===================================================================
@@ -75,16 +75,182 @@
* @{
*/
+ /**
+ * @mainpage Mathematical Special Functions
+ *
+ * @section intro Introduction and History
+ * The first significant library upgrade on the road to C++2011,
+ * <a href="http://www.open-std.org/JTC1/SC22/WG21/docs/papers/2005/n1836.pdf">
+ * TR1</a>, included a set of 23 mathematical functions that significntly
+ * extended the standard trancendental functions inherited from C and declared
+ * in @<cmath@>.
+ *
+ * Although most components from TR1 were eventually adopted for C++11 these
+ * math function were left behind out of concern for implementability.
+ * The math functions were published as a separate international standard
+ * <a href="http://www.open-std.org/JTC1/SC22/WG21/docs/papers/2010/n3060.pdf">
+ * IS 29124 - Extensions to the C++ Library to Support Mathematical Special
+ * Functions</a>.
+ *
+ * For C++17 these functions were incorporated into the main standard.
+ *
+ * @section contents Contents
+ * The folowing functions are implemented in namespace @c std:
+ * - @ref assoc_laguerre "assoc_laguerre - Associated Laguerre functions"
+ * - @ref assoc_legendre "assoc_legendre - Associated Legendre functions"
+ * - @ref beta "beta - Beta functions"
+ * - @ref comp_ellint_1 "comp_ellint_1 - Complete elliptic functions of the first kind"
+ * - @ref comp_ellint_2 "comp_ellint_2 - Complete elliptic functions of the second kind"
+ * - @ref comp_ellint_3 "comp_ellint_3 - Complete elliptic functions of the third kind"
+ * - @ref cyl_bessel_i "cyl_bessel_i - Regular modified cylindrical Bessel functions"
+ * - @ref cyl_bessel_j "cyl_bessel_j - Cylindrical Bessel functions of the first kind"
+ * - @ref cyl_bessel_k "cyl_bessel_k - Irregular modified cylindrical Bessel functions"
+ * - @ref cyl_neumann "cyl_neumann - Cylindrical Neumann functions or Cylindrical Bessel functions of the second kind"
+ * - @ref ellint_1 "ellint_1 - Incomplete elliptic functions of the first kind"
+ * - @ref ellint_2 "ellint_2 - Incomplete elliptic functions of the second kind"
+ * - @ref ellint_3 "ellint_3 - Incomplete elliptic functions of the third kind"
+ * - @ref expint "expint - The exponential integral"
+ * - @ref hermite "hermite - Hermite polynomials"
+ * - @ref laguerre "laguerre - Laguerre functions"
+ * - @ref legendre "legendre - Legendre polynomials"
+ * - @ref riemann_zeta "riemann_zeta - The Riemann zeta function"
+ * - @ref sph_bessel "sph_bessel - Spherical Bessel functions"
+ * - @ref sph_legendre "sph_legendre - Spherical Legendre functions"
+ * - @ref sph_neumann "sph_neumann - Spherical Neumann functions"
+ *
+ * The hypergeometric functions were stricken from the TR29124 and C++17
+ * versions of this math library because of implementation concerns.
+ * However, since they were in the TR1 version and since they are popular
+ * we kept them as an extension in namespace @c __gnu_cxx:
+ * - @ref conf_hyperg "conf_hyperg - Confluent hypergeometric functions"
+ * - @ref hyperg "hyperg - Hypergeometric functions"
+ *
+ * @section general General Features
+ *
+ * @subsection "Argument Promotion"
+ * The arguments suppled to the non-suffixed functions will be promoted
+ * according to the following rules:
+ * 1. If any argument intended to be floating opint is given an integral value
+ * That integral value is promoted to double.
+ * 2. All floating point arguments are promoted up to the largest floating
+ * point precision among them.
+ *
+ * @subsection NaN NaN Arguments
+ * If any of the floating point arguments supplied to these functions is
+ * invalid or NaN (std::numeric_limits<Tp>::quiet_NaN),
+ * the value NaN is returned.
+ *
+ * @section impl Implementation
+ *
+ * We strive to implement the underlying math with type generic algorithms
+ * to the greatest extent possible. In practice, the function are thin
+ * wrappers that dispatch to function templates. Type dependence is
+ * controlled with std::numeric_limits and functions thereof.
+ *
+ * We don't promote *c float to *c double or *c double to <tt>long double</tt>
+ * reflexively. The goal is for float functions to operate more quickly,
+ * at the cost of float accuracy and possibly a smaller domain of validity.
+ * Similaryly, <tt>long double</tt> should give you more dynamic range
+ * and slightly more pecision than @c double on many systems.
+ *
+ * @section testing Testing
+ *
+ * These functions have been tested against equivalent implementations
+ * from the <a href="http://www.gnu.org/software/gsl">
+ * Gnu Scientific Library, GSL</a> and
+ * <a href="http://www.boost.org/doc/libs/1_60_0/libs/math/doc/html/index.html>Boost</a>
+ * and the ratio
+ * @f[
+ * \frac{|f - f_{test}|}{|f_{test}|}
+ * @f]
+ * is generally found to be within 10^-15 for 64-bit double on linux-x86_64 systems
+ * over most of the ranges of validity.
+ *
+ * @todo Provide accuracy comparisons on a per-function basis for a small
+ * number of targets.
+ *
+ * @section bibliography General Bibliography
+ *
+ * @see Abramowitz and Stegun: Handbook of Mathematical Functions,
+ * with Formulas, Graphs, and Mathematical Tables
+ * Edited by Milton Abramowitz and Irene A. Stegun,
+ * National Bureau of Standards Applied Mathematics Series - 55
+ * Issued June 1964, Tenth Printing, December 1972, with corrections
+ * Electronic versions of A&S abound including both pdf and navigable html.
+ * @see for example http://people.math.sfu.ca/~cbm/aands/
+ *
+ * @see The old A&S has been redone as the
+ * NIST Digital Library of Mathematical Functions: http://dlmf.nist.gov/
+ * This version is far more navigable and includes more recent work.
+ *
+ * @see An Atlas of Functions: with Equator, the Atlas Function Calculator
+ * 2nd Edition, by Oldham, Keith B., Myland, Jan, Spanier, Jerome
+ *
+ * @see Asymptotics and Special Functions by Frank W. J. Olver,
+ * Academic Press, 1974
+ *
+ * @see Numerical Recipes in C, The Art of Scientific Computing,
+ * by William H. Press, Second Ed., Saul A. Teukolsky,
+ * William T. Vetterling, and Brian P. Flannery,
+ * Cambridge University Press, 1992
+ *
+ * @see The Special Functions and Their Approximations: Volumes 1 and 2,
+ * by Yudell L. Luke, Academic Press, 1969
+ */
+
// Associated Laguerre polynomials
+ /**
+ * Return the associated Laguerre polynomial of order @c n,
+ * degree @c m: @f$ L_n^m(x) @f$ for @c float argument.
+ *
+ * @see assoc_laguerre for more details.
+ */
inline float
assoc_laguerref(unsigned int __n, unsigned int __m, float __x)
{ return __detail::__assoc_laguerre<float>(__n, __m, __x); }
+ /**
+ * Return the associated Laguerre polynomial of order @c n,
+ * degree @c m: @f$ L_n^m(x) @f$.
+ *
+ * @see assoc_laguerre for more details.
+ */
inline long double
assoc_laguerrel(unsigned int __n, unsigned int __m, long double __x)
{ return __detail::__assoc_laguerre<long double>(__n, __m, __x); }
+ /**
+ * Return the associated Laguerre polynomial of nonnegative order @c n,
+ * nonnegative degree @c m and real argument @c x: @f$ L_n^m(x) @f$.
+ *
+ * The associated Laguerre function of real degree @f$ \alpha @f$,
+ * @f$ L_n^\alpha(x) @f$, is defined by
+ * @f[
+ * L_n^\alpha(x) = \frac{(\alpha + 1)_n}{n!}
+ * {}_1F_1(-n; \alpha + 1; x)
+ * @f]
+ * where @f$ (\alpha)_n @f$ is the Pochhammer symbol and
+ * @f$ {}_1F_1(a; c; x) @f$ is the confluent hypergeometric function.
+ *
+ * The associated Laguerre polynomial is defined for integral
+ * degree @f$ \alpha = m @f$ by:
+ * @f[
+ * L_n^m(x) = (-1)^m \frac{d^m}{dx^m} L_{n + m}(x)
+ * @f]
+ * where the Laguerre polynomial is defined by:
+ * @f[
+ * L_n(x) = \frac{e^x}{n!} \frac{d^n}{dx^n} (x^ne^{-x})
+ * @f]
+ * and @f$ x >= 0 @f$.
+ * @see laguerre for details of the Laguerre function of degree @c n
+ *
+ * @tparam _Tp The floating-point type of the argument @c __x.
+ * @param __n The order of the Laguerre function, <tt>__n >= 0</tt>.
+ * @param __m The degree of the Laguerre function, <tt>__m >= 0</tt>.
+ * @param __x The argument of the Laguerre function, <tt>__x >= 0</tt>.
+ * @throw std::domain_error if <tt>__x < 0</tt>.
+ */
template<typename _Tp>
inline typename __gnu_cxx::__promote<_Tp>::__type
assoc_laguerre(unsigned int __n, unsigned int __m, _Tp __x)
@@ -95,14 +261,42 @@
// Associated Legendre functions
+ /**
+ * Return the associated Legendre function of degree @c l and order @c m
+ * for @c float argument.
+ *
+ * @see assoc_legendre for more details.
+ */
inline float
assoc_legendref(unsigned int __l, unsigned int __m, float __x)
{ return __detail::__assoc_legendre_p<float>(__l, __m, __x); }
+ /**
+ * Return the associated Legendre function of degree @c l and order @c m.
+ *
+ * @see assoc_legendre for more details.
+ */
inline long double
assoc_legendrel(unsigned int __l, unsigned int __m, long double __x)
{ return __detail::__assoc_legendre_p<long double>(__l, __m, __x); }
+
+ /**
+ * Return the associated Legendre function of degree @c l and order @c m.
+ *
+ * The associated Legendre function is derived from the Legendre function
+ * @f$ P_l(x) @f$ by the Rodrigues formula:
+ * @f[
+ * P_l^m(x) = (1 - x^2)^{m/2}\frac{d^m}{dx^m}P_l(x)
+ * @f]
+ * @see legendre for details of the Legendre function of degree @c l
+ *
+ * @tparam _Tp The floating-point type of the argument @c __x.
+ * @param __l The degree <tt>__l >= 0</tt>.
+ * @param __m The order <tt>__m <= l</tt>.
+ * @param __x The argument, <tt>abs(__x) <= 1</tt>.
+ * @throw std::domain_error if <tt>abs(__x) > 1</tt>.
+ */
template<typename _Tp>
inline typename __gnu_cxx::__promote<_Tp>::__type
assoc_legendre(unsigned int __l, unsigned int __m, _Tp __x)
@@ -113,32 +307,89 @@
// Beta functions
+ /**
+ * Return the beta function, @f$ B(a,b) @f$, for @c float parameters @c a, @c b.
+ *
+ * @see beta for more details.
+ */
inline float
- betaf(float __x, float __y)
- { return __detail::__beta<float>(__x, __y); }
+ betaf(float __a, float __b)
+ { return __detail::__beta<float>(__a, __b); }
+ /**
+ * Return the beta function, @f$B(a,b)@f$, for long double
+ * parameters @c a, @c b.
+ *
+ * @see beta for more details.
+ */
inline long double
- betal(long double __x, long double __y)
- { return __detail::__beta<long double>(__x, __y); }
+ betal(long double __a, long double __b)
+ { return __detail::__beta<long double>(__a, __b); }
- template<typename _Tpx, typename _Tpy>
- inline typename __gnu_cxx::__promote_2<_Tpx, _Tpy>::__type
- beta(_Tpx __x, _Tpy __y)
+ /**
+ * Return the beta function, @f$B(a,b)@f$, for real parameters @c a, @c b.
+ *
+ * The beta function is defined by
+ * @f[
+ * B(a,b) = \int_0^1 t^{a - 1} (1 - t)^{b - 1} dt
+ * = \frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)}
+ * @f]
+ * where @f$ x > 0 @f$ and @f$ y > 0 @f$
+ *
+ * @tparam _Tpa The floating-point type of the parameter @c __a.
+ * @tparam _Tpb The floating-point type of the parameter @c __b.
+ * @param __a The first argument of the beta function, <tt> __a > 0 </tt>.
+ * @param __b The second argument of the beta function, <tt> __b > 0 </tt>.
+ * @throw std::domain_error if <tt> __a < 0 </tt> or <tt> __b < 0 </tt>.
+ */
+ template<typename _Tpa, typename _Tpb>
+ inline typename __gnu_cxx::__promote_2<_Tpa, _Tpb>::__type
+ beta(_Tpa __a, _Tpb __b)
{
- typedef typename __gnu_cxx::__promote_2<_Tpx, _Tpy>::__type __type;
- return __detail::__beta<__type>(__x, __y);
+ typedef typename __gnu_cxx::__promote_2<_Tpa, _Tpb>::__type __type;
+ return __detail::__beta<__type>(__a, __b);
}
// Complete elliptic integrals of the first kind
+ /**
+ * Return the complete elliptic integral of the first kind @f$ E(k) @f$
+ * for @c float modulus @c k.
+ *
+ * @see comp_ellint_1 for details.
+ */
inline float
comp_ellint_1f(float __k)
{ return __detail::__comp_ellint_1<float>(__k); }
+ /**
+ * Return the complete elliptic integral of the first kind @f$ E(k) @f$
+ * for long double modulus @c k.
+ *
+ * @see comp_ellint_1 for details.
+ */
inline long double
comp_ellint_1l(long double __k)
{ return __detail::__comp_ellint_1<long double>(__k); }
+ /**
+ * Return the complete elliptic integral of the first kind
+ * @f$ K(k) @f$ for real modulus @c k.
+ *
+ * The complete elliptic integral of the first kind is defined as
+ * @f[
+ * K(k) = F(k,\pi/2) = \int_0^{\pi/2}\frac{d\theta}
+ * {\sqrt{1 - k^2 sin^2\theta}}
+ * @f]
+ * where @f$ F(k,\phi) @f$ is the incomplete elliptic integral of the
+ * first kind and the modulus @f$ |k| <= 1 @f$.
+ * @see ellint_1 for details of the incomplete elliptic function
+ * of the first kind.
+ *
+ * @tparam _Tp The floating-point type of the modulus @c __k.
+ * @param __k The modulus, <tt> abs(__k) <= 1 </tt>
+ * @throw std::domain_error if <tt> abs(__k) > 1 </tt>.
+ */
template<typename _Tp>
inline typename __gnu_cxx::__promote<_Tp>::__type
comp_ellint_1(_Tp __k)
@@ -149,14 +400,43 @@
// Complete elliptic integrals of the second kind
+ /**
+ * Return the complete elliptic integral of the second kind @f$ E(k) @f$
+ * for @c float modulus @c k.
+ *
+ * @see comp_ellint_2 for details.
+ */
inline float
comp_ellint_2f(float __k)
{ return __detail::__comp_ellint_2<float>(__k); }
+ /**
+ * Return the complete elliptic integral of the second kind @f$ E(k) @f$
+ * for long double modulus @c k.
+ *
+ * @see comp_ellint_2 for details.
+ */
inline long double
comp_ellint_2l(long double __k)
{ return __detail::__comp_ellint_2<long double>(__k); }
+ /**
+ * Return the complete elliptic integral of the second kind @f$ E(k) @f$
+ * for real modulus @c k.
+ *
+ * The complete elliptic integral of the second kind is defined as
+ * @f[
+ * E(k) = E(k,\pi/2) = \int_0^{\pi/2}\sqrt{1 - k^2 sin^2\theta}
+ * @f]
+ * where @f$ E(k,\phi) @f$ is the incomplete elliptic integral of the
+ * second kind and the modulus @f$ |k| <= 1 @f$.
+ * @see ellint_2 for details of the incomplete elliptic function
+ * of the second kind.
+ *
+ * @tparam _Tp The floating-point type of the modulus @c __k.
+ * @param __k The modulus, @c abs(__k) <= 1
+ * @throw std::domain_error if @c abs(__k) > 1.
+ */
template<typename _Tp>
inline typename __gnu_cxx::__promote<_Tp>::__type
comp_ellint_2(_Tp __k)
@@ -167,14 +447,47 @@
// Complete elliptic integrals of the third kind
+ /**
+ * @brief Return the complete elliptic integral of the third kind
+ * @f$ \Pi(k,\nu) @f$ for @c float modulus @c k.
+ *
+ * @see comp_ellint_3 for details.
+ */
inline float
comp_ellint_3f(float __k, float __nu)
{ return __detail::__comp_ellint_3<float>(__k, __nu); }
+ /**
+ * @brief Return the complete elliptic integral of the third kind
+ * @f$ \Pi(k,\nu) @f$ for <tt>long double</tt> modulus @c k.
+ *
+ * @see comp_ellint_3 for details.
+ */
inline long double
comp_ellint_3l(long double __k, long double __nu)
{ return __detail::__comp_ellint_3<long double>(__k, __nu); }
+ /**
+ * Return the complete elliptic integral of the third kind
+ * @f$ \Pi(k,\nu) = \Pi(k,\nu,\pi/2) @f$ for real modulus @c k.
+ *
+ * The complete elliptic integral of the third kind is defined as
+ * @f[
+ * \Pi(k,\nu) = \Pi(k,\nu,\pi/2) = \int_0^{\pi/2}
+ * \frac{d\theta}
+ * {(1 - \nu \sin^2\theta)\sqrt{1 - k^2 \sin^2\theta}}
+ * @f]
+ * where @f$ \Pi(k,\nu,\phi) @f$ is the incomplete elliptic integral of the
+ * second kind and the modulus @f$ |k| <= 1 @f$.
+ * @see ellint_3 for details of the incomplete elliptic function
+ * of the third kind.
+ *
+ * @tparam _Tp The floating-point type of the modulus @c __k.
+ * @tparam _Tpn The floating-point type of the argument @c __nu.
+ * @param __k The modulus, @c abs(__k) <= 1
+ * @param __nu The argument
+ * @throw std::domain_error if @c abs(__k) > 1.
+ */
template<typename _Tp, typename _Tpn>
inline typename __gnu_cxx::__promote_2<_Tp, _Tpn>::__type
comp_ellint_3(_Tp __k, _Tpn __nu)
@@ -185,14 +498,42 @@
// Regular modified cylindrical Bessel functions
+ /**
+ * Return the regular modified Bessel function @f$ I_{\nu}(x) @f$
+ * for @c float order @f$ \nu @f$ and argument @f$ x >= 0 @f$.
+ *
+ * @see cyl_bessel_i for setails.
+ */
inline float
cyl_bessel_if(float __nu, float __x)
{ return __detail::__cyl_bessel_i<float>(__nu, __x); }
+ /**
+ * Return the regular modified Bessel function @f$ I_{\nu}(x) @f$
+ * for <tt>long double</tt> order @f$ \nu @f$ and argument @f$ x >= 0 @f$.
+ *
+ * @see cyl_bessel_i for setails.
+ */
inline long double
cyl_bessel_il(long double __nu, long double __x)
{ return __detail::__cyl_bessel_i<long double>(__nu, __x); }
+ /**
+ * Return the regular modified Bessel function @f$ I_{\nu}(x) @f$
+ * for real order @f$ \nu @f$ and argument @f$ x >= 0 @f$.
+ *
+ * The regular modified cylindrical Bessel function is:
+ * @f[
+ * I_{\nu}(x) = i^{-\nu}J_\nu(ix) = \sum_{k=0}^{\infty}
+ * \frac{(x/2)^{\nu + 2k}}{k!\Gamma(\nu+k+1)}
+ * @f]
+ *
+ * @tparam _Tpnu The floating-point type of the order @c __nu.
+ * @tparam _Tp The floating-point type of the argument @c __x.
+ * @param __nu The order
+ * @param __x The argument, <tt> __x >= 0 </tt>
+ * @throw std::domain_error if <tt> __x < 0 </tt>.
+ */
template<typename _Tpnu, typename _Tp>
inline typename __gnu_cxx::__promote_2<_Tpnu, _Tp>::__type
cyl_bessel_i(_Tpnu __nu, _Tp __x)
@@ -203,14 +544,42 @@
// Cylindrical Bessel functions (of the first kind)
+ /**
+ * Return the Bessel function of the first kind @f$ J_{\nu}(x) @f$
+ * for @c float order @f$ \nu @f$ and argument @f$ x >= 0 @f$.
+ *
+ * @see cyl_bessel_j for setails.
+ */
inline float
cyl_bessel_jf(float __nu, float __x)
{ return __detail::__cyl_bessel_j<float>(__nu, __x); }
+ /**
+ * Return the Bessel function of the first kind @f$ J_{\nu}(x) @f$
+ * for <tt>long double</tt> order @f$ \nu @f$ and argument @f$ x >= 0 @f$.
+ *
+ * @see cyl_bessel_j for setails.
+ */
inline long double
cyl_bessel_jl(long double __nu, long double __x)
{ return __detail::__cyl_bessel_j<long double>(__nu, __x); }
+ /**
+ * Return the Bessel function @f$ J_{\nu}(x) @f$ of real order @f$ \nu @f$
+ * and argument @f$ x >= 0 @f$.
+ *
+ * The cylindrical Bessel function is:
+ * @f[
+ * J_{\nu}(x) = \sum_{k=0}^{\infty}
+ * \frac{(-1)^k (x/2)^{\nu + 2k}}{k!\Gamma(\nu+k+1)}
+ * @f]
+ *
+ * @tparam _Tpnu The floating-point type of the order @c __nu.
+ * @tparam _Tp The floating-point type of the argument @c __x.
+ * @param __nu The order
+ * @param __x The argument, <tt> __x >= 0 </tt>
+ * @throw std::domain_error if <tt> __x < 0 </tt>.
+ */
template<typename _Tpnu, typename _Tp>
inline typename __gnu_cxx::__promote_2<_Tpnu, _Tp>::__type
cyl_bessel_j(_Tpnu __nu, _Tp __x)
@@ -221,14 +590,48 @@
// Irregular modified cylindrical Bessel functions
+ /**
+ * Return the irregular modified Bessel function @f$ K_{\nu}(x) @f$
+ * for @c float order @f$ \nu @f$ and argument @f$ x >= 0 @f$.
+ *
+ * @see cyl_bessel_k for setails.
+ */
inline float
cyl_bessel_kf(float __nu, float __x)
{ return __detail::__cyl_bessel_k<float>(__nu, __x); }
+ /**
+ * Return the irregular modified Bessel function @f$ K_{\nu}(x) @f$
+ * for <tt>long double</tt> order @f$ \nu @f$ and argument @f$ x >= 0 @f$.
+ *
+ * @see cyl_bessel_k for setails.
+ */
inline long double
cyl_bessel_kl(long double __nu, long double __x)
{ return __detail::__cyl_bessel_k<long double>(__nu, __x); }
+ /**
+ * Return the irregular modified Bessel function @f$ K_{\nu}(x) @f$
+ * of real order @f$ \nu @f$ and argument @f$ x @f$.
+ *
+ * The irregular modified Bessel function is defined by:
+ * @f[
+ * K_{\nu}(x) = \frac{\pi}{2}
+ * \frac{I_{-\nu}(x) - I_{\nu}(x)}{\sin \nu\pi}
+ * @f]
+ * where for integral @f$ \nu = n @f$ a limit is taken:
+ * @f$ lim_{\nu \to n} @f$.
+ * For negative argument we have simply:
+ * @f[
+ * K_{-\nu}(x) = K_{\nu}(x)
+ * @f]
+ *
+ * @tparam _Tpnu The floating-point type of the order @c __nu.
+ * @tparam _Tp The floating-point type of the argument @c __x.
+ * @param __nu The order
+ * @param __x The argument, <tt> __x >= 0 </tt>
+ * @throw std::domain_error if <tt> __x < 0 </tt>.
+ */
template<typename _Tpnu, typename _Tp>
inline typename __gnu_cxx::__promote_2<_Tpnu, _Tp>::__type
cyl_bessel_k(_Tpnu __nu, _Tp __x)
@@ -239,14 +642,44 @@
// Cylindrical Neumann functions
+ /**
+ * Return the Neumann function @f$ N_{\nu}(x) @f$
+ * of @c float order @f$ \nu @f$ and argument @f$ x @f$.
+ *
+ * @see cyl_neumann for setails.
+ */
inline float
cyl_neumannf(float __nu, float __x)
{ return __detail::__cyl_neumann_n<float>(__nu, __x); }
+ /**
+ * Return the Neumann function @f$ N_{\nu}(x) @f$
+ * of <tt>long double</tt> order @f$ \nu @f$ and argument @f$ x @f$.
+ *
+ * @see cyl_neumann for setails.
+ */
inline long double
cyl_neumannl(long double __nu, long double __x)
{ return __detail::__cyl_neumann_n<long double>(__nu, __x); }
+ /**
+ * Return the Neumann function @f$ N_{\nu}(x) @f$
+ * of real order @f$ \nu @f$ and argument @f$ x >= 0 @f$.
+ *
+ * The Neumann function is defined by:
+ * @f[
+ * N_{\nu}(x) = \frac{J_{\nu}(x) \cos \nu\pi - J_{-\nu}(x)}
+ * {\sin \nu\pi}
+ * @f]
+ * where @f$ x >= 0 @f$ and for integral order @f$ \nu = n @f$
+ * a limit is taken: @f$ lim_{\nu \to n} @f$.
+ *
+ * @tparam _Tpnu The floating-point type of the order @c __nu.
+ * @tparam _Tp The floating-point type of the argument @c __x.
+ * @param __nu The order
+ * @param __x The argument, <tt> __x >= 0 </tt>
+ * @throw std::domain_error if <tt> __x < 0 </tt>.
+ */
template<typename _Tpnu, typename _Tp>
inline typename __gnu_cxx::__promote_2<_Tpnu, _Tp>::__type
cyl_neumann(_Tpnu __nu, _Tp __x)
@@ -257,14 +690,44 @@
// Incomplete elliptic integrals of the first kind
+ /**
+ * Return the incomplete elliptic integral of the first kind @f$ E(k,\phi) @f$
+ * for @c float modulus @f$ k @f$ and angle @f$ \phi @f$.
+ *
+ * @see ellint_1 for details.
+ */
inline float
ellint_1f(float __k, float __phi)
{ return __detail::__ellint_1<float>(__k, __phi); }
+ /**
+ * Return the incomplete elliptic integral of the first kind @f$ E(k,\phi) @f$
+ * for <tt>long double</tt> modulus @f$ k @f$ and angle @f$ \phi @f$.
+ *
+ * @see ellint_1 for details.
+ */
inline long double
ellint_1l(long double __k, long double __phi)
{ return __detail::__ellint_1<long double>(__k, __phi); }
+ /**
+ * Return the incomplete elliptic integral of the first kind @f$ F(k,\phi) @f$
+ * for @c real modulus @f$ k @f$ and angle @f$ \phi @f$.
+ *
+ * The incomplete elliptic integral of the first kind is defined as
+ * @f[
+ * F(k,\phi) = \int_0^{\phi}\frac{d\theta}
+ * {\sqrt{1 - k^2 sin^2\theta}}
+ * @f]
+ * For @f$ \phi= \pi/2 @f$ this becomes the complete elliptic integral of
+ * the first kind, @f$ K(k) @f$. @see comp_ellint_1.
+ *
+ * @tparam _Tp The floating-point type of the modulus @c __k.
+ * @tparam _Tpp The floating-point type of the angle @c __phi.
+ * @param __k The modulus, <tt> abs(__k) <= 1 </tt>
+ * @param __phi The integral limit argument in radians
+ * @throw std::domain_error if <tt> abs(__k) > 1 </tt>.
+ */
template<typename _Tp, typename _Tpp>
inline typename __gnu_cxx::__promote_2<_Tp, _Tpp>::__type
ellint_1(_Tp __k, _Tpp __phi)
@@ -275,14 +738,44 @@
// Incomplete elliptic integrals of the second kind
+ /**
+ * @brief Return the incomplete elliptic integral of the second kind
+ * @f$ E(k,\phi) @f$ for @c float argument.
+ *
+ * @see ellint_2 for details.
+ */
inline float
ellint_2f(float __k, float __phi)
{ return __detail::__ellint_2<float>(__k, __phi); }
+ /**
+ * @brief Return the incomplete elliptic integral of the second kind
+ * @f$ E(k,\phi) @f$.
+ *
+ * @see ellint_2 for details.
+ */
inline long double
ellint_2l(long double __k, long double __phi)
{ return __detail::__ellint_2<long double>(__k, __phi); }
+ /**
+ * Return the incomplete elliptic integral of the second kind
+ * @f$ E(k,\phi) @f$.
+ *
+ * The incomplete elliptic integral of the second kind is defined as
+ * @f[
+ * E(k,\phi) = \int_0^{\phi} \sqrt{1 - k^2 sin^2\theta}
+ * @f]
+ * For @f$ \phi= \pi/2 @f$ this becomes the complete elliptic integral of
+ * the second kind, @f$ E(k) @f$. @see comp_ellint_2.
+ *
+ * @tparam _Tp The floating-point type of the modulus @c __k.
+ * @tparam _Tpp The floating-point type of the angle @c __phi.
+ * @param __k The modulus, <tt> abs(__k) <= 1 </tt>
+ * @param __phi The integral limit argument in radians
+ * @return The elliptic function of the second kind.
+ * @throw std::domain_error if <tt> abs(__k) > 1 </tt>.
+ */
template<typename _Tp, typename _Tpp>
inline typename __gnu_cxx::__promote_2<_Tp, _Tpp>::__type
ellint_2(_Tp __k, _Tpp __phi)
@@ -293,14 +786,49 @@
// Incomplete elliptic integrals of the third kind
+ /**
+ * @brief Return the incomplete elliptic integral of the third kind
+ * @f$ \Pi(k,\nu,\phi) @f$ for @c float argument.
+ *
+ * @see ellint_3 for details.
+ */
inline float
ellint_3f(float __k, float __nu, float __phi)
{ return __detail::__ellint_3<float>(__k, __nu, __phi); }
+ /**
+ * @brief Return the incomplete elliptic integral of the third kind
+ * @f$ \Pi(k,\nu,\phi) @f$.
+ *
+ * @see ellint_3 for details.
+ */
inline long double
ellint_3l(long double __k, long double __nu, long double __phi)
{ return __detail::__ellint_3<long double>(__k, __nu, __phi); }
+ /**
+ * @brief Return the incomplete elliptic integral of the third kind
+ * @f$ \Pi(k,\nu,\phi) @f$.
+ *
+ * The incomplete elliptic integral of the third kind is defined by:
+ * @f[
+ * \Pi(k,\nu,\phi) = \int_0^{\phi}
+ * \frac{d\theta}
+ * {(1 - \nu \sin^2\theta)
+ * \sqrt{1 - k^2 \sin^2\theta}}
+ * @f]
+ * For @f$ \phi= \pi/2 @f$ this becomes the complete elliptic integral of
+ * the third kind, @f$ \Pi(k,\nu) @f$. @see comp_ellint_3.
+ *
+ * @tparam _Tp The floating-point type of the modulus @c __k.
+ * @tparam _Tpn The floating-point type of the argument @c __nu.
+ * @tparam _Tpp The floating-point type of the angle @c __phi.
+ * @param __k The modulus, <tt> abs(__k) <= 1 </tt>
+ * @param __nu The second argument
+ * @param __phi The integral limit argument in radians
+ * @return The elliptic function of the third kind.
+ * @throw std::domain_error if <tt> abs(__k) > 1 </tt>.
+ */
template<typename _Tp, typename _Tpn, typename _Tpp>
inline typename __gnu_cxx::__promote_3<_Tp, _Tpn, _Tpp>::__type
ellint_3(_Tp __k, _Tpn __nu, _Tpp __phi)
@@ -311,14 +839,36 @@
// Exponential integrals
+ /**
+ * Return the exponential integral @f$ Ei(x) @f$ for @c float argument @c x.
+ *
+ * @see expint for details.
+ */
inline float
expintf(float __x)
{ return __detail::__expint<float>(__x); }
+ /**
+ * Return the exponential integral @f$ Ei(x) @f$
+ * for <tt>long double</tt> argument @c x.
+ *
+ * @see expint for details.
+ */
inline long double
expintl(long double __x)
{ return __detail::__expint<long double>(__x); }
+ /**
+ * Return the exponential integral @f$ Ei(x) @f$ for @c real argument @c x.
+ *
+ * The exponential integral is given by
+ * \f[
+ * Ei(x) = -\int_{-x}^\infty \frac{e^t}{t} dt
+ * \f]
+ *
+ * @tparam _Tp The floating-point type of the argument @c __x.
+ * @param __x The argument of the exponential integral function.
+ */
template<typename _Tp>
inline typename __gnu_cxx::__promote<_Tp>::__type
expint(_Tp __x)
@@ -329,14 +879,44 @@
// Hermite polynomials
+ /**
+ * Return the Hermite polynomial @f$ H_n(x) @f$ of nonnegative order n
+ * and float argument @c x.
+ *
+ * @see hermite for details.
+ */
inline float
hermitef(unsigned int __n, float __x)
{ return __detail::__poly_hermite<float>(__n, __x); }
+ /**
+ * Return the Hermite polynomial @f$ H_n(x) @f$ of nonnegative order n
+ * and <tt>long double</tt> argument @c x.
+ *
+ * @see hermite for details.
+ */
inline long double
hermitel(unsigned int __n, long double __x)
{ return __detail::__poly_hermite<long double>(__n, __x); }
+ /**
+ * Return the Hermite polynomial @f$ H_n(x) @f$ of order n
+ * and @c real argument @c x.
+ *
+ * The Hermite polynomial is defined by:
+ * @f[
+ * H_n(x) = (-1)^n e^{x^2} \frac{d^n}{dx^n} e^{-x^2}
+ * @f]
+ *
+ * The Hermite polynomial obeys a reflection formula:
+ * @f[
+ * H_n(-x) = (-1)^n H_n(x)
+ * @f]
+ *
+ * @tparam _Tp The floating-point type of the argument @c __x.
+ * @param __n The order
+ * @param __x The argument
+ */
template<typename _Tp>
inline typename __gnu_cxx::__promote<_Tp>::__type
hermite(unsigned int __n, _Tp __x)
@@ -347,14 +927,40 @@
// Laguerre polynomials
+ /**
+ * Returns the Laguerre polynomial @f$ L_n(x) @f$ of nonnegative degree @c n
+ * and @c float argument @f$ x >= 0 @f$.
+ *
+ * @see laguerre for more details.
+ */
inline float
laguerref(unsigned int __n, float __x)
{ return __detail::__laguerre<float>(__n, __x); }
+ /**
+ * Returns the Laguerre polynomial @f$ L_n(x) @f$ of nonnegative degree @c n
+ * and <tt>long double</tt> argument @f$ x >= 0 @f$.
+ *
+ * @see laguerre for more details.
+ */
inline long double
laguerrel(unsigned int __n, long double __x)
{ return __detail::__laguerre<long double>(__n, __x); }
+ /**
+ * Returns the Laguerre polynomial @f$ L_n(x) @f$
+ * of nonnegative degree @c n and real argument @f$ x >= 0 @f$.
+ *
+ * The Laguerre polynomial is defined by:
+ * @f[
+ * L_n(x) = \frac{e^x}{n!} \frac{d^n}{dx^n} (x^ne^{-x})
+ * @f]
+ *
+ * @tparam _Tp The floating-point type of the argument @c __x.
+ * @param __n The nonnegative order
+ * @param __x The argument <tt> __x >= 0 </tt>
+ * @throw std::domain_error if <tt> __x < 0 </tt>.
+ */
template<typename _Tp>
inline typename __gnu_cxx::__promote<_Tp>::__type
laguerre(unsigned int __n, _Tp __x)
@@ -365,32 +971,92 @@
// Legendre polynomials
+ /**
+ * Return the Legendre polynomial @f$ P_l(x) @f$ of nonnegative
+ * degree @f$ l @f$ and @c float argument @f$ |x| <= 0 @f$.
+ *
+ * @see legendre for more details.
+ */
inline float
- legendref(unsigned int __n, float __x)
- { return __detail::__poly_legendre_p<float>(__n, __x); }
+ legendref(unsigned int __l, float __x)
+ { return __detail::__poly_legendre_p<float>(__l, __x); }
+ /**
+ * Return the Legendre polynomial @f$ P_l(x) @f$ of nonnegative
+ * degree @f$ l @f$ and <tt>long double</tt> argument @f$ |x| <= 0 @f$.
+ *
+ * @see legendre for more details.
+ */
inline long double
- legendrel(unsigned int __n, long double __x)
- { return __detail::__poly_legendre_p<long double>(__n, __x); }
+ legendrel(unsigned int __l, long double __x)
+ { return __detail::__poly_legendre_p<long double>(__l, __x); }
+ /**
+ * Return the Legendre polynomial @f$ P_l(x) @f$ of nonnegative
+ * degree @f$ l @f$ and real argument @f$ |x| <= 0 @f$.
+ *
+ * The Legendre function of order @f$ l @f$ and argument @f$ x @f$,
+ * @f$ P_l(x) @f$, is defined by:
+ * @f[
+ * P_l(x) = \frac{1}{2^l l!}\frac{d^l}{dx^l}(x^2 - 1)^{l}
+ * @f]
+ *
+ * @tparam _Tp The floating-point type of the argument @c __x.
+ * @param __l The degree @f$ l >= 0 @f$
+ * @param __x The argument @c abs(__x) <= 1
+ * @throw std::domain_error if @c abs(__x) > 1
+ */
template<typename _Tp>
inline typename __gnu_cxx::__promote<_Tp>::__type
- legendre(unsigned int __n, _Tp __x)
+ legendre(unsigned int __l, _Tp __x)
{
typedef typename __gnu_cxx::__promote<_Tp>::__type __type;
- return __detail::__poly_legendre_p<__type>(__n, __x);
+ return __detail::__poly_legendre_p<__type>(__l, __x);
}
// Riemann zeta functions
+ /**
+ * Return the Riemann zeta function @f$ \zeta(s) @f$
+ * for @c float argument @f$ s @f$.
+ *
+ * @see riemann_zeta for more details.
+ */
inline float
riemann_zetaf(float __s)
{ return __detail::__riemann_zeta<float>(__s); }
+ /**
+ * Return the Riemann zeta function @f$ \zeta(s) @f$
+ * for <tt>long double</tt> argument @f$ s @f$.
+ *
+ * @see riemann_zeta for more details.
+ */
inline long double
riemann_zetal(long double __s)
{ return __detail::__riemann_zeta<long double>(__s); }
+ /**
+ * Return the Riemann zeta function @f$ \zeta(s) @f$
+ * for real argument @f$ s @f$.
+ *
+ * The Riemann zeta function is defined by:
+ * @f[
+ * \zeta(s) = \sum_{k=1}^{\infty} k^{-s} \hbox{ for } s > 1
+ * @f]
+ * and
+ * @f[
+ * \zeta(s) = \frac{1}{1-2^{1-s}}\sum_{k=1}^{\infty}(-1)^{k-1}k^{-s}
+ * \hbox{ for } 0 <= s <= 1
+ * @f]
+ * For s < 1 use the reflection formula:
+ * @f[
+ * \zeta(s) = 2^s \pi^{s-1} \sin(\frac{\pi s}{2}) \Gamma(1-s) \zeta(1-s)
+ * @f]
+ *
+ * @tparam _Tp The floating-point type of the argument @c __s.
+ * @param __s The argument <tt> s != 1 </tt>
+ */
template<typename _Tp>
inline typename __gnu_cxx::__promote<_Tp>::__type
riemann_zeta(_Tp __s)
@@ -401,14 +1067,40 @@
// Spherical Bessel functions
+ /**
+ * Return the spherical Bessel function @f$ j_n(x) @f$ of nonnegative order n
+ * and @c float argument @f$ x >= 0 @f$.
+ *
+ * @see sph_bessel for more details.
+ */
inline float
sph_besself(unsigned int __n, float __x)
{ return __detail::__sph_bessel<float>(__n, __x); }
+ /**
+ * Return the spherical Bessel function @f$ j_n(x) @f$ of nonnegative order n
+ * and <tt>long double</tt> argument @f$ x >= 0 @f$.
+ *
+ * @see sph_bessel for more details.
+ */
inline long double
sph_bessell(unsigned int __n, long double __x)
{ return __detail::__sph_bessel<long double>(__n, __x); }
+ /**
+ * Return the spherical Bessel function @f$ j_n(x) @f$ of nonnegative order n
+ * and real argument @f$ x >= 0 @f$.
+ *
+ * The spherical Bessel function is defined by:
+ * @f[
+ * j_n(x) = \left(\frac{\pi}{2x} \right) ^{1/2} J_{n+1/2}(x)
+ * @f]
+ *
+ * @tparam _Tp The floating-point type of the argument @c __x.
+ * @param __n The integral order <tt> n >= 0 </tt>
+ * @param __x The real argument <tt> x >= 0 </tt>
+ * @throw std::domain_error if <tt> __x < 0 </tt>.
+ */
template<typename _Tp>
inline typename __gnu_cxx::__promote<_Tp>::__type
sph_bessel(unsigned int __n, _Tp __x)
@@ -419,14 +1111,43 @@
// Spherical associated Legendre functions
+ /**
+ * Return the spherical Legendre function of nonnegative integral
+ * degree @c l and order @c m and float angle @f$ \theta @f$ in radians.
+ *
+ * @see sph_legendre for details.
+ */
inline float
sph_legendref(unsigned int __l, unsigned int __m, float __theta)
{ return __detail::__sph_legendre<float>(__l, __m, __theta); }
+ /**
+ * Return the spherical Legendre function of nonnegative integral
+ * degree @c l and order @c m and <tt>long double</tt> angle @f$ \theta @f$
+ * in radians.
+ *
+ * @see sph_legendre for details.
+ */
inline long double
sph_legendrel(unsigned int __l, unsigned int __m, long double __theta)
{ return __detail::__sph_legendre<long double>(__l, __m, __theta); }
+ /**
+ * Return the spherical Legendre function of nonnegative integral
+ * degree @c l and order @c m and real angle @f$ \theta @f$ in radians.
+ *
+ * The spherical Legendre function is defined by
+ * @f[
+ * Y_l^m(\theta,\phi) = (-1)^m[\frac{(2l+1)}{4\pi}
+ * \frac{(l-m)!}{(l+m)!}]
+ * P_l^m(\cos\theta) \exp^{im\phi}
+ * @f]
+ *
+ * @tparam _Tp The floating-point type of the angle @c __theta.
+ * @param __l The order <tt> __l >= 0 </tt>
+ * @param __m The degree <tt> __m >= 0 </tt> and <tt> __m <= __l </tt>
+ * @param __theta The radian polar angle argument
+ */
template<typename _Tp>
inline typename __gnu_cxx::__promote<_Tp>::__type
sph_legendre(unsigned int __l, unsigned int __m, _Tp __theta)
@@ -437,14 +1158,40 @@
// Spherical Neumann functions
+ /**
+ * Return the spherical Neumann function of integral order @f$ n >= 0 @f$
+ * and @c float argument @f$ x >= 0 @f$.
+ *
+ * @see sph_neumann for details.
+ */
inline float
sph_neumannf(unsigned int __n, float __x)
{ return __detail::__sph_neumann<float>(__n, __x); }
+ /**
+ * Return the spherical Neumann function of integral order @f$ n >= 0 @f$
+ * and <tt>long double</tt> @f$ x >= 0 @f$.
+ *
+ * @see sph_neumann for details.
+ */
inline long double
sph_neumannl(unsigned int __n, long double __x)
{ return __detail::__sph_neumann<long double>(__n, __x); }
+ /**
+ * Return the spherical Neumann function of integral order @f$ n >= 0 @f$
+ * and real argument @f$ x >= 0 @f$.
+ *
+ * The spherical Neumann function is defined by
+ * @f[
+ * n_n(x) = \left(\frac{\pi}{2x} \right) ^{1/2} N_{n+1/2}(x)
+ * @f]
+ *
+ * @tparam _Tp The floating-point type of the argument @c __x.
+ * @param __n The integral order <tt> n >= 0 @f$ </tt>
+ * @param __x The real argument <tt> __x >= 0 </tt>
+ * @throw std::domain_error if <tt> __x < 0 </tt>.
+ */
template<typename _Tp>
inline typename __gnu_cxx::__promote<_Tp>::__type
sph_neumann(unsigned int __n, _Tp __x)
@@ -463,14 +1210,44 @@
// Confluent hypergeometric functions
+ /**
+ * Return the confluent hypergeometric function of @c float
+ * numeratorial parameter @c a, denominatorial parameter @c c,
+ * and argument @c x.
+ *
+ * @see conf_hyperg for details.
+ */
inline float
conf_hypergf(float __a, float __c, float __x)
{ return std::__detail::__conf_hyperg<float>(__a, __c, __x); }
+ /**
+ * Return the confluent hypergeometric function of @c long double
+ * numeratorial parameter @c a, denominatorial parameter @c c,
+ * and argument @c x.
+ *
+ * @see conf_hyperg for details.
+ */
inline long double
conf_hypergl(long double __a, long double __c, long double __x)
{ return std::__detail::__conf_hyperg<long double>(__a, __c, __x); }
+ /**
+ * Return the confluent hypergeometric function @f$ {}_1F_1(a;c;x) @f$
+ * of real numeratorial parameter @c a, denominatorial parameter @c c,
+ * and argument @c x.
+ *
+ * The confluent hypergeometric function is defined by
+ * @f[
+ * {}_1F_1(a;c;x) = \sum_{n=0}^{\infty} \frac{(a)_n x^n}{(c)_n n!}
+ * @f]
+ * where the Pochhammer symbol is @f$ (x)_k = (x)(x+1)...(x+k-1) @f$,
+ * @f$ (x)_0 = 1 @f$
+ *
+ * @param __a The numeratorial parameter
+ * @param __c The denominatorial parameter
+ * @param __x The argument
+ */
template<typename _Tpa, typename _Tpc, typename _Tp>
inline typename __gnu_cxx::__promote_3<_Tpa, _Tpc, _Tp>::__type
conf_hyperg(_Tpa __a, _Tpc __c, _Tp __x)
@@ -481,14 +1258,45 @@
// Hypergeometric functions
+ /**
+ * Return the hypergeometric function @f$ {}_2F_1(a,b;c;x) @f$
+ * of @ float numeratorial parameters @c a and @c b,
+ * denominatorial parameter @c c, and argument @c x.
+ *
+ * @see hyperg for details.
+ */
inline float
hypergf(float __a, float __b, float __c, float __x)
{ return std::__detail::__hyperg<float>(__a, __b, __c, __x); }
+ /**
+ * Return the hypergeometric function @f$ {}_2F_1(a,b;c;x) @f$
+ * of @ long double numeratorial parameters @c a and @c b,
+ * denominatorial parameter @c c, and argument @c x.
+ *
+ * @see hyperg for details.
+ */
inline long double
hypergl(long double __a, long double __b, long double __c, long double __x)
{ return std::__detail::__hyperg<long double>(__a, __b, __c, __x); }
+ /**
+ * Return the hypergeometric function @f$ {}_2F_1(a,b;c;x) @f$
+ * of real numeratorial parameters @c a and @c b,
+ * denominatorial parameter @c c, and argument @c x.
+ *
+ * The hypergeometric function is defined by
+ * @f[
+ * {}_2F_1(a;c;x) = \sum_{n=0}^{\infty} \frac{(a)_n (b)_n x^n}{(c)_n n!}
+ * @f]
+ * where the Pochhammer symbol is @f$ (x)_k = (x)(x+1)...(x+k-1) @f$,
+ * @f$ (x)_0 = 1 @f$
+ *
+ * @param __a The first numeratorial parameter
+ * @param __b The second numeratorial parameter
+ * @param __c The denominatorial parameter
+ * @param __x The argument
+ */
template<typename _Tpa, typename _Tpb, typename _Tpc, typename _Tp>
inline typename __gnu_cxx::__promote_4<_Tpa, _Tpb, _Tpc, _Tp>::__type
hyperg(_Tpa __a, _Tpb __b, _Tpc __c, _Tp __x)