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Edouard DUPIN
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@@ -1,11 +1,11 @@
<html>
<head>
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>The Mathematical Constants</title>
<link rel="stylesheet" href="../math.css" type="text/css">
<meta name="generator" content="DocBook XSL Stylesheets V1.79.1">
<link rel="home" href="../index.html" title="Math Toolkit 2.6.0">
<link rel="up" href="../constants.html" title="Chapter&#160;4.&#160;Mathematical Constants">
<link rel="home" href="../index.html" title="Math Toolkit 3.0.0">
<link rel="up" href="../constants.html" title="Chapter 4. Mathematical Constants">
<link rel="prev" href="tutorial/user_def.html" title="Use With User-Defined Types">
<link rel="next" href="new_const.html" title="Defining New Constants">
</head>
@@ -31,7 +31,7 @@
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>
<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>
@@ -172,7 +172,7 @@
</td>
<td>
<p>
&#8730;2
√2
</p>
</td>
<td>
@@ -181,6 +181,9 @@
</p>
</td>
<td>
<p>
Equivalent to POSIX constant M_SQRT2
</p>
</td>
</tr>
<tr>
@@ -191,7 +194,7 @@
</td>
<td>
<p>
&#8730;3
√3
</p>
</td>
<td>
@@ -210,7 +213,7 @@
</td>
<td>
<p>
&#8730;2 /2
√2 /2
</p>
</td>
<td>
@@ -238,6 +241,9 @@
</p>
</td>
<td>
<p>
Equivalent to POSIX constant M_LN2
</p>
</td>
</tr>
<tr>
@@ -257,6 +263,9 @@
</p>
</td>
<td>
<p>
Equivalent to POSIX constant M_LN10
</p>
</td>
</tr>
<tr>
@@ -289,7 +298,7 @@
</td>
<td>
<p>
&#8730;ln(4)
√ln(4)
</p>
</td>
<td>
@@ -308,7 +317,7 @@
</td>
<td>
<p>
1/&#8730;2
1/√2
</p>
</td>
<td>
@@ -317,12 +326,15 @@
</p>
</td>
<td>
<p>
Equivalent to POSIX constant M_SQRT1_2
</p>
</td>
</tr>
<tr>
<td>
<p>
<span class="bold"><strong>&#960; and related</strong></span>
<span class="bold"><strong>π and related</strong></span>
</p>
</td>
<td>
@@ -340,7 +352,7 @@
</td>
<td>
<p>
pi
π
</p>
</td>
<td>
@@ -350,7 +362,8 @@
</td>
<td>
<p>
Ubiquitous. Archimedes constant <a href="http://en.wikipedia.org/wiki/Pi" target="_top">&#960;</a>
Ubiquitous. Archimedes constant <a href="http://en.wikipedia.org/wiki/Pi" target="_top">π</a>.
Equivalent to POSIX constant M_PI
</p>
</td>
</tr>
@@ -362,7 +375,7 @@
</td>
<td>
<p>
&#960;/2
π/2
</p>
</td>
<td>
@@ -371,6 +384,9 @@
</p>
</td>
<td>
<p>
Equivalent to POSIX constant M_PI2
</p>
</td>
</tr>
<tr>
@@ -381,7 +397,7 @@
</td>
<td>
<p>
&#960;/3
π/3
</p>
</td>
<td>
@@ -393,6 +409,28 @@
</td>
</tr>
<tr>
<td>
<p>
quarter_pi
</p>
</td>
<td>
<p>
π/4
</p>
</td>
<td>
<p>
0.78539816
</p>
</td>
<td>
<p>
Equivalent to POSIX constant M_PI_4
</p>
</td>
</tr>
<tr>
<td>
<p>
sixth_pi
@@ -400,7 +438,7 @@
</td>
<td>
<p>
&#960;/6
π/6
</p>
</td>
<td>
@@ -419,7 +457,7 @@
</td>
<td>
<p>
2&#960;
2π
</p>
</td>
<td>
@@ -441,7 +479,7 @@
</td>
<td>
<p>
2/3 &#960;
2/3 π
</p>
</td>
<td>
@@ -452,7 +490,7 @@
<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;
of a hemi-sphere</a> = 4/3 π r³
</p>
</td>
</tr>
@@ -464,7 +502,7 @@
</td>
<td>
<p>
3/4 &#960;
3/4 π
</p>
</td>
<td>
@@ -474,7 +512,7 @@
</td>
<td>
<p>
= 3/4 &#960;
= 3/4 π
</p>
</td>
</tr>
@@ -486,7 +524,7 @@
</td>
<td>
<p>
4/3 &#960;
4/3 π
</p>
</td>
<td>
@@ -497,7 +535,7 @@
<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;
of a sphere</a> = 4/3 π r³
</p>
</td>
</tr>
@@ -509,7 +547,7 @@
</td>
<td>
<p>
1/(2&#960;)
1/(2π)
</p>
</td>
<td>
@@ -531,7 +569,7 @@
</td>
<td>
<p>
&#8730;&#960;
√π
</p>
</td>
<td>
@@ -553,7 +591,7 @@
</td>
<td>
<p>
&#8730; &#960;/2
√ π/2
</p>
</td>
<td>
@@ -575,7 +613,7 @@
</td>
<td>
<p>
&#8730; &#960;*2
√ π*2
</p>
</td>
<td>
@@ -590,6 +628,50 @@
</td>
</tr>
<tr>
<td>
<p>
one_div_pi
</p>
</td>
<td>
<p>
1/π
</p>
</td>
<td>
<p>
0.31830988
</p>
</td>
<td>
<p>
Equivalent to POSIX constant M_1_PI
</p>
</td>
</tr>
<tr>
<td>
<p>
two_div_pi
</p>
</td>
<td>
<p>
2/π
</p>
</td>
<td>
<p>
0.63661977
</p>
</td>
<td>
<p>
Equivalent to POSIX constant M_2_PI
</p>
</td>
</tr>
<tr>
<td>
<p>
one_div_root_pi
@@ -597,7 +679,7 @@
</td>
<td>
<p>
1/&#8730;&#960;
1/√π
</p>
</td>
<td>
@@ -609,6 +691,28 @@
</td>
</tr>
<tr>
<td>
<p>
two_div_root_pi
</p>
</td>
<td>
<p>
2/√π
</p>
</td>
<td>
<p>
1.128379
</p>
</td>
<td>
<p>
Equivalent to POSIX constant M_2_SQRTPI
</p>
</td>
</tr>
<tr>
<td>
<p>
one_div_root_two_pi
@@ -616,7 +720,7 @@
</td>
<td>
<p>
1/&#8730;(2&#960;)
1/√(2π)
</p>
</td>
<td>
@@ -635,7 +739,7 @@
</td>
<td>
<p>
&#8730;(1/&#960;
√(1/π
</p>
</td>
<td>
@@ -654,7 +758,7 @@
</td>
<td>
<p>
&#960;-3
π-3
</p>
</td>
<td>
@@ -673,7 +777,7 @@
</td>
<td>
<p>
4 -&#960;
4 -π
</p>
</td>
<td>
@@ -692,7 +796,7 @@
</td>
<td>
<p>
&#960;<sup>e</sup>
π<sup>e</sup>
</p>
</td>
<td>
@@ -711,7 +815,7 @@
</td>
<td>
<p>
&#960;<sup>2</sup>
π<sup>2</sup>
</p>
</td>
<td>
@@ -730,7 +834,7 @@
</td>
<td>
<p>
&#960;<sup>2</sup>/6
π<sup>2</sup>/6
</p>
</td>
<td>
@@ -749,7 +853,7 @@
</td>
<td>
<p>
&#960;<sup>3</sup>
π<sup>3</sup>
</p>
</td>
<td>
@@ -768,7 +872,7 @@
</td>
<td>
<p>
&#8730;<sup>3</sup> &#960;
√<sup>3</sup> π
</p>
</td>
<td>
@@ -787,7 +891,7 @@
</td>
<td>
<p>
1/&#8730;<sup>3</sup> &#960;
1/√<sup>3</sup> π
</p>
</td>
<td>
@@ -830,7 +934,7 @@
<td>
<p>
<a href="http://en.wikipedia.org/wiki/E_(mathematical_constant)" target="_top">Euler's
constant e</a>
constant e</a>, equivalent to POSIX constant M_E
</p>
</td>
</tr>
@@ -861,7 +965,7 @@
</td>
<td>
<p>
e <sup>&#960;</sup>
e <sup>π</sup>
</p>
</td>
<td>
@@ -880,7 +984,7 @@
</td>
<td>
<p>
&#8730; e
√ e
</p>
</td>
<td>
@@ -908,6 +1012,9 @@
</p>
</td>
<td>
<p>
Equivalent to POSIX constant M_LOG10E
</p>
</td>
</tr>
<tr>
@@ -930,6 +1037,28 @@
</td>
</tr>
<tr>
<td>
<p>
log2_e
</p>
</td>
<td>
<p>
log<sub>2</sub>(e)
</p>
</td>
<td>
<p>
1.442695
</p>
</td>
<td>
<p>
This is the same as 1/ln(2) and is equivalent to POSIX constant M_LOG2E
</p>
</td>
</tr>
<tr>
<td>
<p>
<span class="bold"><strong>Trigonometric</strong></span>
@@ -950,7 +1079,7 @@
</td>
<td>
<p>
radians = &#960; / 180
radians = π / 180
</p>
</td>
<td>
@@ -969,7 +1098,7 @@
</td>
<td>
<p>
degrees = 180 / &#960;
degrees = 180 / π
</p>
</td>
<td>
@@ -1084,7 +1213,7 @@
</td>
<td>
<p>
(1 + &#8730;5) /2
(1 + √5) /2
</p>
</td>
<td>
@@ -1106,7 +1235,7 @@
</td>
<td>
<p>
ln(&#966;)
ln(φ)
</p>
</td>
<td>
@@ -1125,7 +1254,7 @@
</td>
<td>
<p>
1/ln(&#966;)
1/ln(φ)
</p>
</td>
<td>
@@ -1231,7 +1360,7 @@
</td>
<td>
<p>
&#950;(2)
ζ(2)
</p>
</td>
<td>
@@ -1254,7 +1383,7 @@
</td>
<td>
<p>
&#950;(3)
ζ(3)
</p>
</td>
<td>
@@ -1346,7 +1475,7 @@
</td>
<td>
<p>
12&#8730;6 &#950;(3)/ &#960;<sup>3</sup>
12√6 ζ(3)/ π<sup>3</sup>
</p>
</td>
<td>
@@ -1368,7 +1497,7 @@
</td>
<td>
<p>
2&#8730;&#960;(&#960;-3)/(4 - &#960;)<sup>3/2</sup>
2√π(π-3)/(4 - π)<sup>3/2</sup>
</p>
</td>
<td>
@@ -1390,7 +1519,7 @@
</td>
<td>
<p>
-(6&#960;<sup>2</sup>-24&#960;+16)/(4-&#960;)<sup>2</sup>
-(6π<sup>2</sup>-24π+16)/(4-π)<sup>2</sup>
</p>
</td>
<td>
@@ -1413,7 +1542,7 @@
</td>
<td>
<p>
3+(6&#960;<sup>2</sup>-24&#960;+16)/(4-&#960;)<sup>2</sup>
3+(6π<sup>2</sup>-24π+16)/(4-π)<sup>2</sup>
</p>
</td>
<td>
@@ -1427,6 +1556,138 @@
</p>
</td>
</tr>
<tr>
<td>
<p>
first_feigenbaum
</p>
</td>
<td>
</td>
<td>
<p>
4.6692016
</p>
</td>
<td>
<p>
<a href="https://en.wikipedia.org/wiki/Feigenbaum_constants" target="_top">First
Feigenbaum constant</a>
</p>
</td>
</tr>
<tr>
<td>
<p>
plastic
</p>
</td>
<td>
<p>
Real solution of x<sup>3</sup> = x + 1
</p>
</td>
<td>
<p>
1.324717957
</p>
</td>
<td>
<p>
<a href="https://en.wikipedia.org/wiki/Plastic_number" target="_top">Plastic
constant</a>
</p>
</td>
</tr>
<tr>
<td>
<p>
gauss
</p>
</td>
<td>
<p>
Reciprocal of agm(1, √2)
</p>
</td>
<td>
<p>
0.8346268
</p>
</td>
<td>
<p>
<a href="https://en.wikipedia.org/wiki/Gauss%27s_constant" target="_top">Gauss's
constant</a>
</p>
</td>
</tr>
<tr>
<td>
<p>
dottie
</p>
</td>
<td>
<p>
Solution of cos(x) = x
</p>
</td>
<td>
<p>
0.739085
</p>
</td>
<td>
<p>
<a href="https://en.wikipedia.org/wiki/Dottie_number" target="_top">Dottie's
number</a>
</p>
</td>
</tr>
<tr>
<td>
<p>
reciprocal_fibonacci
</p>
</td>
<td>
<p>
Sum of reciprocals of Fibonacci numbers
</p>
</td>
<td>
<p>
3.359885666
</p>
</td>
<td>
<p>
<a href="https://en.wikipedia.org/wiki/Reciprocal_Fibonacci_constant" target="_top">Reciprocal
Fibonacci constant</a>
</p>
</td>
</tr>
<tr>
<td>
<p>
laplace_limit
</p>
</td>
<td>
</td>
<td>
<p>
.6627434193
</p>
</td>
<td>
<p>
<a href="https://en.wikipedia.org/wiki/Laplace_limit" target="_top">Laplace
Limit</a>
</p>
</td>
</tr>
</tbody>
</table></div>
</div>
@@ -1473,11 +1734,11 @@
</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, 2017 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, Nicholas Thompson, Thijs van den Berg, Daryle Walker
and Xiaogang Zhang<p>
<td align="right"><div class="copyright-footer">Copyright © 2006-2021 Nikhar Agrawal, Anton Bikineev, Matthew Borland,
Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert Holin, Bruno
Lalande, John Maddock, Evan Miller, Jeremy Murphy, Matthew Pulver, Johan Råde,
Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, 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>