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<head>
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<title>The Mathematical Constants</title>
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<div class="section">
<div class="titlepage"><div><div><h2 class="title" style="clear: both">
<a name="math_toolkit.constants"></a><a class="link" href="constants.html" title="The Mathematical Constants">The Mathematical Constants</a>
</h2></div></div></div>
<p>
This section lists the mathematical constants, their use(s) (and sometimes
rationale for their inclusion).
</p>
<div class="table">
<a name="math_toolkit.constants.mathematical_constants"></a><p class="title"><b>Table&#160;4.1.&#160;Mathematical Constants</b></p>
<div class="table-contents"><table class="table" summary="Mathematical Constants">
<colgroup>
<col>
<col>
<col>
<col>
</colgroup>
<thead><tr>
<th>
<p>
name
</p>
</th>
<th>
<p>
formula
</p>
</th>
<th>
<p>
Value (6 decimals)
</p>
</th>
<th>
<p>
Uses and Rationale
</p>
</th>
</tr></thead>
<tbody>
<tr>
<td>
<p>
<span class="bold"><strong>Rational fractions</strong></span>
</p>
</td>
<td>
</td>
<td>
</td>
<td>
</td>
</tr>
<tr>
<td>
<p>
half
</p>
</td>
<td>
<p>
1/2
</p>
</td>
<td>
<p>
0.5
</p>
</td>
<td>
</td>
</tr>
<tr>
<td>
<p>
third
</p>
</td>
<td>
<p>
1/3
</p>
</td>
<td>
<p>
0.333333
</p>
</td>
<td>
</td>
</tr>
<tr>
<td>
<p>
two_thirds
</p>
</td>
<td>
<p>
2/3
</p>
</td>
<td>
<p>
0.66667
</p>
</td>
<td>
</td>
</tr>
<tr>
<td>
<p>
three_quarters
</p>
</td>
<td>
<p>
3/4
</p>
</td>
<td>
<p>
0.75
</p>
</td>
<td>
</td>
</tr>
<tr>
<td>
<p>
<span class="bold"><strong>two and related</strong></span>
</p>
</td>
<td>
</td>
<td>
</td>
<td>
</td>
</tr>
<tr>
<td>
<p>
root_two
</p>
</td>
<td>
<p>
&#8730;2
</p>
</td>
<td>
<p>
1.41421
</p>
</td>
<td>
</td>
</tr>
<tr>
<td>
<p>
root_three
</p>
</td>
<td>
<p>
&#8730;3
</p>
</td>
<td>
<p>
1.73205
</p>
</td>
<td>
</td>
</tr>
<tr>
<td>
<p>
half_root_two
</p>
</td>
<td>
<p>
&#8730;2 /2
</p>
</td>
<td>
<p>
0.707106
</p>
</td>
<td>
</td>
</tr>
<tr>
<td>
<p>
ln_two
</p>
</td>
<td>
<p>
ln(2)
</p>
</td>
<td>
<p>
0.693147
</p>
</td>
<td>
</td>
</tr>
<tr>
<td>
<p>
ln_ten
</p>
</td>
<td>
<p>
ln(10)
</p>
</td>
<td>
<p>
2.30258
</p>
</td>
<td>
</td>
</tr>
<tr>
<td>
<p>
ln_ln_two
</p>
</td>
<td>
<p>
ln(ln(2))
</p>
</td>
<td>
<p>
-0.366512
</p>
</td>
<td>
<p>
Gumbel distribution median
</p>
</td>
</tr>
<tr>
<td>
<p>
root_ln_four
</p>
</td>
<td>
<p>
&#8730;ln(4)
</p>
</td>
<td>
<p>
1.177410
</p>
</td>
<td>
</td>
</tr>
<tr>
<td>
<p>
one_div_root_two
</p>
</td>
<td>
<p>
1/&#8730;2
</p>
</td>
<td>
<p>
0.707106
</p>
</td>
<td>
</td>
</tr>
<tr>
<td>
<p>
<span class="bold"><strong>&#960; and related</strong></span>
</p>
</td>
<td>
</td>
<td>
</td>
<td>
</td>
</tr>
<tr>
<td>
<p>
pi
</p>
</td>
<td>
<p>
pi
</p>
</td>
<td>
<p>
3.14159
</p>
</td>
<td>
<p>
Ubiquitous. Archimedes constant <a href="http://en.wikipedia.org/wiki/Pi" target="_top">&#960;</a>
</p>
</td>
</tr>
<tr>
<td>
<p>
half_pi
</p>
</td>
<td>
<p>
&#960;/2
</p>
</td>
<td>
<p>
1.570796
</p>
</td>
<td>
</td>
</tr>
<tr>
<td>
<p>
third_pi
</p>
</td>
<td>
<p>
&#960;/3
</p>
</td>
<td>
<p>
1.04719
</p>
</td>
<td>
</td>
</tr>
<tr>
<td>
<p>
sixth_pi
</p>
</td>
<td>
<p>
&#960;/6
</p>
</td>
<td>
<p>
0.523598
</p>
</td>
<td>
</td>
</tr>
<tr>
<td>
<p>
two_pi
</p>
</td>
<td>
<p>
2&#960;
</p>
</td>
<td>
<p>
6.28318
</p>
</td>
<td>
<p>
Many uses, most simply, circumference of a circle
</p>
</td>
</tr>
<tr>
<td>
<p>
two_thirds_pi
</p>
</td>
<td>
<p>
2/3 &#960;
</p>
</td>
<td>
<p>
2.09439
</p>
</td>
<td>
<p>
<a href="http://en.wikipedia.org/wiki/Sphere#Volume_of_a_sphere" target="_top">volume
of a hemi-sphere</a> = 4/3 &#960; r&#179;
</p>
</td>
</tr>
<tr>
<td>
<p>
three_quarters_pi
</p>
</td>
<td>
<p>
3/4 &#960;
</p>
</td>
<td>
<p>
2.35619
</p>
</td>
<td>
<p>
= 3/4 &#960;
</p>
</td>
</tr>
<tr>
<td>
<p>
four_thirds_pi
</p>
</td>
<td>
<p>
4/3 &#960;
</p>
</td>
<td>
<p>
4.18879
</p>
</td>
<td>
<p>
<a href="http://en.wikipedia.org/wiki/Sphere#Volume_of_a_sphere" target="_top">volume
of a sphere</a> = 4/3 &#960; r&#179;
</p>
</td>
</tr>
<tr>
<td>
<p>
one_div_two_pi
</p>
</td>
<td>
<p>
1/(2&#960;)
</p>
</td>
<td>
<p>
1.59155
</p>
</td>
<td>
<p>
Widely used
</p>
</td>
</tr>
<tr>
<td>
<p>
root_pi
</p>
</td>
<td>
<p>
&#8730;&#960;
</p>
</td>
<td>
<p>
1.77245
</p>
</td>
<td>
<p>
Widely used
</p>
</td>
</tr>
<tr>
<td>
<p>
root_half_pi
</p>
</td>
<td>
<p>
&#8730; &#960;/2
</p>
</td>
<td>
<p>
1.25331
</p>
</td>
<td>
<p>
Widely used
</p>
</td>
</tr>
<tr>
<td>
<p>
root_two_pi
</p>
</td>
<td>
<p>
&#8730; &#960;*2
</p>
</td>
<td>
<p>
2.50662
</p>
</td>
<td>
<p>
Widely used
</p>
</td>
</tr>
<tr>
<td>
<p>
one_div_root_pi
</p>
</td>
<td>
<p>
1/&#8730;&#960;
</p>
</td>
<td>
<p>
0.564189
</p>
</td>
<td>
</td>
</tr>
<tr>
<td>
<p>
one_div_root_two_pi
</p>
</td>
<td>
<p>
1/&#8730;(2&#960;)
</p>
</td>
<td>
<p>
0.398942
</p>
</td>
<td>
</td>
</tr>
<tr>
<td>
<p>
root_one_div_pi
</p>
</td>
<td>
<p>
&#8730;(1/&#960;
</p>
</td>
<td>
<p>
0.564189
</p>
</td>
<td>
</td>
</tr>
<tr>
<td>
<p>
pi_minus_three
</p>
</td>
<td>
<p>
&#960;-3
</p>
</td>
<td>
<p>
0.141593
</p>
</td>
<td>
</td>
</tr>
<tr>
<td>
<p>
four_minus_pi
</p>
</td>
<td>
<p>
4 -&#960;
</p>
</td>
<td>
<p>
0.858407
</p>
</td>
<td>
</td>
</tr>
<tr>
<td>
<p>
pi_pow_e
</p>
</td>
<td>
<p>
&#960;<sup>e</sup>
</p>
</td>
<td>
<p>
22.4591
</p>
</td>
<td>
</td>
</tr>
<tr>
<td>
<p>
pi_sqr
</p>
</td>
<td>
<p>
&#960;<sup>2</sup>
</p>
</td>
<td>
<p>
9.86960
</p>
</td>
<td>
</td>
</tr>
<tr>
<td>
<p>
pi_sqr_div_six
</p>
</td>
<td>
<p>
&#960;<sup>2</sup>/6
</p>
</td>
<td>
<p>
1.64493
</p>
</td>
<td>
</td>
</tr>
<tr>
<td>
<p>
pi_cubed
</p>
</td>
<td>
<p>
&#960;<sup>3</sup>
</p>
</td>
<td>
<p>
31.00627
</p>
</td>
<td>
</td>
</tr>
<tr>
<td>
<p>
cbrt_pi
</p>
</td>
<td>
<p>
&#8730;<sup>3</sup> &#960;
</p>
</td>
<td>
<p>
1.46459
</p>
</td>
<td>
</td>
</tr>
<tr>
<td>
<p>
one_div_cbrt_pi
</p>
</td>
<td>
<p>
1/&#8730;<sup>3</sup> &#960;
</p>
</td>
<td>
<p>
0.682784
</p>
</td>
<td>
</td>
</tr>
<tr>
<td>
<p>
<span class="bold"><strong>Euler's e and related</strong></span>
</p>
</td>
<td>
</td>
<td>
</td>
<td>
</td>
</tr>
<tr>
<td>
<p>
e
</p>
</td>
<td>
<p>
e
</p>
</td>
<td>
<p>
2.71828
</p>
</td>
<td>
<p>
<a href="http://en.wikipedia.org/wiki/E_(mathematical_constant)" target="_top">Euler's
constant e</a>
</p>
</td>
</tr>
<tr>
<td>
<p>
exp_minus_half
</p>
</td>
<td>
<p>
e <sup>-1/2</sup>
</p>
</td>
<td>
<p>
0.606530
</p>
</td>
<td>
</td>
</tr>
<tr>
<td>
<p>
e_pow_pi
</p>
</td>
<td>
<p>
e <sup>&#960;</sup>
</p>
</td>
<td>
<p>
23.14069
</p>
</td>
<td>
</td>
</tr>
<tr>
<td>
<p>
root_e
</p>
</td>
<td>
<p>
&#8730; e
</p>
</td>
<td>
<p>
1.64872
</p>
</td>
<td>
</td>
</tr>
<tr>
<td>
<p>
log10_e
</p>
</td>
<td>
<p>
log10(e)
</p>
</td>
<td>
<p>
0.434294
</p>
</td>
<td>
</td>
</tr>
<tr>
<td>
<p>
one_div_log10_e
</p>
</td>
<td>
<p>
1/log10(e)
</p>
</td>
<td>
<p>
2.30258
</p>
</td>
<td>
</td>
</tr>
<tr>
<td>
<p>
<span class="bold"><strong>Trigonometric</strong></span>
</p>
</td>
<td>
</td>
<td>
</td>
<td>
</td>
</tr>
<tr>
<td>
<p>
degree
</p>
</td>
<td>
<p>
radians = &#960; / 180
</p>
</td>
<td>
<p>
0.017453
</p>
</td>
<td>
</td>
</tr>
<tr>
<td>
<p>
radian
</p>
</td>
<td>
<p>
degrees = 180 / &#960;
</p>
</td>
<td>
<p>
57.2957
</p>
</td>
<td>
</td>
</tr>
<tr>
<td>
<p>
sin_one
</p>
</td>
<td>
<p>
sin(1)
</p>
</td>
<td>
<p>
0.841470
</p>
</td>
<td>
</td>
</tr>
<tr>
<td>
<p>
cos_one
</p>
</td>
<td>
<p>
cos(1)
</p>
</td>
<td>
<p>
0.54030
</p>
</td>
<td>
</td>
</tr>
<tr>
<td>
<p>
sinh_one
</p>
</td>
<td>
<p>
sinh(1)
</p>
</td>
<td>
<p>
1.17520
</p>
</td>
<td>
</td>
</tr>
<tr>
<td>
<p>
cosh_one
</p>
</td>
<td>
<p>
cosh(1)
</p>
</td>
<td>
<p>
1.54308
</p>
</td>
<td>
</td>
</tr>
<tr>
<td>
<p>
<span class="bold"><strong>Phi</strong></span>
</p>
</td>
<td>
<p>
Phidias golden ratio
</p>
</td>
<td>
<p>
<a href="http://en.wikipedia.org/wiki/Golden_ratio" target="_top">Phidias golden
ratio</a>
</p>
</td>
<td>
</td>
</tr>
<tr>
<td>
<p>
phi
</p>
</td>
<td>
<p>
(1 + &#8730;5) /2
</p>
</td>
<td>
<p>
1.61803
</p>
</td>
<td>
<p>
finance
</p>
</td>
</tr>
<tr>
<td>
<p>
ln_phi
</p>
</td>
<td>
<p>
ln(&#966;)
</p>
</td>
<td>
<p>
0.48121
</p>
</td>
<td>
</td>
</tr>
<tr>
<td>
<p>
one_div_ln_phi
</p>
</td>
<td>
<p>
1/ln(&#966;)
</p>
</td>
<td>
<p>
2.07808
</p>
</td>
<td>
</td>
</tr>
<tr>
<td>
<p>
<span class="bold"><strong>Euler's Gamma</strong></span>
</p>
</td>
<td>
</td>
<td>
</td>
<td>
</td>
</tr>
<tr>
<td>
<p>
euler
</p>
</td>
<td>
<p>
euler
</p>
</td>
<td>
<p>
0.577215
</p>
</td>
<td>
<p>
<a href="http://en.wikipedia.org/wiki/Euler%E2%80%93Mascheroni_constant" target="_top">Euler-Mascheroni
gamma constant</a>
</p>
</td>
</tr>
<tr>
<td>
<p>
one_div_euler
</p>
</td>
<td>
<p>
1/euler
</p>
</td>
<td>
<p>
1.73245
</p>
</td>
<td>
</td>
</tr>
<tr>
<td>
<p>
euler_sqr
</p>
</td>
<td>
<p>
euler<sup>2</sup>
</p>
</td>
<td>
<p>
0.333177
</p>
</td>
<td>
</td>
</tr>
<tr>
<td>
<p>
<span class="bold"><strong>Misc</strong></span>
</p>
</td>
<td>
</td>
<td>
</td>
<td>
</td>
</tr>
<tr>
<td>
<p>
zeta_two
</p>
</td>
<td>
<p>
&#950;(2)
</p>
</td>
<td>
<p>
1.64493
</p>
</td>
<td>
<p>
<a href="http://en.wikipedia.org/wiki/Riemann_zeta_function" target="_top">Riemann
zeta function</a>
</p>
</td>
</tr>
<tr>
<td>
<p>
zeta_three
</p>
</td>
<td>
<p>
&#950;(3)
</p>
</td>
<td>
<p>
1.20205
</p>
</td>
<td>
<p>
<a href="http://en.wikipedia.org/wiki/Riemann_zeta_function" target="_top">Riemann
zeta function</a>
</p>
</td>
</tr>
<tr>
<td>
<p>
catalan
</p>
</td>
<td>
<p>
<span class="emphasis"><em>K</em></span>
</p>
</td>
<td>
<p>
0.915965
</p>
</td>
<td>
<p>
<a href="http://mathworld.wolfram.com/CatalansConstant.html" target="_top">Catalan
(or Glaisher) combinatorial constant</a>
</p>
</td>
</tr>
<tr>
<td>
<p>
glaisher
</p>
</td>
<td>
<p>
<span class="emphasis"><em>A</em></span>
</p>
</td>
<td>
<p>
1.28242
</p>
</td>
<td>
<p>
<a href="https://oeis.org/A074962/constant" target="_top">Decimal expansion
of Glaisher-Kinkelin constant</a>
</p>
</td>
</tr>
<tr>
<td>
<p>
khinchin
</p>
</td>
<td>
<p>
<span class="emphasis"><em>k</em></span>
</p>
</td>
<td>
<p>
2.685452
</p>
</td>
<td>
<p>
<a href="https://oeis.org/A002210/constant" target="_top">Decimal expansion
of Khinchin constant</a>
</p>
</td>
</tr>
<tr>
<td>
<p>
extreme_value_skewness
</p>
</td>
<td>
<p>
12&#8730;6 &#950;(3)/ &#960;<sup>3</sup>
</p>
</td>
<td>
<p>
1.139547
</p>
</td>
<td>
<p>
Extreme value distribution
</p>
</td>
</tr>
<tr>
<td>
<p>
rayleigh_skewness
</p>
</td>
<td>
<p>
2&#8730;&#960;(&#960;-3)/(4 - &#960;)<sup>3/2</sup>
</p>
</td>
<td>
<p>
0.631110
</p>
</td>
<td>
<p>
Rayleigh distribution skewness
</p>
</td>
</tr>
<tr>
<td>
<p>
rayleigh_kurtosis_excess
</p>
</td>
<td>
<p>
-(6&#960;<sup>2</sup>-24&#960;+16)/(4-&#960;)<sup>2</sup>
</p>
</td>
<td>
<p>
0.245089
</p>
</td>
<td>
<p>
<a href="http://en.wikipedia.org/wiki/Rayleigh_distribution" target="_top">Rayleigh
distribution kurtosis excess</a>
</p>
</td>
</tr>
<tr>
<td>
<p>
rayleigh_kurtosis
</p>
</td>
<td>
<p>
3+(6&#960;<sup>2</sup>-24&#960;+16)/(4-&#960;)<sup>2</sup>
</p>
</td>
<td>
<p>
3.245089
</p>
</td>
<td>
<p>
Rayleigh distribution kurtosis
</p>
</td>
</tr>
</tbody>
</table></div>
</div>
<br class="table-break"><div class="note"><table border="0" summary="Note">
<tr>
<td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../doc/src/images/note.png"></td>
<th align="left">Note</th>
</tr>
<tr><td align="left" valign="top"><p>
Integer values are <span class="bold"><strong>not included</strong></span> in this
list of math constants, however interesting, because they can be so easily
and exactly constructed, even for UDT, for example: <code class="computeroutput"><span class="keyword">static_cast</span><span class="special">&lt;</span><span class="identifier">cpp_float</span><span class="special">&gt;(</span><span class="number">42</span><span class="special">)</span></code>.
</p></td></tr>
</table></div>
<div class="tip"><table border="0" summary="Tip">
<tr>
<td rowspan="2" align="center" valign="top" width="25"><img alt="[Tip]" src="../../../../../doc/src/images/tip.png"></td>
<th align="left">Tip</th>
</tr>
<tr><td align="left" valign="top"><p>
If you know the approximate value of the constant, you can search for the
value to find Boost.Math chosen name in this table.
</p></td></tr>
</table></div>
<div class="tip"><table border="0" summary="Tip">
<tr>
<td rowspan="2" align="center" valign="top" width="25"><img alt="[Tip]" src="../../../../../doc/src/images/tip.png"></td>
<th align="left">Tip</th>
</tr>
<tr><td align="left" valign="top"><p>
Bernoulli numbers are available at <a class="link" href="number_series/bernoulli_numbers.html" title="Bernoulli Numbers">Bernoulli
numbers</a>.
</p></td></tr>
</table></div>
<div class="tip"><table border="0" summary="Tip">
<tr>
<td rowspan="2" align="center" valign="top" width="25"><img alt="[Tip]" src="../../../../../doc/src/images/tip.png"></td>
<th align="left">Tip</th>
</tr>
<tr><td align="left" valign="top"><p>
Factorials are available at <a class="link" href="factorials/sf_factorial.html" title="Factorial">factorial</a>.
</p></td></tr>
</table></div>
</div>
<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
<td align="left"></td>
<td align="right"><div class="copyright-footer">Copyright &#169; 2006-2010, 2012-2014 Nikhar Agrawal,
Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan R&#229;de, Gautam Sewani,
Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p>
Distributed under the Boost Software License, Version 1.0. (See accompanying
file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
</p>
</div></td>
</tr></table>
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