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<html> <head> <meta http-equiv="Content-Type" content="text/html; charset=US-ASCII"> <title>The Mathematical Constants</title> <link rel="stylesheet" href="../math.css" type="text/css"> <meta name="generator" content="DocBook XSL Stylesheets V1.78.1"> <link rel="home" href="../index.html" title="Math Toolkit 2.1.0"> <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> <body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"> <table cellpadding="2" width="100%"><tr> <td valign="top"><img alt="Boost C++ Libraries" width="277" height="86" src="../../../../../boost.png"></td> <td align="center"><a href="../../../../../index.html">Home</a></td> <td align="center"><a href="../../../../../libs/libraries.htm">Libraries</a></td> <td align="center"><a href="http://www.boost.org/users/people.html">People</a></td> <td align="center"><a href="http://www.boost.org/users/faq.html">FAQ</a></td> <td align="center"><a href="../../../../../more/index.htm">More</a></td> </tr></table> <hr> <div class="spirit-nav"> <a accesskey="p" href="tutorial/user_def.html"><img src="../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../constants.html"><img src="../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../index.html"><img src="../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="new_const.html"><img src="../../../../../doc/src/images/next.png" alt="Next"></a> </div> <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> This section lists the mathematical constants, their use(s) (and sometimes rationale for their inclusion). <div class="table"> <a name="math_toolkit.constants.mathematical_constants"></a>Table 4.1. Mathematical Constants <div class="table-contents"><table class="table" summary="Mathematical Constants"> <colgroup> <col> <col> <col> <col> </colgroup> <thead><tr> <th> name </th> <th> formula </th> <th> Value (6 decimals) </th> <th> Uses and Rationale </th> </tr></thead> <tbody> <tr> <td> Rational fractions </td> <td> </td> <td> </td> <td> </td> </tr> <tr> <td> half </td> <td> 1/2 </td> <td> 0.5 </td> <td> </td> </tr> <tr> <td> third </td> <td> 1/3 </td> <td> 0.333333 </td> <td> </td> </tr> <tr> <td> two_thirds </td> <td> 2/3 </td> <td> 0.66667 </td> <td> </td> </tr> <tr> <td> three_quarters </td> <td> 3/4 </td> <td> 0.75 </td> <td> </td> </tr> <tr> <td> two and related </td> <td> </td> <td> </td> <td> </td> </tr> <tr> <td> root_two </td> <td> √2 </td> <td> 1.41421 </td> <td> </td> </tr> <tr> <td> root_three </td> <td> √3 </td> <td> 1.73205 </td> <td> </td> </tr> <tr> <td> half_root_two </td> <td> √2 /2 </td> <td> 0.707106 </td> <td> </td> </tr> <tr> <td> ln_two </td> <td> ln(2) </td> <td> 0.693147 </td> <td> </td> </tr> <tr> <td> ln_ten </td> <td> ln(10) </td> <td> 2.30258 </td> <td> </td> </tr> <tr> <td> ln_ln_two </td> <td> ln(ln(2)) </td> <td> -0.366512 </td> <td> Gumbel distribution median </td> </tr> <tr> <td> root_ln_four </td> <td> √ln(4) </td> <td> 1.177410 </td> <td> </td> </tr> <tr> <td> one_div_root_two </td> <td> 1/√2 </td> <td> 0.707106 </td> <td> </td> </tr> <tr> <td> π and related </td> <td> </td> <td> </td> <td> </td> </tr> <tr> <td> pi </td> <td> pi </td> <td> 3.14159 </td> <td> Ubiquitous. Archimedes constant <a href="http://en.wikipedia.org/wiki/Pi" target="_top">π</a> </td> </tr> <tr> <td> half_pi </td> <td> π/2 </td> <td> 1.570796 </td> <td> </td> </tr> <tr> <td> third_pi </td> <td> π/3 </td> <td> 1.04719 </td> <td> </td> </tr> <tr> <td> sixth_pi </td> <td> π/6 </td> <td> 0.523598 </td> <td> </td> </tr> <tr> <td> two_pi </td> <td> 2π </td> <td> 6.28318 </td> <td> Many uses, most simply, circumference of a circle </td> </tr> <tr> <td> two_thirds_pi </td> <td> 2/3 π </td> <td> 2.09439 </td> <td> <a href="http://en.wikipedia.org/wiki/Sphere#Volume_of_a_sphere" target="_top">volume of a hemi-sphere</a> = 4/3 π r³ </td> </tr> <tr> <td> three_quarters_pi </td> <td> 3/4 π </td> <td> 2.35619 </td> <td> = 3/4 π </td> </tr> <tr> <td> four_thirds_pi </td> <td> 4/3 π </td> <td> 4.18879 </td> <td> <a href="http://en.wikipedia.org/wiki/Sphere#Volume_of_a_sphere" target="_top">volume of a sphere</a> = 4/3 π r³ </td> </tr> <tr> <td> one_div_two_pi </td> <td> 1/(2π) </td> <td> 1.59155 </td> <td> Widely used </td> </tr> <tr> <td> root_pi </td> <td> √π </td> <td> 1.77245 </td> <td> Widely used </td> </tr> <tr> <td> root_half_pi </td> <td> √ π/2 </td> <td> 1.25331 </td> <td> Widely used </td> </tr> <tr> <td> root_two_pi </td> <td> √ π*2 </td> <td> 2.50662 </td> <td> Widely used </td> </tr> <tr> <td> one_div_root_pi </td> <td> 1/√π </td> <td> 0.564189 </td> <td> </td> </tr> <tr> <td> one_div_root_two_pi </td> <td> 1/√(2π) </td> <td> 0.398942 </td> <td> </td> </tr> <tr> <td> root_one_div_pi </td> <td> √(1/π </td> <td> 0.564189 </td> <td> </td> </tr> <tr> <td> pi_minus_three </td> <td> π-3 </td> <td> 0.141593 </td> <td> </td> </tr> <tr> <td> four_minus_pi </td> <td> 4 -π </td> <td> 0.858407 </td> <td> </td> </tr> <tr> <td> pi_pow_e </td> <td> πe </td> <td> 22.4591 </td> <td> </td> </tr> <tr> <td> pi_sqr </td> <td> π2 </td> <td> 9.86960 </td> <td> </td> </tr> <tr> <td> pi_sqr_div_six </td> <td> π2/6 </td> <td> 1.64493 </td> <td> </td> </tr> <tr> <td> pi_cubed </td> <td> π3 </td> <td> 31.00627 </td> <td> </td> </tr> <tr> <td> cbrt_pi </td> <td> √3 π </td> <td> 1.46459 </td> <td> </td> </tr> <tr> <td> one_div_cbrt_pi </td> <td> 1/√3 π </td> <td> 0.682784 </td> <td> </td> </tr> <tr> <td> Euler's e and related </td> <td> </td> <td> </td> <td> </td> </tr> <tr> <td> e </td> <td> e </td> <td> 2.71828 </td> <td> <a href="http://en.wikipedia.org/wiki/E_(mathematical_constant)" target="_top">Euler's constant e</a> </td> </tr> <tr> <td> exp_minus_half </td> <td> e -1/2 </td> <td> 0.606530 </td> <td> </td> </tr> <tr> <td> e_pow_pi </td> <td> e π </td> <td> 23.14069 </td> <td> </td> </tr> <tr> <td> root_e </td> <td> √ e </td> <td> 1.64872 </td> <td> </td> </tr> <tr> <td> log10_e </td> <td> log10(e) </td> <td> 0.434294 </td> <td> </td> </tr> <tr> <td> one_div_log10_e </td> <td> 1/log10(e) </td> <td> 2.30258 </td> <td> </td> </tr> <tr> <td> Trigonometric </td> <td> </td> <td> </td> <td> </td> </tr> <tr> <td> degree </td> <td> radians = π / 180 </td> <td> 0.017453 </td> <td> </td> </tr> <tr> <td> radian </td> <td> degrees = 180 / π </td> <td> 57.2957 </td> <td> </td> </tr> <tr> <td> sin_one </td> <td> sin(1) </td> <td> 0.841470 </td> <td> </td> </tr> <tr> <td> cos_one </td> <td> cos(1) </td> <td> 0.54030 </td> <td> </td> </tr> <tr> <td> sinh_one </td> <td> sinh(1) </td> <td> 1.17520 </td> <td> </td> </tr> <tr> <td> cosh_one </td> <td> cosh(1) </td> <td> 1.54308 </td> <td> </td> </tr> <tr> <td> Phi </td> <td> Phidias golden ratio </td> <td> <a href="http://en.wikipedia.org/wiki/Golden_ratio" target="_top">Phidias golden ratio</a> </td> <td> </td> </tr> <tr> <td> phi </td> <td> (1 + √5) /2 </td> <td> 1.61803 </td> <td> finance </td> </tr> <tr> <td> ln_phi </td> <td> ln(φ) </td> <td> 0.48121 </td> <td> </td> </tr> <tr> <td> one_div_ln_phi </td> <td> 1/ln(φ) </td> <td> 2.07808 </td> <td> </td> </tr> <tr> <td> Euler's Gamma </td> <td> </td> <td> </td> <td> </td> </tr> <tr> <td> euler </td> <td> euler </td> <td> 0.577215 </td> <td> <a href="http://en.wikipedia.org/wiki/Euler%E2%80%93Mascheroni_constant" target="_top">Euler-Mascheroni gamma constant</a> </td> </tr> <tr> <td> one_div_euler </td> <td> 1/euler </td> <td> 1.73245 </td> <td> </td> </tr> <tr> <td> euler_sqr </td> <td> euler2 </td> <td> 0.333177 </td> <td> </td> </tr> <tr> <td> Misc </td> <td> </td> <td> </td> <td> </td> </tr> <tr> <td> zeta_two </td> <td> ζ(2) </td> <td> 1.64493 </td> <td> <a href="http://en.wikipedia.org/wiki/Riemann_zeta_function" target="_top">Riemann zeta function</a> </td> </tr> <tr> <td> zeta_three </td> <td> ζ(3) </td> <td> 1.20205 </td> <td> <a href="http://en.wikipedia.org/wiki/Riemann_zeta_function" target="_top">Riemann zeta function</a> </td> </tr> <tr> <td> catalan </td> <td> K </td> <td> 0.915965 </td> <td> <a href="http://mathworld.wolfram.com/CatalansConstant.html" target="_top">Catalan (or Glaisher) combinatorial constant</a> </td> </tr> <tr> <td> glaisher </td> <td> A </td> <td> 1.28242 </td> <td> <a href="https://oeis.org/A074962/constant" target="_top">Decimal expansion of Glaisher-Kinkelin constant</a> </td> </tr> <tr> <td> khinchin </td> <td> k </td> <td> 2.685452 </td> <td> <a href="https://oeis.org/A002210/constant" target="_top">Decimal expansion of Khinchin constant</a> </td> </tr> <tr> <td> extreme_value_skewness </td> <td> 12√6 ζ(3)/ π3 </td> <td> 1.139547 </td> <td> Extreme value distribution </td> </tr> <tr> <td> rayleigh_skewness </td> <td> 2√π(π-3)/(4 - π)3/2 </td> <td> 0.631110 </td> <td> Rayleigh distribution skewness </td> </tr> <tr> <td> rayleigh_kurtosis_excess </td> <td> -(6π2-24π+16)/(4-π)2 </td> <td> 0.245089 </td> <td> <a href="http://en.wikipedia.org/wiki/Rayleigh_distribution" target="_top">Rayleigh distribution kurtosis excess</a> </td> </tr> <tr> <td> rayleigh_kurtosis </td> <td> 3+(6π2-24π+16)/(4-π)2 </td> <td> 3.245089 </td> <td> Rayleigh distribution kurtosis </td> </tr> </tbody> </table></div> </div> <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"> Integer values are not included 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">static_cast<cpp_float>(42)</code>. </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"> If you know the approximate value of the constant, you can search for the value to find Boost.Math chosen name in this table. </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"> Bernoulli numbers are available at <a class="link" href="number_series/bernoulli_numbers.html" title="Bernoulli Numbers">Bernoulli numbers</a>. </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"> Factorials are available at <a class="link" href="factorials/sf_factorial.html" title="Factorial">factorial</a>. </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, Johan Råde, Gautam Sewani, Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang 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>) </div></td> </tr></table> <hr> <div class="spirit-nav"> <a accesskey="p" href="tutorial/user_def.html"><img src="../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../constants.html"><img src="../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../index.html"><img src="../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="new_const.html"><img src="../../../../../doc/src/images/next.png" alt="Next"></a> </div> </body> </html>