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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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<li><a class="reference internal" href="#conversions-to-and-from-polar-coordinates">Conversions to and from polar coordinates</a></li>
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<section id="module-cmath">
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<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>
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<hr class="docutils" />
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<p>This module provides access to mathematical functions for complex numbers. The
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functions in this module accept integers, floating-point numbers or complex
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numbers as arguments. They will also accept any Python object that has either a
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<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
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convert the object to a complex or floating-point number, respectively, and
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the function is then applied to the result of the conversion.</p>
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<div class="admonition note">
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<p class="admonition-title">Note</p>
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<p>For functions involving branch cuts, we have the problem of deciding how to
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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
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standards, we use the sign of zero to distinguish one side of the branch cut
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||
from the other: for a branch cut along (a portion of) the real axis we look
|
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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">complex(-2.0,</span> <span class="pre">-0.0)</span></code> is treated as
|
||
though it lies <em>below</em> the branch cut, and so gives a result on the negative
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||
imaginary axis:</p>
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||
<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="nb">complex</span><span class="p">(</span><span class="o">-</span><span class="mf">2.0</span><span class="p">,</span> <span class="o">-</span><span class="mf">0.0</span><span class="p">))</span>
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<span class="go">-1.4142135623730951j</span>
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||
</pre></div>
|
||
</div>
|
||
<p>But an argument of <code class="docutils literal notranslate"><span class="pre">complex(-2.0,</span> <span class="pre">0.0)</span></code> is treated as though it lies above
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||
the branch cut:</p>
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||
<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="nb">complex</span><span class="p">(</span><span class="o">-</span><span class="mf">2.0</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">))</span>
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<span class="go">1.4142135623730951j</span>
|
||
</pre></div>
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</div>
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</div>
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<table class="docutils align-default">
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<tbody>
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<tr class="row-odd"><td colspan="2"><p><strong>Conversions to and from polar coordinates</strong></p></td>
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</tr>
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<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>
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<td><p>Return the phase of <em>z</em></p></td>
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</tr>
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<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>
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<td><p>Return the representation of <em>z</em> in polar coordinates</p></td>
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</tr>
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<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>
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<td><p>Return the complex number <em>z</em> with polar coordinates <em>r</em> and <em>phi</em></p></td>
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</tr>
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<tr class="row-odd"><td colspan="2"><p><strong>Power and logarithmic functions</strong></p></td>
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</tr>
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<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>
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<td><p>Return <em>e</em> raised to the power <em>z</em></p></td>
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</tr>
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<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>
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<td><p>Return the logarithm of <em>z</em> to the given <em>base</em> (<em>e</em> by default)</p></td>
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</tr>
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<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>
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<td><p>Return the base-10 logarithm of <em>z</em></p></td>
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</tr>
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<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>
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<td><p>Return the square root of <em>z</em></p></td>
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||
</tr>
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<tr class="row-even"><td colspan="2"><p><strong>Trigonometric functions</strong></p></td>
|
||
</tr>
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||
<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>
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||
<td><p>Return the arc cosine of <em>z</em></p></td>
|
||
</tr>
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||
<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>
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||
<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>
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<td><p>Return the arc tangent of <em>z</em></p></td>
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</tr>
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<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>
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<td><p>Return the cosine of <em>z</em></p></td>
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</tr>
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<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>
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<td><p>Return the sine of <em>z</em></p></td>
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||
</tr>
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<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>
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<td><p>Return the tangent of <em>z</em></p></td>
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||
</tr>
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<tr class="row-odd"><td colspan="2"><p><strong>Hyperbolic functions</strong></p></td>
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||
</tr>
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||
<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>
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<td><p>Return the inverse hyperbolic cosine of <em>z</em></p></td>
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</tr>
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<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>
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<td><p>Return the inverse hyperbolic sine of <em>z</em></p></td>
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</tr>
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<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>
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<td><p>Return the inverse hyperbolic tangent of <em>z</em></p></td>
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</tr>
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<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>
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<td><p>Return the hyperbolic cosine of <em>z</em></p></td>
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</tr>
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<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>
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<td><p>Return the hyperbolic sine of <em>z</em></p></td>
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</tr>
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<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>
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<td><p>Return the hyperbolic tangent of <em>z</em></p></td>
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</tr>
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<tr class="row-even"><td colspan="2"><p><strong>Classification functions</strong></p></td>
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</tr>
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<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="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">3.141592653589793</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">-3.141592653589793</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 <a class="reference internal" href="#module-cmath" title="cmath: Mathematical functions for complex numbers."><code class="xref py py-mod docutils literal notranslate"><span class="pre">cmath</span></code></a> 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>.</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 <a class="reference internal" href="#module-cmath" title="cmath: Mathematical functions for complex numbers."><code class="xref py py-mod docutils literal notranslate"><span class="pre">cmath</span></code></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 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>
|
||
</section>
|
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|
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<ul>
|
||
<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>
|
||
<li><a class="reference internal" href="#classification-functions">Classification functions</a></li>
|
||
<li><a class="reference internal" href="#constants">Constants</a></li>
|
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|
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