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   4 <title>Chapter&#160;8.&#160;Special Functions</title>
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  26 <div class="titlepage"><div><div><h1 class="title">
  27 <a name="special"></a>Chapter&#160;8.&#160;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 </dl>
 221 </div>
 222 </div>
 223 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 224 <td align="left"></td>
 225 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2019 Nikhar
 226       Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
 227       Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
 228       R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
 229       Daryle Walker and Xiaogang Zhang<p>
 230         Distributed under the Boost Software License, Version 1.0. (See accompanying
 231         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>)
 232       </p>
 233 </div></td>
 234 </tr></table>
 235 <hr>
 236 <div class="spirit-nav">
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