code> along the real axis to <code class="docutils literal notranslate"><span class="pre">-∞</span></code>.</p>
	</dd></dl>

	<dl class="py function">
	<dt class="sig sig-object py" id="cmath.cosh">
	<span class="sig-prename descclassname"><span class="pre">cmath.</span></span><span class="sig-name descname"><span class="pre">cosh</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">z</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#cmath.cosh" title="Link to this definition">¶</a></dt>
	<dd><p>Return the hyperbolic cosine of <em>z</em>.</p>
	</dd></dl>

	<dl class="py function">
	<dt class="sig sig-object py" id="cmath.sinh">
	<span class="sig-prename descclassname"><span class="pre">cmath.</span></span><span class="sig-name descname"><span class="pre">sinh</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">z</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#cmath.sinh" title="Link to this definition">¶</a></dt>
	<dd><p>Return the hyperbolic sine of <em>z</em>.</p>
	</dd></dl>

	<dl class="py function">
	<dt class="sig sig-object py" id="cmath.tanh">
	<span class="sig-prename descclassname"><span class="pre">cmath.</span></span><span class="sig-name descname"><span class="pre">tanh</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">z</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#cmath.tanh" title="Link to this definition">¶</a></dt>
	<dd><p>Return the hyperbolic tangent of <em>z</em>.</p>
	</dd></dl>

	</section>
	<section id="classification-functions">
	<h2>Classification functions<a class="headerlink" href="#classification-functions" title="Link to this heading">¶</a></h2>
	<dl class="py function">
	<dt class="sig sig-object py" id="cmath.isfinite">
	<span class="sig-prename descclassname"><span class="pre">cmath.</span></span><span class="sig-name descname"><span class="pre">isfinite</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">z</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#cmath.isfinite" title="Link to this definition">¶</a></dt>
	<dd><p>Return <code class="docutils literal notranslate"><span class="pre">True</span></code> if both the real and imaginary parts of <em>z</em> are finite, and
	<code class="docutils literal notranslate"><span class="pre">False</span></code> otherwise.</p>
	<div class="versionadded">
	<p><span class="versionmodified added">Added in version 3.2.</span></p>
	</div>
	</dd></dl>

	<dl class="py function">
	<dt class="sig sig-object py" id="cmath.isinf">
	<span class="sig-prename descclassname"><span class="pre">cmath.</span></span><span class="sig-name descname"><span class="pre">isinf</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">z</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#cmath.isinf" title="Link to this definition">¶</a></dt>
	<dd><p>Return <code class="docutils literal notranslate"><span class="pre">True</span></code> if either the real or the imaginary part of <em>z</em> is an
	infinity, and <code class="docutils literal notranslate"><span class="pre">False</span></code> otherwise.</p>
	</dd></dl>

	<dl class="py function">
	<dt class="sig sig-object py" id="cmath.isnan">
	<span class="sig-prename descclassname"><span class="pre">cmath.</span></span><span class="sig-name descname"><span class="pre">isnan</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">z</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#cmath.isnan" title="Link to this definition">¶</a></dt>
	<dd><p>Return <code class="docutils literal notranslate"><span class="pre">True</span></code> if either the real or the imaginary part of <em>z</em> is a NaN,
	and <code class="docutils literal notranslate"><span class="pre">False</span></code> otherwise.</p>
	</dd></dl>

	<dl class="py function">
	<dt class="sig sig-object py" id="cmath.isclose">
	<span class="sig-prename descclassname"><span class="pre">cmath.</span></span><span class="sig-name descname"><span class="pre">isclose</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">a</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">b</span></span></em>, <em class="sig-param"><span class="keyword-only-separator o"><abbr title="Keyword-only parameters separator (PEP 3102)"><span class="pre">*</span></abbr></span></em>, <em class="sig-param"><span class="n"><span class="pre">rel_tol</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">1e-09</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">abs_tol</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">0.0</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#cmath.isclose" title="Link to this definition">¶</a></dt>
	<dd><p>Return <code class="docutils literal notranslate"><span class="pre">True</span></code> if the values <em>a</em> and <em>b</em> are close to each other and
	<code class="docutils literal notranslate"><span class="pre">False</span></code> otherwise.</p>
	<p>Whether or not two values are considered close is determined according to
	given absolute and relative tolerances. If no errors occur, the result will
	be: <code class="docutils literal notranslate"><span class="pre">abs(a-b)</span> <span class="pre"><=</span> <span class="pre">max(rel_tol</span> <span class="pre">*</span> <span class="pre">max(abs(a),</span> <span class="pre">abs(b)),</span> <span class="pre">abs_tol)</span></code>.</p>
	<p><em>rel_tol</em> is the relative tolerance – it is the maximum allowed difference
	between <em>a</em> and <em>b</em>, relative to the larger absolute value of <em>a</em> or <em>b</em>.
	For example, to set a tolerance of 5%, pass <code class="docutils literal notranslate"><span class="pre">rel_tol=0.05</span></code>. The default
	tolerance is <code class="docutils literal notranslate"><span class="pre">1e-09</span></code>, which assures that the two values are the same
	within about 9 decimal digits. <em>rel_tol</em> must be nonnegative and less
	than <code class="docutils literal notranslate"><span class="pre">1.0</span></code>.</p>
	<p><em>abs_tol</em> is the absolute tolerance; it defaults to <code class="docutils literal notranslate"><span class="pre">0.0</span></code> and it must be
	nonnegative. When comparing <code class="docutils literal notranslate"><span class="pre">x</span></code> to <code class="docutils literal notranslate"><span class="pre">0.0</span></code>, <code class="docutils literal notranslate"><span class="pre">isclose(x,</span> <span class="pre">0)</span></code> is computed
	as <code class="docutils literal notranslate"><span class="pre">abs(x)</span> <span class="pre"><=</span> <span class="pre">rel_tol</span>  <span class="pre">*</span> <span class="pre">abs(x)</span></code>, which is <code class="docutils literal notranslate"><span class="pre">False</span></code> for any <code class="docutils literal notranslate"><span class="pre">x</span></code> and
	rel_tol less than <code class="docutils literal notranslate"><span class="pre">1.0</span></code>. So add an appropriate positive abs_tol argument
	to the call.</p>
	<p>The IEEE 754 special values of <code class="docutils literal notranslate"><span class="pre">NaN</span></code>, <code class="docutils literal notranslate"><span class="pre">inf</span></code>, and <code class="docutils literal notranslate"><span class="pre">-inf</span></code> will be
	handled according to IEEE rules. Specifically, <code class="docutils literal notranslate"><span class="pre">NaN</span></code> is not considered
	close to any other value, including <code class="docutils literal notranslate"><span class="pre">NaN</span></code>. <code class="docutils literal notranslate"><span class="pre">inf</span></code> and <code class="docutils literal notranslate"><span class="pre">-inf</span></code> are only
	considered close to themselves.</p>
	<div class="versionadded">
	<p><span class="versionmodified added">Added in version 3.5.</span></p>
	</div>
	<div class="admonition seealso">
	<p class="admonition-title">See also</p>
	<p><span class="target" id="index-0"></span><a class="pep reference external" href="https://peps.python.org/pep-0485/"><strong>PEP 485</strong></a> – A function for testing approximate equality</p>
	</div>
	</dd></dl>

	</section>
	<section id="constants">
	<h2>Constants<a class="headerlink" href="#constants" title="Link to this heading">¶</a></h2>
	<dl class="py data">
	<dt class="sig sig-object py" id="cmath.pi">
	<span class="sig-prename descclassname"><span class="pre">cmath.</span></span><span class="sig-name descname"><span class="pre">pi</span></span><a class="headerlink" href="#cmath.pi" title="Link to this definition">¶</a></dt>
	<dd><p>The mathematical constant <em>π</em>, as a float.</p>
	</dd></dl>

	<dl class="py data">
	<dt class="sig sig-object py" id="cmath.e">
	<span class="sig-prename descclassname"><span class="pre">cmath.</span></span><span class="sig-name descname"><span class="pre">e</span></span><a class="headerlink" href="#cmath.e" title="Link to this definition">¶</a></dt>
	<dd><p>The mathematical constant <em>e</em>, as a float.</p>
	</dd></dl>

	<dl class="py data">
	<dt class="sig sig-object py" id="cmath.tau">
	<span class="sig-prename descclassname"><span class="pre">cmath.</span></span><span class="sig-name descname"><span class="pre">tau</span></span><a class="headerlink" href="#cmath.tau" title="Link to this definition">¶</a></dt>
	<dd><p>The mathematical constant <em>τ</em>, as a float.</p>
	<div class="versionadded">
	<p><span class="versionmodified added">Added in version 3.6.</span></p>
	</div>
	</dd></dl>

	<dl class="py data">
	<dt class="sig sig-object py" id="cmath.inf">
	<span class="sig-prename descclassname"><span class="pre">cmath.</span></span><span class="sig-name descname"><span class="pre">inf</span></span><a class="headerlink" href="#cmath.inf" title="Link to this definition">¶</a></dt>
	<dd><p>Floating-point positive infinity. Equivalent to <code class="docutils literal notranslate"><span class="pre">float('inf')</span></code>.</p>
	<div class="versionadded">
	<p><span class="versionmodified added">Added in version 3.6.</span></p>
	</div>
	</dd></dl>

	<dl class="py data">
	<dt class="sig sig-object py" id="cmath.infj">
	<span class="sig-prename descclassname"><span class="pre">cmath.</span></span><span class="sig-name descname"><span class="pre">infj</span></span><a class="headerlink" href="#cmath.infj" title="Link to this definition">¶</a></dt>
	<dd><p>Complex number with zero real part and positive infinity imaginary
	part. Equivalent to <code class="docutils literal notranslate"><span class="pre">complex(0.0,</span> <span class="pre">float('inf'))</span></code>.</p>
	<div class="versionadded">
	<p><span class="versionmodified added">Added in version 3.6.</span></p>
	</div>
	</dd></dl>

	<dl class="py data">
	<dt class="sig sig-object py" id="cmath.nan">
	<span class="sig-prename descclassname"><span class="pre">cmath.</span></span><span class="sig-name descname"><span class="pre">nan</span></span><a class="headerlink" href="#cmath.nan" title="Link to this definition">¶</a></dt>
	<dd><p>A floating-point “not a number” (NaN) value. Equivalent to
	<code class="docutils literal notranslate"><span class="pre">float('nan')</span></code>. See also <a class="reference internal" href="math.html#math.nan" title="math.nan"><code class="xref py py-data docutils literal notranslate"><span class="pre">math.nan</span></code></a>.</p>
	<div class="versionadded">
	<p><span class="versionmodified added">Added in version 3.6.</span></p>
	</div>
	</dd></dl>

	<dl class="py data">
	<dt class="sig sig-object py" id="cmath.nanj">
	<span class="sig-prename descclassname"><span class="pre">cmath.</span></span><span class="sig-name descname"><span class="pre">nanj</span></span><a class="headerlink" href="#cmath.nanj" title="Link to this definition">¶</a></dt>
	<dd><p>Complex number with zero real part and NaN imaginary part. Equivalent to
	<code class="docutils literal notranslate"><span class="pre">complex(0.0,</span> <span class="pre">float('nan'))</span></code>.</p>
	<div class="versionadded">
	<p><span class="versionmodified added">Added in version 3.6.</span></p>
	</div>
	</dd></dl>

	<p id="index-1">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.)."><code class="xref py py-mod docutils literal notranslate"><span class="pre">math</span></code></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 <code class="docutils literal notranslate"><span class="pre">math.sqrt(-1)</span></code> raise an exception than return a complex
	number. Also note that the functions defined in <code class="xref py py-mod docutils literal notranslate"><span class="pre">cmath</span></code> 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 seealso">
	<p class="admonition-title">See also</p>
	<p>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>
	</section>
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	<li><a class="reference internal" href="#"><code class="xref py py-mod docutils literal notranslate"><span class="pre">cmath</span></code> — Mathematical functions for complex numbers</a><ul>
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