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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 4.1. 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>
√2
</p>
</td>
<td>
<p>
1.41421
</p>
</td>
<td>
</td>
</tr>
<tr>
<td>
<p>
root_three
</p>
</td>
<td>
<p>
√3
</p>
</td>
<td>
<p>
1.73205
</p>
</td>
<td>
</td>
</tr>
<tr>
<td>
<p>
half_root_two
</p>
</td>
<td>
<p>
√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>
√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/√2
</p>
</td>
<td>
<p>
0.707106
</p>
</td>
<td>
</td>
</tr>
<tr>
<td>
<p>
<span class="bold"><strong>π 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">π</a>
</p>
</td>
</tr>
<tr>
<td>
<p>
half_pi
</p>
</td>
<td>
<p>
π/2
</p>
</td>
<td>
<p>
1.570796
</p>
</td>
<td>
</td>
</tr>
<tr>
<td>
<p>
third_pi
</p>
</td>
<td>
<p>
π/3
</p>
</td>
<td>
<p>
1.04719
</p>
</td>
<td>
</td>
</tr>
<tr>
<td>
<p>
sixth_pi
</p>
</td>
<td>
<p>
π/6
</p>
</td>
<td>
<p>
0.523598
</p>
</td>
<td>
</td>
</tr>
<tr>
<td>
<p>
two_pi
</p>
</td>
<td>
<p>
2π
</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 π
</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 π r³
</p>
</td>
</tr>
<tr>
<td>
<p>
three_quarters_pi
</p>
</td>
<td>
<p>
3/4 π
</p>
</td>
<td>
<p>
2.35619
</p>
</td>
<td>
<p>
= 3/4 π
</p>
</td>
</tr>
<tr>
<td>
<p>
four_thirds_pi
</p>
</td>
<td>
<p>
4/3 π
</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 π r³
</p>
</td>
</tr>
<tr>
<td>
<p>
one_div_two_pi
</p>
</td>
<td>
<p>
1/(2π)
</p>
</td>
<td>
<p>
1.59155
</p>
</td>
<td>
<p>
Widely used
</p>
</td>
</tr>
<tr>
<td>
<p>
root_pi
</p>
</td>
<td>
<p>
√π
</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>
√ π/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>
√ π*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/√π
</p>
</td>
<td>
<p>
0.564189
</p>
</td>
<td>
</td>
</tr>
<tr>
<td>
<p>
one_div_root_two_pi
</p>
</td>
<td>
<p>
1/√(2π)
</p>
</td>
<td>
<p>
0.398942
</p>
</td>
<td>
</td>
</tr>
<tr>
<td>
<p>
root_one_div_pi
</p>
</td>
<td>
<p>
√(1/π
</p>
</td>
<td>
<p>
0.564189
</p>
</td>
<td>
</td>
</tr>
<tr>
<td>
<p>
pi_minus_three
</p>
</td>
<td>
<p>
π-3
</p>
</td>
<td>
<p>
0.141593
</p>
</td>
<td>
</td>
</tr>
<tr>
<td>
<p>
four_minus_pi
</p>
</td>
<td>
<p>
4 -π
</p>
</td>
<td>
<p>
0.858407
</p>
</td>
<td>
</td>
</tr>
<tr>
<td>
<p>
pi_pow_e
</p>
</td>
<td>
<p>
π<sup>e</sup>
</p>
</td>
<td>
<p>
22.4591
</p>
</td>
<td>
</td>
</tr>
<tr>
<td>
<p>
pi_sqr
</p>
</td>
<td>
<p>
π<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>
π<sup>2</sup>/6
</p>
</td>
<td>
<p>
1.64493
</p>
</td>
<td>
</td>
</tr>
<tr>
<td>
<p>
pi_cubed
</p>
</td>
<td>
<p>
π<sup>3</sup>
</p>
</td>
<td>
<p>
31.00627
</p>
</td>
<td>
</td>
</tr>
<tr>
<td>
<p>
cbrt_pi
</p>
</td>
<td>
<p>
√<sup>3</sup> π
</p>
</td>
<td>
<p>
1.46459
</p>
</td>
<td>
</td>
</tr>
<tr>
<td>
<p>
one_div_cbrt_pi
</p>
</td>
<td>
<p>
1/√<sup>3</sup> π
</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>π</sup>
</p>
</td>
<td>
<p>
23.14069
</p>
</td>
<td>
</td>
</tr>
<tr>
<td>
<p>
root_e
</p>
</td>
<td>
<p>
√ 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 = π / 180
</p>
</td>
<td>
<p>
0.017453
</p>
</td>
<td>
</td>
</tr>
<tr>
<td>
<p>
radian
</p>
</td>
<td>
<p>
degrees = 180 / π
</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 + √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(φ)
</p>
</td>
<td>
<p>
0.48121
</p>
</td>
<td>
</td>
</tr>
<tr>
<td>
<p>
one_div_ln_phi
</p>
</td>
<td>
<p>
1/ln(φ)
</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>
ζ(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>
ζ(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√6 ζ(3)/ π<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√π(π-3)/(4 - π)<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>
-(6π<sup>2</sup>-24π+16)/(4-π)<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+(6π<sup>2</sup>-24π+16)/(4-π)<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"><</span><span class="identifier">cpp_float</span><span class="special">>(</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 © 2006-2010, 2012-2014 Nikhar Agrawal,
Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan Råde, 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>
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