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Code Editor : math.html

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 <section id="module-math">
<h1>9.2. <a class="reference internal" href="#module-math" title="math: Mathematical functions (sin() etc.)."><code class="xref py py-mod docutils literal notranslate">math</code></a> — Mathematical functions<a class="headerlink" href="#module-math" title="Permalink to this headline">¶</a></h1>
This module is always available. It provides access to the mathematical
functions defined by the C standard.
These functions cannot be used with complex numbers; use the functions of the
same name from the <a class="reference internal" href="cmath.html#module-cmath" title="cmath: Mathematical functions for complex numbers."><code class="xref py py-mod docutils literal notranslate">cmath</code></a> module if you require support for complex
numbers. The distinction between functions which support complex numbers and
those which don’t is made since most users do not want to learn quite as much
mathematics as required to understand complex numbers. Receiving an exception
instead of a complex result allows earlier detection of the unexpected complex
number used as a parameter, so that the programmer can determine how and why it
was generated in the first place.
The following functions are provided by this module. Except when explicitly
noted otherwise, all return values are floats.
<section id="number-theoretic-and-representation-functions">
<h2>9.2.1. Number-theoretic and representation functions<a class="headerlink" href="#number-theoretic-and-representation-functions" title="Permalink to this headline">¶</a></h2>
<dl class="py function">
<dt class="sig sig-object py" id="math.ceil">
math.ceil(x)<a class="headerlink" href="#math.ceil" title="Permalink to this definition">¶</a></dt>
<dd>Return the ceiling of x as a float, the smallest integer value greater than or
equal to x.
</dd></dl>

<dl class="py function">
<dt class="sig sig-object py" id="math.copysign">
math.copysign(x, y)<a class="headerlink" href="#math.copysign" title="Permalink to this definition">¶</a></dt>
<dd>Return x with the sign of y. On a platform that supports
signed zeros, <code class="docutils literal notranslate">copysign(1.0, -0.0)</code> returns -1.0.
<div class="versionadded">
New in version 2.6.
</div>
</dd></dl>

<dl class="py function">
<dt class="sig sig-object py" id="math.fabs">
math.fabs(x)<a class="headerlink" href="#math.fabs" title="Permalink to this definition">¶</a></dt>
<dd>Return the absolute value of x.
</dd></dl>

<dl class="py function">
<dt class="sig sig-object py" id="math.factorial">
math.factorial(x)<a class="headerlink" href="#math.factorial" title="Permalink to this definition">¶</a></dt>
<dd>Return x factorial. Raises <code class="xref py py-exc docutils literal notranslate">ValueError</code> if x is not integral or
is negative.
<div class="versionadded">
New in version 2.6.
</div>
</dd></dl>

<dl class="py function">
<dt class="sig sig-object py" id="math.floor">
math.floor(x)<a class="headerlink" href="#math.floor" title="Permalink to this definition">¶</a></dt>
<dd>Return the floor of x as a float, the largest integer value less than or equal
to x.
</dd></dl>

<dl class="py function">
<dt class="sig sig-object py" id="math.fmod">
math.fmod(x, y)<a class="headerlink" href="#math.fmod" title="Permalink to this definition">¶</a></dt>
<dd>Return <code class="docutils literal notranslate">fmod(x, y)</code>, as defined by the platform C library. Note that the
Python expression <code class="docutils literal notranslate">x % y</code> may not return the same result. The intent of the C
standard is that <code class="docutils literal notranslate">fmod(x, y)</code> be exactly (mathematically; to infinite
precision) equal to <code class="docutils literal notranslate">x - n*y</code> for some integer n such that the result has
the same sign as x and magnitude less than <code class="docutils literal notranslate">abs(y)</code>. Python’s <code class="docutils literal notranslate">x % y</code>
returns a result with the sign of y instead, and may not be exactly computable
for float arguments. For example, <code class="docutils literal notranslate">fmod(-1e-100, 1e100)</code> is <code class="docutils literal notranslate">-1e-100</code>, but
the result of Python’s <code class="docutils literal notranslate">-1e-100 % 1e100</code> is <code class="docutils literal notranslate">1e100-1e-100</code>, which cannot be
represented exactly as a float, and rounds to the surprising <code class="docutils literal notranslate">1e100</code>. For
this reason, function <a class="reference internal" href="#math.fmod" title="math.fmod"><code class="xref py py-func docutils literal notranslate">fmod()</code></a> is generally preferred when working with
floats, while Python’s <code class="docutils literal notranslate">x % y</code> is preferred when working with integers.
</dd></dl>

<dl class="py function">
<dt class="sig sig-object py" id="math.frexp">
math.frexp(x)<a class="headerlink" href="#math.frexp" title="Permalink to this definition">¶</a></dt>
<dd>Return the mantissa and exponent of x as the pair <code class="docutils literal notranslate">(m, e)</code>. m is a float
and e is an integer such that <code class="docutils literal notranslate">x == m * 2**e</code> exactly. If x is zero,
returns <code class="docutils literal notranslate">(0.0, 0)</code>, otherwise <code class="docutils literal notranslate">0.5 &lt;= abs(m) &lt; 1</code>. This is used to “pick
apart” the internal representation of a float in a portable way.
</dd></dl>

<dl class="py function">
<dt class="sig sig-object py" id="math.fsum">
math.fsum(iterable)<a class="headerlink" href="#math.fsum" title="Permalink to this definition">¶</a></dt>
<dd>Return an accurate floating point sum of values in the iterable. Avoids
loss of precision by tracking multiple intermediate partial sums:
<div class="highlight-default notranslate"><div class="highlight"><pre>&gt;&gt;&gt; sum([.1, .1, .1, .1, .1, .1, .1, .1, .1, .1])
0.9999999999999999
&gt;&gt;&gt; fsum([.1, .1, .1, .1, .1, .1, .1, .1, .1, .1])
1.0
</pre></div>
</div>
The algorithm’s accuracy depends on IEEE-754 arithmetic guarantees and the
typical case where the rounding mode is half-even. On some non-Windows
builds, the underlying C library uses extended precision addition and may
occasionally double-round an intermediate sum causing it to be off in its
least significant bit.
For further discussion and two alternative approaches, see the <a class="reference external" href="https://code.activestate.com/recipes/393090/">ASPN cookbook
recipes for accurate floating point summation</a>.
<div class="versionadded">
New in version 2.6.
</div>
</dd></dl>

<dl class="py function">
<dt class="sig sig-object py" id="math.isinf">
math.isinf(x)<a class="headerlink" href="#math.isinf" title="Permalink to this definition">¶</a></dt>
<dd>Check if the float x is positive or negative infinity.
<div class="versionadded">
New in version 2.6.
</div>
</dd></dl>

<dl class="py function">
<dt class="sig sig-object py" id="math.isnan">
math.isnan(x)<a class="headerlink" href="#math.isnan" title="Permalink to this definition">¶</a></dt>
<dd>Check if the float x is a NaN (not a number). For more information
on NaNs, see the IEEE 754 standards.
<div class="versionadded">
New in version 2.6.
</div>
</dd></dl>

<dl class="py function">
<dt class="sig sig-object py" id="math.ldexp">
math.ldexp(x, i)<a class="headerlink" href="#math.ldexp" title="Permalink to this definition">¶</a></dt>
<dd>Return <code class="docutils literal notranslate">x * (2**i)</code>. This is essentially the inverse of function
<a class="reference internal" href="#math.frexp" title="math.frexp"><code class="xref py py-func docutils literal notranslate">frexp()</code></a>.
</dd></dl>

<dl class="py function">
<dt class="sig sig-object py" id="math.modf">
math.modf(x)<a class="headerlink" href="#math.modf" title="Permalink to this definition">¶</a></dt>
<dd>Return the fractional and integer parts of x. Both results carry the sign
of x and are floats.
</dd></dl>

<dl class="py function">
<dt class="sig sig-object py" id="math.trunc">
math.trunc(x)<a class="headerlink" href="#math.trunc" title="Permalink to this definition">¶</a></dt>
<dd>Return the <a class="reference internal" href="numbers.html#numbers.Real" title="numbers.Real"><code class="xref py py-class docutils literal notranslate">Real</code></a> value x truncated to an
<a class="reference internal" href="numbers.html#numbers.Integral" title="numbers.Integral"><code class="xref py py-class docutils literal notranslate">Integral</code></a> (usually a long integer). Uses the
<code class="docutils literal notranslate">__trunc__</code> method.
<div class="versionadded">
New in version 2.6.
</div>
</dd></dl>

Note that <a class="reference internal" href="#math.frexp" title="math.frexp"><code class="xref py py-func docutils literal notranslate">frexp()</code></a> and <a class="reference internal" href="#math.modf" title="math.modf"><code class="xref py py-func docutils literal notranslate">modf()</code></a> have a different call/return pattern
than their C equivalents: they take a single argument and return a pair of
values, rather than returning their second return value through an ‘output
parameter’ (there is no such thing in Python).
For the <a class="reference internal" href="#math.ceil" title="math.ceil"><code class="xref py py-func docutils literal notranslate">ceil()</code></a>, <a class="reference internal" href="#math.floor" title="math.floor"><code class="xref py py-func docutils literal notranslate">floor()</code></a>, and <a class="reference internal" href="#math.modf" title="math.modf"><code class="xref py py-func docutils literal notranslate">modf()</code></a> functions, note that all
floating-point numbers of sufficiently large magnitude are exact integers.
Python floats typically carry no more than 53 bits of precision (the same as the
platform C double type), in which case any float x with <code class="docutils literal notranslate">abs(x) &gt;= 2**52</code>
necessarily has no fractional bits.
</section>
<section id="power-and-logarithmic-functions">
<h2>9.2.2. Power and logarithmic functions<a class="headerlink" href="#power-and-logarithmic-functions" title="Permalink to this headline">¶</a></h2>
<dl class="py function">
<dt class="sig sig-object py" id="math.exp">
math.exp(x)<a class="headerlink" href="#math.exp" title="Permalink to this definition">¶</a></dt>
<dd>Return <code class="docutils literal notranslate">e**x</code>.
</dd></dl>

<dl class="py function">
<dt class="sig sig-object py" id="math.expm1">
math.expm1(x)<a class="headerlink" href="#math.expm1" title="Permalink to this definition">¶</a></dt>
<dd>Return <code class="docutils literal notranslate">e**x - 1</code>. For small floats x, the subtraction in
<code class="docutils literal notranslate">exp(x) - 1</code> can result in a significant loss of precision; the
<a class="reference internal" href="#math.expm1" title="math.expm1"><code class="xref py py-func docutils literal notranslate">expm1()</code></a> function provides a way to compute this quantity to
full precision:
<div class="highlight-default notranslate"><div class="highlight"><pre>&gt;&gt;&gt; from math import exp, expm1
&gt;&gt;&gt; exp(1e-5) - 1 # gives result accurate to 11 places
1.0000050000069649e-05
&gt;&gt;&gt; expm1(1e-5) # result accurate to full precision
1.0000050000166668e-05
</pre></div>
</div>
<div class="versionadded">
New in version 2.7.
</div>
</dd></dl>

<dl class="py function">
<dt class="sig sig-object py" id="math.log">
math.log(x[, base])<a class="headerlink" href="#math.log" title="Permalink to this definition">¶</a></dt>
<dd>With one argument, return the natural logarithm of x (to base e).
With two arguments, return the logarithm of x to the given base,
calculated as <code class="docutils literal notranslate">log(x)/log(base)</code>.
<div class="versionchanged">
Changed in version 2.3: base argument added.
</div>
</dd></dl>

<dl class="py function">
<dt class="sig sig-object py" id="math.log1p">
math.log1p(x)<a class="headerlink" href="#math.log1p" title="Permalink to this definition">¶</a></dt>
<dd>Return the natural logarithm of 1+x (base e). The
result is calculated in a way which is accurate for x near zero.
<div class="versionadded">
New in version 2.6.
</div>
</dd></dl>

<dl class="py function">
<dt class="sig sig-object py" id="math.log10">
math.log10(x)<a class="headerlink" href="#math.log10" title="Permalink to this definition">¶</a></dt>
<dd>Return the base-10 logarithm of x. This is usually more accurate
than <code class="docutils literal notranslate">log(x, 10)</code>.
</dd></dl>

<dl class="py function">
<dt class="sig sig-object py" id="math.pow">
math.pow(x, y)<a class="headerlink" href="#math.pow" title="Permalink to this definition">¶</a></dt>
<dd>Return <code class="docutils literal notranslate">x</code> raised to the power <code class="docutils literal notranslate">y</code>. Exceptional cases follow
Annex ‘F’ of the C99 standard as far as possible. In particular,
<code class="docutils literal notranslate">pow(1.0, x)</code> and <code class="docutils literal notranslate">pow(x, 0.0)</code> always return <code class="docutils literal notranslate">1.0</code>, even
when <code class="docutils literal notranslate">x</code> is a zero or a NaN. If both <code class="docutils literal notranslate">x</code> and <code class="docutils literal notranslate">y</code> are finite,
<code class="docutils literal notranslate">x</code> is negative, and <code class="docutils literal notranslate">y</code> is not an integer then <code class="docutils literal notranslate">pow(x, y)</code>
is undefined, and raises <code class="xref py py-exc docutils literal notranslate">ValueError</code>.
Unlike the built-in <code class="docutils literal notranslate">**</code> operator, <a class="reference internal" href="#math.pow" title="math.pow"><code class="xref py py-func docutils literal notranslate">math.pow()</code></a> converts both
its arguments to type <a class="reference internal" href="functions.html#float" title="float"><code class="xref py py-class docutils literal notranslate">float</code></a>. Use <code class="docutils literal notranslate">**</code> or the built-in
<a class="reference internal" href="functions.html#pow" title="pow"><code class="xref py py-func docutils literal notranslate">pow()</code></a> function for computing exact integer powers.
<div class="versionchanged">
Changed in version 2.6: The outcome of <code class="docutils literal notranslate">1**nan</code> and <code class="docutils literal notranslate">nan**0</code> was undefined.
</div>
</dd></dl>

<dl class="py function">
<dt class="sig sig-object py" id="math.sqrt">
math.sqrt(x)<a class="headerlink" href="#math.sqrt" title="Permalink to this definition">¶</a></dt>
<dd>Return the square root of x.
</dd></dl>

</section>
<section id="trigonometric-functions">
<h2>9.2.3. Trigonometric functions<a class="headerlink" href="#trigonometric-functions" title="Permalink to this headline">¶</a></h2>
<dl class="py function">
<dt class="sig sig-object py" id="math.acos">
math.acos(x)<a class="headerlink" href="#math.acos" title="Permalink to this definition">¶</a></dt>
<dd>Return the arc cosine of x, in radians.
</dd></dl>

<dl class="py function">
<dt class="sig sig-object py" id="math.asin">
math.asin(x)<a class="headerlink" href="#math.asin" title="Permalink to this definition">¶</a></dt>
<dd>Return the arc sine of x, in radians.
</dd></dl>

<dl class="py function">
<dt class="sig sig-object py" id="math.atan">
math.atan(x)<a class="headerlink" href="#math.atan" title="Permalink to this definition">¶</a></dt>
<dd>Return the arc tangent of x, in radians.
</dd></dl>

<dl class="py function">
<dt class="sig sig-object py" id="math.atan2">
math.atan2(y, x)<a class="headerlink" href="#math.atan2" title="Permalink to this definition">¶</a></dt>
<dd>Return <code class="docutils literal notranslate">atan(y / x)</code>, in radians. The result is between <code class="docutils literal notranslate">-pi</code> and <code class="docutils literal notranslate">pi</code>.
The vector in the plane from the origin to point <code class="docutils literal notranslate">(x, y)</code> makes this angle
with the positive X axis. The point of <a class="reference internal" href="#math.atan2" title="math.atan2"><code class="xref py py-func docutils literal notranslate">atan2()</code></a> is that the signs of both
inputs are known to it, so it can compute the correct quadrant for the angle.
For example, <code class="docutils literal notranslate">atan(1)</code> and <code class="docutils literal notranslate">atan2(1, 1)</code> are both <code class="docutils literal notranslate">pi/4</code>, but <code class="docutils literal notranslate">atan2(-1,
-1)</code> is <code class="docutils literal notranslate">-3*pi/4</code>.
</dd></dl>

<dl class="py function">
<dt class="sig sig-object py" id="math.cos">
math.cos(x)<a class="headerlink" href="#math.cos" title="Permalink to this definition">¶</a></dt>
<dd>Return the cosine of x radians.
</dd></dl>

<dl class="py function">
<dt class="sig sig-object py" id="math.hypot">
math.hypot(x, y)<a class="headerlink" href="#math.hypot" title="Permalink to this definition">¶</a></dt>
<dd>Return the Euclidean norm, <code class="docutils literal notranslate">sqrt(x*x + y*y)</code>. This is the length of the vector
from the origin to point <code class="docutils literal notranslate">(x, y)</code>.
</dd></dl>

<dl class="py function">
<dt class="sig sig-object py" id="math.sin">
math.sin(x)<a class="headerlink" href="#math.sin" title="Permalink to this definition">¶</a></dt>
<dd>Return the sine of x radians.
</dd></dl>

<dl class="py function">
<dt class="sig sig-object py" id="math.tan">
math.tan(x)<a class="headerlink" href="#math.tan" title="Permalink to this definition">¶</a></dt>
<dd>Return the tangent of x radians.
</dd></dl>

</section>
<section id="angular-conversion">
<h2>9.2.4. Angular conversion<a class="headerlink" href="#angular-conversion" title="Permalink to this headline">¶</a></h2>
<dl class="py function">
<dt class="sig sig-object py" id="math.degrees">
math.degrees(x)<a class="headerlink" href="#math.degrees" title="Permalink to this definition">¶</a></dt>
<dd>Convert angle x from radians to degrees.
</dd></dl>

<dl class="py function">
<dt class="sig sig-object py" id="math.radians">
math.radians(x)<a class="headerlink" href="#math.radians" title="Permalink to this definition">¶</a></dt>
<dd>Convert angle x from degrees to radians.
</dd></dl>

</section>
<section id="hyperbolic-functions">
<h2>9.2.5. Hyperbolic functions<a class="headerlink" href="#hyperbolic-functions" title="Permalink to this headline">¶</a></h2>
<dl class="py function">
<dt class="sig sig-object py" id="math.acosh">
math.acosh(x)<a class="headerlink" href="#math.acosh" title="Permalink to this definition">¶</a></dt>
<dd>Return the inverse hyperbolic cosine of x.
<div class="versionadded">
New in version 2.6.
</div>
</dd></dl>

<dl class="py function">
<dt class="sig sig-object py" id="math.asinh">
math.asinh(x)<a class="headerlink" href="#math.asinh" title="Permalink to this definition">¶</a></dt>
<dd>Return the inverse hyperbolic sine of x.
<div class="versionadded">
New in version 2.6.
</div>
</dd></dl>

<dl class="py function">
<dt class="sig sig-object py" id="math.atanh">
math.atanh(x)<a class="headerlink" href="#math.atanh" title="Permalink to this definition">¶</a></dt>
<dd>Return the inverse hyperbolic tangent of x.
<div class="versionadded">
New in version 2.6.
</div>
</dd></dl>

<dl class="py function">
<dt class="sig sig-object py" id="math.cosh">
math.cosh(x)<a class="headerlink" href="#math.cosh" title="Permalink to this definition">¶</a></dt>
<dd>Return the hyperbolic cosine of x.
</dd></dl>

<dl class="py function">
<dt class="sig sig-object py" id="math.sinh">
math.sinh(x)<a class="headerlink" href="#math.sinh" title="Permalink to this definition">¶</a></dt>
<dd>Return the hyperbolic sine of x.
</dd></dl>

<dl class="py function">
<dt class="sig sig-object py" id="math.tanh">
math.tanh(x)<a class="headerlink" href="#math.tanh" title="Permalink to this definition">¶</a></dt>
<dd>Return the hyperbolic tangent of x.
</dd></dl>

</section>
<section id="special-functions">
<h2>9.2.6. Special functions<a class="headerlink" href="#special-functions" title="Permalink to this headline">¶</a></h2>
<dl class="py function">
<dt class="sig sig-object py" id="math.erf">
math.erf(x)<a class="headerlink" href="#math.erf" title="Permalink to this definition">¶</a></dt>
<dd>Return the error function at x.
<div class="versionadded">
New in version 2.7.
</div>
</dd></dl>

<dl class="py function">
<dt class="sig sig-object py" id="math.erfc">
math.erfc(x)<a class="headerlink" href="#math.erfc" title="Permalink to this definition">¶</a></dt>
<dd>Return the complementary error function at x.
<div class="versionadded">
New in version 2.7.
</div>
</dd></dl>

<dl class="py function">
<dt class="sig sig-object py" id="math.gamma">
math.gamma(x)<a class="headerlink" href="#math.gamma" title="Permalink to this definition">¶</a></dt>
<dd>Return the Gamma function at x.
<div class="versionadded">
New in version 2.7.
</div>
</dd></dl>

<dl class="py function">
<dt class="sig sig-object py" id="math.lgamma">
math.lgamma(x)<a class="headerlink" href="#math.lgamma" title="Permalink to this definition">¶</a></dt>
<dd>Return the natural logarithm of the absolute value of the Gamma
function at x.
<div class="versionadded">
New in version 2.7.
</div>
</dd></dl>

</section>
<section id="constants">
<h2>9.2.7. Constants<a class="headerlink" href="#constants" title="Permalink to this headline">¶</a></h2>
<dl class="py data">
<dt class="sig sig-object py" id="math.pi">
math.pi<a class="headerlink" href="#math.pi" title="Permalink to this definition">¶</a></dt>
<dd>The mathematical constant π = 3.141592…, to available precision.
</dd></dl>

<dl class="py data">
<dt class="sig sig-object py" id="math.e">
math.e<a class="headerlink" href="#math.e" title="Permalink to this definition">¶</a></dt>
<dd>The mathematical constant e = 2.718281…, to available precision.
</dd></dl>

<div class="impl-detail compound">
CPython implementation detail: The <a class="reference internal" href="#module-math" title="math: Mathematical functions (sin() etc.)."><code class="xref py py-mod docutils literal notranslate">math</code></a> module consists mostly of thin wrappers around the platform C
math library functions. Behavior in exceptional cases follows Annex F of
the C99 standard where appropriate. The current implementation will raise
<code class="xref py py-exc docutils literal notranslate">ValueError</code> for invalid operations like <code class="docutils literal notranslate">sqrt(-1.0)</code> or <code class="docutils literal notranslate">log(0.0)</code>
(where C99 Annex F recommends signaling invalid operation or divide-by-zero),
and <code class="xref py py-exc docutils literal notranslate">OverflowError</code> for results that overflow (for example,
<code class="docutils literal notranslate">exp(1000.0)</code>). A NaN will not be returned from any of the functions
above unless one or more of the input arguments was a NaN; in that case,
most functions will return a NaN, but (again following C99 Annex F) there
are some exceptions to this rule, for example <code class="docutils literal notranslate">pow(float('nan'), 0.0)</code> or
<code class="docutils literal notranslate">hypot(float('nan'), float('inf'))</code>.
Note that Python makes no effort to distinguish signaling NaNs from
quiet NaNs, and behavior for signaling NaNs remains unspecified.
Typical behavior is to treat all NaNs as though they were quiet.
<div class="compound-last versionchanged">
Changed in version 2.6: Behavior in special cases now aims to follow C99 Annex F. In earlier
versions of Python the behavior in special cases was loosely specified.
</div>
</div>
<div class="admonition seealso">
See also
<dl class="simple">
<dt>Module <a class="reference internal" href="cmath.html#module-cmath" title="cmath: Mathematical functions for complex numbers."><code class="xref py py-mod docutils literal notranslate">cmath</code></a></dt><dd>Complex number versions of many of these functions.
</dd>
</dl>
</div>
</section>
</section>

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 <h3><a href="../contents.html">Table of Contents</a></h3>
 <ul>
<li><a class="reference internal" href="#">9.2. <code class="xref py py-mod docutils literal notranslate">math</code> — Mathematical functions</a><ul>
<li><a class="reference internal" href="#number-theoretic-and-representation-functions">9.2.1. Number-theoretic and representation functions</a></li>
<li><a class="reference internal" href="#power-and-logarithmic-functions">9.2.2. Power and logarithmic functions</a></li>
<li><a class="reference internal" href="#trigonometric-functions">9.2.3. Trigonometric functions</a></li>
<li><a class="reference internal" href="#angular-conversion">9.2.4. Angular conversion</a></li>
<li><a class="reference internal" href="#hyperbolic-functions">9.2.5. Hyperbolic functions</a></li>
<li><a class="reference internal" href="#special-functions">9.2.6. Special functions</a></li>
<li><a class="reference internal" href="#constants">9.2.7. Constants</a></li>
</ul>
</li>
</ul>

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