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<div class="section" id="module-cmath">
<span id="cmath-mathematical-functions-for-complex-numbers"></span><h1>9.3. <a class="reference internal" href="#module-cmath" title="cmath: Mathematical functions for complex numbers."><code class="xref py py-mod docutils literal"><span class="pre">cmath</span></code></a> — Mathematical functions for complex numbers<a class="headerlink" href="#module-cmath" title="Permalink to this headline">¶</a></h1>
<hr class="docutils" />
<p>This module is always available. It provides access to mathematical functions
for complex numbers. The functions in this module accept integers,
floating-point numbers or complex numbers as arguments. They will also accept
any Python object that has either a <a class="reference internal" href="../reference/datamodel.html#object.__complex__" title="object.__complex__"><code class="xref py py-meth docutils literal"><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"><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="first admonition-title">Note</p>
<p class="last">On platforms with hardware and system-level support for signed
zeros, functions involving branch cuts are continuous on <em>both</em>
sides of the branch cut: the sign of the zero distinguishes one
side of the branch cut from the other. On platforms that do not
support signed zeros the continuity is as specified below.</p>
</div>
<div class="section" id="conversions-to-and-from-polar-coordinates">
<h2>9.3.1. Conversions to and from polar coordinates<a class="headerlink" href="#conversions-to-and-from-polar-coordinates" title="Permalink to this headline">¶</a></h2>
<p>A Python complex number <code class="docutils literal"><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"><span class="pre">z.real</span></code> and its <em>imaginary part</em> <code class="docutils literal"><span class="pre">z.imag</span></code>. In other
words:</p>
<div class="highlight-python3"><div class="highlight"><pre><span></span><span class="n">z</span> <span class="o">==</span> <span class="n">z</span><span class="o">.</span><span class="n">real</span> <span class="o">+</span> <span class="n">z</span><span class="o">.</span><span class="n">imag</span><span class="o">*</span><span class="mi">1</span><span class="n">j</span>
</pre></div>
</div>
<p><em>Polar coordinates</em> give an alternative way to represent a complex
number. In polar coordinates, a complex number <em>z</em> is defined by the
modulus <em>r</em> and the phase angle <em>phi</em>. The modulus <em>r</em> is the distance
from <em>z</em> to the origin, while the phase <em>phi</em> is the counterclockwise
angle, measured in radians, from the positive x-axis to the line
segment that joins the origin to <em>z</em>.</p>
<p>The following functions can be used to convert from the native
rectangular coordinates to polar coordinates and back.</p>
<dl class="function">
<dt id="cmath.phase">
<code class="descclassname">cmath.</code><code class="descname">phase</code><span class="sig-paren">(</span><em>x</em><span class="sig-paren">)</span><a class="headerlink" href="#cmath.phase" title="Permalink to this definition">¶</a></dt>
<dd><p>Return the phase of <em>x</em> (also known as the <em>argument</em> of <em>x</em>), as a
float. <code class="docutils literal"><span class="pre">phase(x)</span></code> is equivalent to <code class="docutils literal"><span class="pre">math.atan2(x.imag,</span>
<span class="pre">x.real)</span></code>. The result lies in the range [-π, π], and the branch
cut for this operation lies along the negative real axis,
continuous from above. On systems with support for signed zeros
(which includes most systems in current use), this means that the
sign of the result is the same as the sign of <code class="docutils literal"><span class="pre">x.imag</span></code>, even when
<code class="docutils literal"><span class="pre">x.imag</span></code> is zero:</p>
<div class="highlight-python3"><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="first admonition-title">Note</p>
<p class="last">The modulus (absolute value) of a complex number <em>x</em> can be
computed using the built-in <a class="reference internal" href="functions.html#abs" title="abs"><code class="xref py py-func docutils literal"><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"><span class="pre">cmath</span></code></a> module function for this operation.</p>
</div>
<dl class="function">
<dt id="cmath.polar">
<code class="descclassname">cmath.</code><code class="descname">polar</code><span class="sig-paren">(</span><em>x</em><span class="sig-paren">)</span><a class="headerlink" href="#cmath.polar" title="Permalink to this definition">¶</a></dt>
<dd><p>Return the representation of <em>x</em> in polar coordinates. Returns a
pair <code class="docutils literal"><span class="pre">(r,</span> <span class="pre">phi)</span></code> where <em>r</em> is the modulus of <em>x</em> and phi is the
phase of <em>x</em>. <code class="docutils literal"><span class="pre">polar(x)</span></code> is equivalent to <code class="docutils literal"><span class="pre">(abs(x),</span>
<span class="pre">phase(x))</span></code>.</p>
</dd></dl>
<dl class="function">
<dt id="cmath.rect">
<code class="descclassname">cmath.</code><code class="descname">rect</code><span class="sig-paren">(</span><em>r</em>, <em>phi</em><span class="sig-paren">)</span><a class="headerlink" href="#cmath.rect" title="Permalink to this definition">¶</a></dt>
<dd><p>Return the complex number <em>x</em> with polar coordinates <em>r</em> and <em>phi</em>.
Equivalent to <code class="docutils literal"><span class="pre">r</span> <span class="pre">*</span> <span class="pre">(math.cos(phi)</span> <span class="pre">+</span> <span class="pre">math.sin(phi)*1j)</span></code>.</p>
</dd></dl>
</div>
<div class="section" id="power-and-logarithmic-functions">
<h2>9.3.2. Power and logarithmic functions<a class="headerlink" href="#power-and-logarithmic-functions" title="Permalink to this headline">¶</a></h2>
<dl class="function">
<dt id="cmath.exp">
<code class="descclassname">cmath.</code><code class="descname">exp</code><span class="sig-paren">(</span><em>x</em><span class="sig-paren">)</span><a class="headerlink" href="#cmath.exp" title="Permalink to this definition">¶</a></dt>
<dd><p>Return the exponential value <code class="docutils literal"><span class="pre">e**x</span></code>.</p>
</dd></dl>
<dl class="function">
<dt id="cmath.log">
<code class="descclassname">cmath.</code><code class="descname">log</code><span class="sig-paren">(</span><em>x</em><span class="optional">[</span>, <em>base</em><span class="optional">]</span><span class="sig-paren">)</span><a class="headerlink" href="#cmath.log" title="Permalink to this definition">¶</a></dt>
<dd><p>Returns the logarithm of <em>x</em> to the given <em>base</em>. If the <em>base</em> is not
specified, returns the natural logarithm of <em>x</em>. There is one branch cut, from 0
along the negative real axis to -∞, continuous from above.</p>
</dd></dl>
<dl class="function">
<dt id="cmath.log10">
<code class="descclassname">cmath.</code><code class="descname">log10</code><span class="sig-paren">(</span><em>x</em><span class="sig-paren">)</span><a class="headerlink" href="#cmath.log10" title="Permalink to this definition">¶</a></dt>
<dd><p>Return the base-10 logarithm of <em>x</em>. This has the same branch cut as
<a class="reference internal" href="#cmath.log" title="cmath.log"><code class="xref py py-func docutils literal"><span class="pre">log()</span></code></a>.</p>
</dd></dl>
<dl class="function">
<dt id="cmath.sqrt">
<code class="descclassname">cmath.</code><code class="descname">sqrt</code><span class="sig-paren">(</span><em>x</em><span class="sig-paren">)</span><a class="headerlink" href="#cmath.sqrt" title="Permalink to this definition">¶</a></dt>
<dd><p>Return the square root of <em>x</em>. This has the same branch cut as <a class="reference internal" href="#cmath.log" title="cmath.log"><code class="xref py py-func docutils literal"><span class="pre">log()</span></code></a>.</p>
</dd></dl>
</div>
<div class="section" id="trigonometric-functions">
<h2>9.3.3. Trigonometric functions<a class="headerlink" href="#trigonometric-functions" title="Permalink to this headline">¶</a></h2>
<dl class="function">
<dt id="cmath.acos">
<code class="descclassname">cmath.</code><code class="descname">acos</code><span class="sig-paren">(</span><em>x</em><span class="sig-paren">)</span><a class="headerlink" href="#cmath.acos" title="Permalink to this definition">¶</a></dt>
<dd><p>Return the arc cosine of <em>x</em>. There are two branch cuts: One extends right from
1 along the real axis to ∞, continuous from below. The other extends left from
-1 along the real axis to -∞, continuous from above.</p>
</dd></dl>
<dl class="function">
<dt id="cmath.asin">
<code class="descclassname">cmath.</code><code class="descname">asin</code><span class="sig-paren">(</span><em>x</em><span class="sig-paren">)</span><a class="headerlink" href="#cmath.asin" title="Permalink to this definition">¶</a></dt>
<dd><p>Return the arc sine of <em>x</em>. This has the same branch cuts as <a class="reference internal" href="#cmath.acos" title="cmath.acos"><code class="xref py py-func docutils literal"><span class="pre">acos()</span></code></a>.</p>
</dd></dl>
<dl class="function">
<dt id="cmath.atan">
<code class="descclassname">cmath.</code><code class="descname">atan</code><span class="sig-paren">(</span><em>x</em><span class="sig-paren">)</span><a class="headerlink" href="#cmath.atan" title="Permalink to this definition">¶</a></dt>
<dd><p>Return the arc tangent of <em>x</em>. There are two branch cuts: One extends from
<code class="docutils literal"><span class="pre">1j</span></code> along the imaginary axis to <code class="docutils literal"><span class="pre">∞j</span></code>, continuous from the right. The
other extends from <code class="docutils literal"><span class="pre">-1j</span></code> along the imaginary axis to <code class="docutils literal"><span class="pre">-∞j</span></code>, continuous
from the left.</p>
</dd></dl>
<dl class="function">
<dt id="cmath.cos">
<code class="descclassname">cmath.</code><code class="descname">cos</code><span class="sig-paren">(</span><em>x</em><span class="sig-paren">)</span><a class="headerlink" href="#cmath.cos" title="Permalink to this definition">¶</a></dt>
<dd><p>Return the cosine of <em>x</em>.</p>
</dd></dl>
<dl class="function">
<dt id="cmath.sin">
<code class="descclassname">cmath.</code><code class="descname">sin</code><span class="sig-paren">(</span><em>x</em><span class="sig-paren">)</span><a class="headerlink" href="#cmath.sin" title="Permalink to this definition">¶</a></dt>
<dd><p>Return the sine of <em>x</em>.</p>
</dd></dl>
<dl class="function">
<dt id="cmath.tan">
<code class="descclassname">cmath.</code><code class="descname">tan</code><span class="sig-paren">(</span><em>x</em><span class="sig-paren">)</span><a class="headerlink" href="#cmath.tan" title="Permalink to this definition">¶</a></dt>
<dd><p>Return the tangent of <em>x</em>.</p>
</dd></dl>
</div>
<div class="section" id="hyperbolic-functions">
<h2>9.3.4. Hyperbolic functions<a class="headerlink" href="#hyperbolic-functions" title="Permalink to this headline">¶</a></h2>
<dl class="function">
<dt id="cmath.acosh">
<code class="descclassname">cmath.</code><code class="descname">acosh</code><span class="sig-paren">(</span><em>x</em><span class="sig-paren">)</span><a class="headerlink" href="#cmath.acosh" title="Permalink to this definition">¶</a></dt>
<dd><p>Return the inverse hyperbolic cosine of <em>x</em>. There is one branch cut,
extending left from 1 along the real axis to -∞, continuous from above.</p>
</dd></dl>
<dl class="function">
<dt id="cmath.asinh">
<code class="descclassname">cmath.</code><code class="descname">asinh</code><span class="sig-paren">(</span><em>x</em><span class="sig-paren">)</span><a class="headerlink" href="#cmath.asinh" title="Permalink to this definition">¶</a></dt>
<dd><p>Return the inverse hyperbolic sine of <em>x</em>. There are two branch cuts:
One extends from <code class="docutils literal"><span class="pre">1j</span></code> along the imaginary axis to <code class="docutils literal"><span class="pre">∞j</span></code>,
continuous from the right. The other extends from <code class="docutils literal"><span class="pre">-1j</span></code> along
the imaginary axis to <code class="docutils literal"><span class="pre">-∞j</span></code>, continuous from the left.</p>
</dd></dl>
<dl class="function">
<dt id="cmath.atanh">
<code class="descclassname">cmath.</code><code class="descname">atanh</code><span class="sig-paren">(</span><em>x</em><span class="sig-paren">)</span><a class="headerlink" href="#cmath.atanh" title="Permalink to this definition">¶</a></dt>
<dd><p>Return the inverse hyperbolic tangent of <em>x</em>. There are two branch cuts: One
extends from <code class="docutils literal"><span class="pre">1</span></code> along the real axis to <code class="docutils literal"><span class="pre">∞</span></code>, continuous from below. The
other extends from <code class="docutils literal"><span class="pre">-1</span></code> along the real axis to <code class="docutils literal"><span class="pre">-∞</span></code>, continuous from
above.</p>
</dd></dl>
<dl class="function">
<dt id="cmath.cosh">
<code class="descclassname">cmath.</code><code class="descname">cosh</code><span class="sig-paren">(</span><em>x</em><span class="sig-paren">)</span><a class="headerlink" href="#cmath.cosh" title="Permalink to this definition">¶</a></dt>
<dd><p>Return the hyperbolic cosine of <em>x</em>.</p>
</dd></dl>
<dl class="function">
<dt id="cmath.sinh">
<code class="descclassname">cmath.</code><code class="descname">sinh</code><span class="sig-paren">(</span><em>x</em><span class="sig-paren">)</span><a class="headerlink" href="#cmath.sinh" title="Permalink to this definition">¶</a></dt>
<dd><p>Return the hyperbolic sine of <em>x</em>.</p>
</dd></dl>
<dl class="function">
<dt id="cmath.tanh">
<code class="descclassname">cmath.</code><code class="descname">tanh</code><span class="sig-paren">(</span><em>x</em><span class="sig-paren">)</span><a class="headerlink" href="#cmath.tanh" title="Permalink to this definition">¶</a></dt>
<dd><p>Return the hyperbolic tangent of <em>x</em>.</p>
</dd></dl>
</div>
<div class="section" id="classification-functions">
<h2>9.3.5. Classification functions<a class="headerlink" href="#classification-functions" title="Permalink to this headline">¶</a></h2>
<dl class="function">
<dt id="cmath.isfinite">
<code class="descclassname">cmath.</code><code class="descname">isfinite</code><span class="sig-paren">(</span><em>x</em><span class="sig-paren">)</span><a class="headerlink" href="#cmath.isfinite" title="Permalink to this definition">¶</a></dt>
<dd><p>Return <code class="docutils literal"><span class="pre">True</span></code> if both the real and imaginary parts of <em>x</em> are finite, and
<code class="docutils literal"><span class="pre">False</span></code> otherwise.</p>
<div class="versionadded">
<p><span class="versionmodified">New in version 3.2.</span></p>
</div>
</dd></dl>
<dl class="function">
<dt id="cmath.isinf">
<code class="descclassname">cmath.</code><code class="descname">isinf</code><span class="sig-paren">(</span><em>x</em><span class="sig-paren">)</span><a class="headerlink" href="#cmath.isinf" title="Permalink to this definition">¶</a></dt>
<dd><p>Return <code class="docutils literal"><span class="pre">True</span></code> if either the real or the imaginary part of <em>x</em> is an
infinity, and <code class="docutils literal"><span class="pre">False</span></code> otherwise.</p>
</dd></dl>
<dl class="function">
<dt id="cmath.isnan">
<code class="descclassname">cmath.</code><code class="descname">isnan</code><span class="sig-paren">(</span><em>x</em><span class="sig-paren">)</span><a class="headerlink" href="#cmath.isnan" title="Permalink to this definition">¶</a></dt>
<dd><p>Return <code class="docutils literal"><span class="pre">True</span></code> if either the real or the imaginary part of <em>x</em> is a NaN,
and <code class="docutils literal"><span class="pre">False</span></code> otherwise.</p>
</dd></dl>
<dl class="function">
<dt id="cmath.isclose">
<code class="descclassname">cmath.</code><code class="descname">isclose</code><span class="sig-paren">(</span><em>a</em>, <em>b</em>, <em>*</em>, <em>rel_tol=1e-09</em>, <em>abs_tol=0.0</em><span class="sig-paren">)</span><a class="headerlink" href="#cmath.isclose" title="Permalink to this definition">¶</a></dt>
<dd><p>Return <code class="docutils literal"><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"><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.</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"><span class="pre">rel_tol=0.05</span></code>. The default
tolerance is <code class="docutils literal"><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 greater than zero.</p>
<p><em>abs_tol</em> is the minimum absolute tolerance – useful for comparisons near
zero. <em>abs_tol</em> must be at least zero.</p>
<p>If no errors occur, the result will be:
<code class="docutils literal"><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>The IEEE 754 special values of <code class="docutils literal"><span class="pre">NaN</span></code>, <code class="docutils literal"><span class="pre">inf</span></code>, and <code class="docutils literal"><span class="pre">-inf</span></code> will be
handled according to IEEE rules. Specifically, <code class="docutils literal"><span class="pre">NaN</span></code> is not considered
close to any other value, including <code class="docutils literal"><span class="pre">NaN</span></code>. <code class="docutils literal"><span class="pre">inf</span></code> and <code class="docutils literal"><span class="pre">-inf</span></code> are only
considered close to themselves.</p>
<div class="versionadded">
<p><span class="versionmodified">New in version 3.5.</span></p>
</div>
<div class="admonition seealso">
<p class="first admonition-title">See also</p>
<p class="last"><span class="target" id="index-0"></span><a class="pep reference external" href="https://www.python.org/dev/peps/pep-0485"><strong>PEP 485</strong></a> – A function for testing approximate equality</p>
</div>
</dd></dl>
</div>
<div class="section" id="constants">
<h2>9.3.6. Constants<a class="headerlink" href="#constants" title="Permalink to this headline">¶</a></h2>
<dl class="data">
<dt id="cmath.pi">
<code class="descclassname">cmath.</code><code class="descname">pi</code><a class="headerlink" href="#cmath.pi" title="Permalink to this definition">¶</a></dt>
<dd><p>The mathematical constant <em>π</em>, as a float.</p>
</dd></dl>
<dl class="data">
<dt id="cmath.e">
<code class="descclassname">cmath.</code><code class="descname">e</code><a class="headerlink" href="#cmath.e" title="Permalink to this definition">¶</a></dt>
<dd><p>The mathematical constant <em>e</em>, as a float.</p>
</dd></dl>
<p id="index-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"><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"><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"><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="first admonition-title">See also</p>
<p class="last">Kahan, W: Branch cuts for complex elementary functions; or, Much ado about
nothing’s sign bit. In Iserles, A., and Powell, M. (eds.), The state of the art
in numerical analysis. Clarendon Press (1987) pp165–211.</p>
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<ul>
<li><a class="reference internal" href="#">9.3. <code class="docutils literal"><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">9.3.1. Conversions to and from polar coordinates</a></li>
<li><a class="reference internal" href="#power-and-logarithmic-functions">9.3.2. Power and logarithmic functions</a></li>
<li><a class="reference internal" href="#trigonometric-functions">9.3.3. Trigonometric functions</a></li>
<li><a class="reference internal" href="#hyperbolic-functions">9.3.4. Hyperbolic functions</a></li>
<li><a class="reference internal" href="#classification-functions">9.3.5. Classification functions</a></li>
<li><a class="reference internal" href="#constants">9.3.6. Constants</a></li>
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