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<title>Chapter 8. Special Functions</title>
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<div class="chapter">
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<div class="titlepage"><div><div><h1 class="title">
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<a name="special"></a>Chapter 8. Special Functions</h1></div></div></div>
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<div class="toc">
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<p><b>Table of Contents</b></p>
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<dl class="toc">
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<dt><span class="section"><a href="math_toolkit/number_series.html">Number Series</a></span></dt>
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<dd><dl>
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<dt><span class="section"><a href="math_toolkit/number_series/bernoulli_numbers.html">Bernoulli
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Numbers</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/number_series/tangent_numbers.html">Tangent Numbers</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/number_series/primes.html">Prime Numbers</a></span></dt>
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</dl></dd>
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<dt><span class="section"><a href="math_toolkit/sf_gamma.html">Gamma Functions</a></span></dt>
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<dd><dl>
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<dt><span class="section"><a href="math_toolkit/sf_gamma/tgamma.html">Gamma</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/sf_gamma/lgamma.html">Log Gamma</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/sf_gamma/digamma.html">Digamma</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/sf_gamma/trigamma.html">Trigamma</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/sf_gamma/polygamma.html">Polygamma</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/sf_gamma/gamma_ratios.html">Ratios of Gamma Functions</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/sf_gamma/igamma.html">Incomplete Gamma Functions</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/sf_gamma/igamma_inv.html">Incomplete Gamma Function
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Inverses</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/sf_gamma/gamma_derivatives.html">Derivative of
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the Incomplete Gamma Function</a></span></dt>
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</dl></dd>
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<dt><span class="section"><a href="math_toolkit/factorials.html">Factorials and Binomial Coefficients</a></span></dt>
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<dd><dl>
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<dt><span class="section"><a href="math_toolkit/factorials/sf_factorial.html">Factorial</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/factorials/sf_double_factorial.html">Double Factorial</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/factorials/sf_rising_factorial.html">Rising Factorial</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/factorials/sf_falling_factorial.html">Falling
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Factorial</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/factorials/sf_binomial.html">Binomial Coefficients</a></span></dt>
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</dl></dd>
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<dt><span class="section"><a href="math_toolkit/sf_beta.html">Beta Functions</a></span></dt>
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<dd><dl>
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<dt><span class="section"><a href="math_toolkit/sf_beta/beta_function.html">Beta</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/sf_beta/ibeta_function.html">Incomplete Beta
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Functions</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/sf_beta/ibeta_inv_function.html">The Incomplete
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Beta Function Inverses</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/sf_beta/beta_derivative.html">Derivative of the
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Incomplete Beta Function</a></span></dt>
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</dl></dd>
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<dt><span class="section"><a href="math_toolkit/sf_erf.html">Error Functions</a></span></dt>
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<dd><dl>
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<dt><span class="section"><a href="math_toolkit/sf_erf/error_function.html">Error Function erf
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and complement erfc</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/sf_erf/error_inv.html">Error Function Inverses</a></span></dt>
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</dl></dd>
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<dt><span class="section"><a href="math_toolkit/sf_poly.html">Polynomials</a></span></dt>
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<dd><dl>
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<dt><span class="section"><a href="math_toolkit/sf_poly/legendre.html">Legendre (and Associated)
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Polynomials</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/sf_poly/legendre_stieltjes.html">Legendre-Stieltjes
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Polynomials</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/sf_poly/laguerre.html">Laguerre (and Associated)
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Polynomials</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/sf_poly/hermite.html">Hermite Polynomials</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/sf_poly/chebyshev.html">Chebyshev Polynomials</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/sf_poly/sph_harm.html">Spherical Harmonics</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/sf_poly/cardinal_b_splines.html">Cardinal B-splines</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/sf_poly/gegenbauer.html">Gegenbauer Polynomials</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/sf_poly/jacobi.html">Jacobi Polynomials</a></span></dt>
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</dl></dd>
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<dt><span class="section"><a href="math_toolkit/bessel.html">Bessel Functions</a></span></dt>
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<dd><dl>
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<dt><span class="section"><a href="math_toolkit/bessel/bessel_over.html">Bessel Function Overview</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/bessel/bessel_first.html">Bessel Functions of
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the First and Second Kinds</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/bessel/bessel_root.html">Finding Zeros of Bessel
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Functions of the First and Second Kinds</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/bessel/mbessel.html">Modified Bessel Functions
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of the First and Second Kinds</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/bessel/sph_bessel.html">Spherical Bessel Functions
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of the First and Second Kinds</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/bessel/bessel_derivatives.html">Derivatives of
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the Bessel Functions</a></span></dt>
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</dl></dd>
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<dt><span class="section"><a href="math_toolkit/hankel.html">Hankel Functions</a></span></dt>
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<dd><dl>
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<dt><span class="section"><a href="math_toolkit/hankel/cyl_hankel.html">Cyclic Hankel Functions</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/hankel/sph_hankel.html">Spherical Hankel Functions</a></span></dt>
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</dl></dd>
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<dt><span class="section"><a href="math_toolkit/airy.html">Airy Functions</a></span></dt>
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<dd><dl>
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<dt><span class="section"><a href="math_toolkit/airy/ai.html">Airy Ai Function</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/airy/bi.html">Airy Bi Function</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/airy/aip.html">Airy Ai' Function</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/airy/bip.html">Airy Bi' Function</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/airy/airy_root.html">Finding Zeros of Airy Functions</a></span></dt>
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</dl></dd>
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<dt><span class="section"><a href="math_toolkit/ellint.html">Elliptic Integrals</a></span></dt>
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<dd><dl>
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<dt><span class="section"><a href="math_toolkit/ellint/ellint_intro.html">Elliptic Integral Overview</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/ellint/ellint_carlson.html">Elliptic Integrals
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- Carlson Form</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/ellint/ellint_1.html">Elliptic Integrals of the
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First Kind - Legendre Form</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/ellint/ellint_2.html">Elliptic Integrals of the
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Second Kind - Legendre Form</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/ellint/ellint_3.html">Elliptic Integrals of the
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Third Kind - Legendre Form</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/ellint/ellint_d.html">Elliptic Integral D - Legendre
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Form</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/ellint/jacobi_zeta.html">Jacobi Zeta Function</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/ellint/heuman_lambda.html">Heuman Lambda Function</a></span></dt>
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</dl></dd>
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<dt><span class="section"><a href="math_toolkit/jacobi.html">Jacobi Elliptic Functions</a></span></dt>
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<dd><dl>
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<dt><span class="section"><a href="math_toolkit/jacobi/jac_over.html">Overview of the Jacobi
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Elliptic Functions</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/jacobi/jacobi_elliptic.html">Jacobi Elliptic
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SN, CN and DN</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/jacobi/jacobi_cd.html">Jacobi Elliptic Function
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cd</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/jacobi/jacobi_cn.html">Jacobi Elliptic Function
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cn</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/jacobi/jacobi_cs.html">Jacobi Elliptic Function
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cs</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/jacobi/jacobi_dc.html">Jacobi Elliptic Function
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dc</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/jacobi/jacobi_dn.html">Jacobi Elliptic Function
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dn</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/jacobi/jacobi_ds.html">Jacobi Elliptic Function
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ds</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/jacobi/jacobi_nc.html">Jacobi Elliptic Function
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nc</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/jacobi/jacobi_nd.html">Jacobi Elliptic Function
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nd</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/jacobi/jacobi_ns.html">Jacobi Elliptic Function
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ns</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/jacobi/jacobi_sc.html">Jacobi Elliptic Function
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sc</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/jacobi/jacobi_sd.html">Jacobi Elliptic Function
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sd</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/jacobi/jacobi_sn.html">Jacobi Elliptic Function
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sn</a></span></dt>
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</dl></dd>
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<dt><span class="section"><a href="math_toolkit/lambert_w.html">Lambert <span class="emphasis"><em>W</em></span>
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function</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/zetas.html">Zeta Functions</a></span></dt>
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<dd><dl><dt><span class="section"><a href="math_toolkit/zetas/zeta.html">Riemann Zeta Function</a></span></dt></dl></dd>
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<dt><span class="section"><a href="math_toolkit/expint.html">Exponential Integrals</a></span></dt>
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<dd><dl>
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<dt><span class="section"><a href="math_toolkit/expint/expint_n.html">Exponential Integral En</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/expint/expint_i.html">Exponential Integral Ei</a></span></dt>
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</dl></dd>
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<dt><span class="section"><a href="math_toolkit/hypergeometric.html">Hypergeometric Functions</a></span></dt>
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<dd><dl>
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<dt><span class="section"><a href="math_toolkit/hypergeometric/hypergeometric_1f0.html">Hypergeometric
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<sub>1</sub><span class="emphasis"><em>F</em></span><sub>0</sub> </a></span></dt>
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<dt><span class="section"><a href="math_toolkit/hypergeometric/hypergeometric_0f1.html">Hypergeometric
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<sub>0</sub><span class="emphasis"><em>F</em></span><sub>1</sub> </a></span></dt>
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<dt><span class="section"><a href="math_toolkit/hypergeometric/hypergeometric_2f0.html">Hypergeometric
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<sub>2</sub><span class="emphasis"><em>F</em></span><sub>0</sub></a></span></dt>
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<dt><span class="section"><a href="math_toolkit/hypergeometric/hypergeometric_1f1.html">Hypergeometric
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<sub>1</sub><span class="emphasis"><em>F</em></span><sub>1</sub></a></span></dt>
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<dt><span class="section"><a href="math_toolkit/hypergeometric/hypergeometric_pfq.html">Hypergeometric
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<sub>p</sub>F<sub>q</sub></a></span></dt>
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<dt><span class="section"><a href="math_toolkit/hypergeometric/hypergeometric_refs.html">Hypergeometric
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References</a></span></dt>
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</dl></dd>
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<dt><span class="section"><a href="math_toolkit/powers.html">Basic Functions</a></span></dt>
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<dd><dl>
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<dt><span class="section"><a href="math_toolkit/powers/sin_pi.html">sin_pi</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/powers/cos_pi.html">cos_pi</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/powers/log1p.html">log1p</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/powers/expm1.html">expm1</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/powers/cbrt.html">cbrt</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/powers/sqrt1pm1.html">sqrt1pm1</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/powers/powm1.html">powm1</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/powers/hypot.html">hypot</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/powers/ct_pow.html">Compile Time Power of a Runtime
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Base</a></span></dt>
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</dl></dd>
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<dt><span class="section"><a href="math_toolkit/sinc.html">Sinus Cardinal and Hyperbolic Sinus
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Cardinal Functions</a></span></dt>
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<dd><dl>
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<dt><span class="section"><a href="math_toolkit/sinc/sinc_overview.html">Sinus Cardinal and Hyperbolic
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Sinus Cardinal Functions Overview</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/sinc/sinc_pi.html">sinc_pi</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/sinc/sinhc_pi.html">sinhc_pi</a></span></dt>
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</dl></dd>
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<dt><span class="section"><a href="math_toolkit/inv_hyper.html">Inverse Hyperbolic Functions</a></span></dt>
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<dd><dl>
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<dt><span class="section"><a href="math_toolkit/inv_hyper/inv_hyper_over.html">Inverse Hyperbolic
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Functions Overview</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/inv_hyper/acosh.html">acosh</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/inv_hyper/asinh.html">asinh</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/inv_hyper/atanh.html">atanh</a></span></dt>
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</dl></dd>
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<dt><span class="section"><a href="math_toolkit/owens_t.html">Owen's T function</a></span></dt>
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</dl>
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</div>
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</div>
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<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
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<td align="left"></td>
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<td align="right"><div class="copyright-footer">Copyright © 2006-2019 Nikhar
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Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
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Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
|
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Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
|
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Daryle Walker and Xiaogang Zhang<p>
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Distributed under the Boost Software License, Version 1.0. (See accompanying
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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>)
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</p>
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</div></td>
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</tr></table>
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<hr>
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<div class="spirit-nav">
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