1<html> 2<head> 3<meta http-equiv="Content-Type" content="text/html; charset=UTF-8"> 4<title>Chapter 8. Special Functions</title> 5<link rel="stylesheet" href="math.css" type="text/css"> 6<meta name="generator" content="DocBook XSL Stylesheets V1.79.1"> 7<link rel="home" href="index.html" title="Math Toolkit 2.12.0"> 8<link rel="up" href="index.html" title="Math Toolkit 2.12.0"> 9<link rel="prev" href="math_toolkit/norms.html" title="Norms"> 10<link rel="next" href="math_toolkit/number_series.html" title="Number Series"> 11</head> 12<body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"> 13<table cellpadding="2" width="100%"><tr> 14<td valign="top"><img alt="Boost C++ Libraries" width="277" height="86" src="../../../../boost.png"></td> 15<td align="center"><a href="../../../../index.html">Home</a></td> 16<td align="center"><a href="../../../../libs/libraries.htm">Libraries</a></td> 17<td align="center"><a href="http://www.boost.org/users/people.html">People</a></td> 18<td align="center"><a href="http://www.boost.org/users/faq.html">FAQ</a></td> 19<td align="center"><a href="../../../../more/index.htm">More</a></td> 20</tr></table> 21<hr> 22<div class="spirit-nav"> 23<a accesskey="p" href="math_toolkit/norms.html"><img src="../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="index.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="math_toolkit/number_series.html"><img src="../../../../doc/src/images/next.png" alt="Next"></a> 24</div> 25<div class="chapter"> 26<div class="titlepage"><div><div><h1 class="title"> 27<a name="special"></a>Chapter 8. Special Functions</h1></div></div></div> 28<div class="toc"> 29<p><b>Table of Contents</b></p> 30<dl class="toc"> 31<dt><span class="section"><a href="math_toolkit/number_series.html">Number Series</a></span></dt> 32<dd><dl> 33<dt><span class="section"><a href="math_toolkit/number_series/bernoulli_numbers.html">Bernoulli 34 Numbers</a></span></dt> 35<dt><span class="section"><a href="math_toolkit/number_series/tangent_numbers.html">Tangent Numbers</a></span></dt> 36<dt><span class="section"><a href="math_toolkit/number_series/primes.html">Prime Numbers</a></span></dt> 37</dl></dd> 38<dt><span class="section"><a href="math_toolkit/sf_gamma.html">Gamma Functions</a></span></dt> 39<dd><dl> 40<dt><span class="section"><a href="math_toolkit/sf_gamma/tgamma.html">Gamma</a></span></dt> 41<dt><span class="section"><a href="math_toolkit/sf_gamma/lgamma.html">Log Gamma</a></span></dt> 42<dt><span class="section"><a href="math_toolkit/sf_gamma/digamma.html">Digamma</a></span></dt> 43<dt><span class="section"><a href="math_toolkit/sf_gamma/trigamma.html">Trigamma</a></span></dt> 44<dt><span class="section"><a href="math_toolkit/sf_gamma/polygamma.html">Polygamma</a></span></dt> 45<dt><span class="section"><a href="math_toolkit/sf_gamma/gamma_ratios.html">Ratios of Gamma Functions</a></span></dt> 46<dt><span class="section"><a href="math_toolkit/sf_gamma/igamma.html">Incomplete Gamma Functions</a></span></dt> 47<dt><span class="section"><a href="math_toolkit/sf_gamma/igamma_inv.html">Incomplete Gamma Function 48 Inverses</a></span></dt> 49<dt><span class="section"><a href="math_toolkit/sf_gamma/gamma_derivatives.html">Derivative of 50 the Incomplete Gamma Function</a></span></dt> 51</dl></dd> 52<dt><span class="section"><a href="math_toolkit/factorials.html">Factorials and Binomial Coefficients</a></span></dt> 53<dd><dl> 54<dt><span class="section"><a href="math_toolkit/factorials/sf_factorial.html">Factorial</a></span></dt> 55<dt><span class="section"><a href="math_toolkit/factorials/sf_double_factorial.html">Double Factorial</a></span></dt> 56<dt><span class="section"><a href="math_toolkit/factorials/sf_rising_factorial.html">Rising Factorial</a></span></dt> 57<dt><span class="section"><a href="math_toolkit/factorials/sf_falling_factorial.html">Falling 58 Factorial</a></span></dt> 59<dt><span class="section"><a href="math_toolkit/factorials/sf_binomial.html">Binomial Coefficients</a></span></dt> 60</dl></dd> 61<dt><span class="section"><a href="math_toolkit/sf_beta.html">Beta Functions</a></span></dt> 62<dd><dl> 63<dt><span class="section"><a href="math_toolkit/sf_beta/beta_function.html">Beta</a></span></dt> 64<dt><span class="section"><a href="math_toolkit/sf_beta/ibeta_function.html">Incomplete Beta 65 Functions</a></span></dt> 66<dt><span class="section"><a href="math_toolkit/sf_beta/ibeta_inv_function.html">The Incomplete 67 Beta Function Inverses</a></span></dt> 68<dt><span class="section"><a href="math_toolkit/sf_beta/beta_derivative.html">Derivative of the 69 Incomplete Beta Function</a></span></dt> 70</dl></dd> 71<dt><span class="section"><a href="math_toolkit/sf_erf.html">Error Functions</a></span></dt> 72<dd><dl> 73<dt><span class="section"><a href="math_toolkit/sf_erf/error_function.html">Error Function erf 74 and complement erfc</a></span></dt> 75<dt><span class="section"><a href="math_toolkit/sf_erf/error_inv.html">Error Function Inverses</a></span></dt> 76</dl></dd> 77<dt><span class="section"><a href="math_toolkit/sf_poly.html">Polynomials</a></span></dt> 78<dd><dl> 79<dt><span class="section"><a href="math_toolkit/sf_poly/legendre.html">Legendre (and Associated) 80 Polynomials</a></span></dt> 81<dt><span class="section"><a href="math_toolkit/sf_poly/legendre_stieltjes.html">Legendre-Stieltjes 82 Polynomials</a></span></dt> 83<dt><span class="section"><a href="math_toolkit/sf_poly/laguerre.html">Laguerre (and Associated) 84 Polynomials</a></span></dt> 85<dt><span class="section"><a href="math_toolkit/sf_poly/hermite.html">Hermite Polynomials</a></span></dt> 86<dt><span class="section"><a href="math_toolkit/sf_poly/chebyshev.html">Chebyshev Polynomials</a></span></dt> 87<dt><span class="section"><a href="math_toolkit/sf_poly/sph_harm.html">Spherical Harmonics</a></span></dt> 88<dt><span class="section"><a href="math_toolkit/sf_poly/cardinal_b_splines.html">Cardinal B-splines</a></span></dt> 89<dt><span class="section"><a href="math_toolkit/sf_poly/gegenbauer.html">Gegenbauer Polynomials</a></span></dt> 90<dt><span class="section"><a href="math_toolkit/sf_poly/jacobi.html">Jacobi Polynomials</a></span></dt> 91</dl></dd> 92<dt><span class="section"><a href="math_toolkit/bessel.html">Bessel Functions</a></span></dt> 93<dd><dl> 94<dt><span class="section"><a href="math_toolkit/bessel/bessel_over.html">Bessel Function Overview</a></span></dt> 95<dt><span class="section"><a href="math_toolkit/bessel/bessel_first.html">Bessel Functions of 96 the First and Second Kinds</a></span></dt> 97<dt><span class="section"><a href="math_toolkit/bessel/bessel_root.html">Finding Zeros of Bessel 98 Functions of the First and Second Kinds</a></span></dt> 99<dt><span class="section"><a href="math_toolkit/bessel/mbessel.html">Modified Bessel Functions 100 of the First and Second Kinds</a></span></dt> 101<dt><span class="section"><a href="math_toolkit/bessel/sph_bessel.html">Spherical Bessel Functions 102 of the First and Second Kinds</a></span></dt> 103<dt><span class="section"><a href="math_toolkit/bessel/bessel_derivatives.html">Derivatives of 104 the Bessel Functions</a></span></dt> 105</dl></dd> 106<dt><span class="section"><a href="math_toolkit/hankel.html">Hankel Functions</a></span></dt> 107<dd><dl> 108<dt><span class="section"><a href="math_toolkit/hankel/cyl_hankel.html">Cyclic Hankel Functions</a></span></dt> 109<dt><span class="section"><a href="math_toolkit/hankel/sph_hankel.html">Spherical Hankel Functions</a></span></dt> 110</dl></dd> 111<dt><span class="section"><a href="math_toolkit/airy.html">Airy Functions</a></span></dt> 112<dd><dl> 113<dt><span class="section"><a href="math_toolkit/airy/ai.html">Airy Ai Function</a></span></dt> 114<dt><span class="section"><a href="math_toolkit/airy/bi.html">Airy Bi Function</a></span></dt> 115<dt><span class="section"><a href="math_toolkit/airy/aip.html">Airy Ai' Function</a></span></dt> 116<dt><span class="section"><a href="math_toolkit/airy/bip.html">Airy Bi' Function</a></span></dt> 117<dt><span class="section"><a href="math_toolkit/airy/airy_root.html">Finding Zeros of Airy Functions</a></span></dt> 118</dl></dd> 119<dt><span class="section"><a href="math_toolkit/ellint.html">Elliptic Integrals</a></span></dt> 120<dd><dl> 121<dt><span class="section"><a href="math_toolkit/ellint/ellint_intro.html">Elliptic Integral Overview</a></span></dt> 122<dt><span class="section"><a href="math_toolkit/ellint/ellint_carlson.html">Elliptic Integrals 123 - Carlson Form</a></span></dt> 124<dt><span class="section"><a href="math_toolkit/ellint/ellint_1.html">Elliptic Integrals of the 125 First Kind - Legendre Form</a></span></dt> 126<dt><span class="section"><a href="math_toolkit/ellint/ellint_2.html">Elliptic Integrals of the 127 Second Kind - Legendre Form</a></span></dt> 128<dt><span class="section"><a href="math_toolkit/ellint/ellint_3.html">Elliptic Integrals of the 129 Third Kind - Legendre Form</a></span></dt> 130<dt><span class="section"><a href="math_toolkit/ellint/ellint_d.html">Elliptic Integral D - Legendre 131 Form</a></span></dt> 132<dt><span class="section"><a href="math_toolkit/ellint/jacobi_zeta.html">Jacobi Zeta Function</a></span></dt> 133<dt><span class="section"><a href="math_toolkit/ellint/heuman_lambda.html">Heuman Lambda Function</a></span></dt> 134</dl></dd> 135<dt><span class="section"><a href="math_toolkit/jacobi.html">Jacobi Elliptic Functions</a></span></dt> 136<dd><dl> 137<dt><span class="section"><a href="math_toolkit/jacobi/jac_over.html">Overview of the Jacobi 138 Elliptic Functions</a></span></dt> 139<dt><span class="section"><a href="math_toolkit/jacobi/jacobi_elliptic.html">Jacobi Elliptic 140 SN, CN and DN</a></span></dt> 141<dt><span class="section"><a href="math_toolkit/jacobi/jacobi_cd.html">Jacobi Elliptic Function 142 cd</a></span></dt> 143<dt><span class="section"><a href="math_toolkit/jacobi/jacobi_cn.html">Jacobi Elliptic Function 144 cn</a></span></dt> 145<dt><span class="section"><a href="math_toolkit/jacobi/jacobi_cs.html">Jacobi Elliptic Function 146 cs</a></span></dt> 147<dt><span class="section"><a href="math_toolkit/jacobi/jacobi_dc.html">Jacobi Elliptic Function 148 dc</a></span></dt> 149<dt><span class="section"><a href="math_toolkit/jacobi/jacobi_dn.html">Jacobi Elliptic Function 150 dn</a></span></dt> 151<dt><span class="section"><a href="math_toolkit/jacobi/jacobi_ds.html">Jacobi Elliptic Function 152 ds</a></span></dt> 153<dt><span class="section"><a href="math_toolkit/jacobi/jacobi_nc.html">Jacobi Elliptic Function 154 nc</a></span></dt> 155<dt><span class="section"><a href="math_toolkit/jacobi/jacobi_nd.html">Jacobi Elliptic Function 156 nd</a></span></dt> 157<dt><span class="section"><a href="math_toolkit/jacobi/jacobi_ns.html">Jacobi Elliptic Function 158 ns</a></span></dt> 159<dt><span class="section"><a href="math_toolkit/jacobi/jacobi_sc.html">Jacobi Elliptic Function 160 sc</a></span></dt> 161<dt><span class="section"><a href="math_toolkit/jacobi/jacobi_sd.html">Jacobi Elliptic Function 162 sd</a></span></dt> 163<dt><span class="section"><a href="math_toolkit/jacobi/jacobi_sn.html">Jacobi Elliptic Function 164 sn</a></span></dt> 165</dl></dd> 166<dt><span class="section"><a href="math_toolkit/lambert_w.html">Lambert <span class="emphasis"><em>W</em></span> 167 function</a></span></dt> 168<dt><span class="section"><a href="math_toolkit/zetas.html">Zeta Functions</a></span></dt> 169<dd><dl><dt><span class="section"><a href="math_toolkit/zetas/zeta.html">Riemann Zeta Function</a></span></dt></dl></dd> 170<dt><span class="section"><a href="math_toolkit/expint.html">Exponential Integrals</a></span></dt> 171<dd><dl> 172<dt><span class="section"><a href="math_toolkit/expint/expint_n.html">Exponential Integral En</a></span></dt> 173<dt><span class="section"><a href="math_toolkit/expint/expint_i.html">Exponential Integral Ei</a></span></dt> 174</dl></dd> 175<dt><span class="section"><a href="math_toolkit/hypergeometric.html">Hypergeometric Functions</a></span></dt> 176<dd><dl> 177<dt><span class="section"><a href="math_toolkit/hypergeometric/hypergeometric_1f0.html">Hypergeometric 178 <sub>1</sub><span class="emphasis"><em>F</em></span><sub>0</sub> </a></span></dt> 179<dt><span class="section"><a href="math_toolkit/hypergeometric/hypergeometric_0f1.html">Hypergeometric 180 <sub>0</sub><span class="emphasis"><em>F</em></span><sub>1</sub> </a></span></dt> 181<dt><span class="section"><a href="math_toolkit/hypergeometric/hypergeometric_2f0.html">Hypergeometric 182 <sub>2</sub><span class="emphasis"><em>F</em></span><sub>0</sub></a></span></dt> 183<dt><span class="section"><a href="math_toolkit/hypergeometric/hypergeometric_1f1.html">Hypergeometric 184 <sub>1</sub><span class="emphasis"><em>F</em></span><sub>1</sub></a></span></dt> 185<dt><span class="section"><a href="math_toolkit/hypergeometric/hypergeometric_pfq.html">Hypergeometric 186 <sub>p</sub>F<sub>q</sub></a></span></dt> 187<dt><span class="section"><a href="math_toolkit/hypergeometric/hypergeometric_refs.html">Hypergeometric 188 References</a></span></dt> 189</dl></dd> 190<dt><span class="section"><a href="math_toolkit/powers.html">Basic Functions</a></span></dt> 191<dd><dl> 192<dt><span class="section"><a href="math_toolkit/powers/sin_pi.html">sin_pi</a></span></dt> 193<dt><span class="section"><a href="math_toolkit/powers/cos_pi.html">cos_pi</a></span></dt> 194<dt><span class="section"><a href="math_toolkit/powers/log1p.html">log1p</a></span></dt> 195<dt><span class="section"><a href="math_toolkit/powers/expm1.html">expm1</a></span></dt> 196<dt><span class="section"><a href="math_toolkit/powers/cbrt.html">cbrt</a></span></dt> 197<dt><span class="section"><a href="math_toolkit/powers/sqrt1pm1.html">sqrt1pm1</a></span></dt> 198<dt><span class="section"><a href="math_toolkit/powers/powm1.html">powm1</a></span></dt> 199<dt><span class="section"><a href="math_toolkit/powers/hypot.html">hypot</a></span></dt> 200<dt><span class="section"><a href="math_toolkit/powers/ct_pow.html">Compile Time Power of a Runtime 201 Base</a></span></dt> 202</dl></dd> 203<dt><span class="section"><a href="math_toolkit/sinc.html">Sinus Cardinal and Hyperbolic Sinus 204 Cardinal Functions</a></span></dt> 205<dd><dl> 206<dt><span class="section"><a href="math_toolkit/sinc/sinc_overview.html">Sinus Cardinal and Hyperbolic 207 Sinus Cardinal Functions Overview</a></span></dt> 208<dt><span class="section"><a href="math_toolkit/sinc/sinc_pi.html">sinc_pi</a></span></dt> 209<dt><span class="section"><a href="math_toolkit/sinc/sinhc_pi.html">sinhc_pi</a></span></dt> 210</dl></dd> 211<dt><span class="section"><a href="math_toolkit/inv_hyper.html">Inverse Hyperbolic Functions</a></span></dt> 212<dd><dl> 213<dt><span class="section"><a href="math_toolkit/inv_hyper/inv_hyper_over.html">Inverse Hyperbolic 214 Functions Overview</a></span></dt> 215<dt><span class="section"><a href="math_toolkit/inv_hyper/acosh.html">acosh</a></span></dt> 216<dt><span class="section"><a href="math_toolkit/inv_hyper/asinh.html">asinh</a></span></dt> 217<dt><span class="section"><a href="math_toolkit/inv_hyper/atanh.html">atanh</a></span></dt> 218</dl></dd> 219<dt><span class="section"><a href="math_toolkit/owens_t.html">Owen's T function</a></span></dt> 220<dt><span class="section"><a href="math_toolkit/daubechies.html">Daubechies Wavelets and Scaling 221 Functions</a></span></dt> 222</dl> 223</div> 224</div> 225<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> 226<td align="left"></td> 227<td align="right"><div class="copyright-footer">Copyright © 2006-2019 Nikhar 228 Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, 229 Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan 230 Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, 231 Daryle Walker and Xiaogang Zhang<p> 232 Distributed under the Boost Software License, Version 1.0. (See accompanying 233 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>) 234 </p> 235</div></td> 236</tr></table> 237<hr> 238<div class="spirit-nav"> 239<a accesskey="p" href="math_toolkit/norms.html"><img src="../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="index.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="math_toolkit/number_series.html"><img src="../../../../doc/src/images/next.png" alt="Next"></a> 240</div> 241</body> 242</html> 243
