Collaboration diagram for Mathematical Special Functions:

Functions
template<typename _Tp >
__gnu_cxx::__promote< _Tp >::__type	__gnu_cxx::airy_ai (_Tp __x)

float	__gnu_cxx::airy_aif (float __x)

long double	__gnu_cxx::airy_ail (long double __x)

template<typename _Tp >
__gnu_cxx::__promote< _Tp >::__type	__gnu_cxx::airy_bi (_Tp __x)

float	__gnu_cxx::airy_bif (float __x)

long double	__gnu_cxx::airy_bil (long double __x)

template<typename _Tp >
__gnu_cxx::__promote< _Tp >::__type	std::assoc_laguerre (unsigned int __n, unsigned int __m, _Tp __x)

float	std::assoc_laguerref (unsigned int __n, unsigned int __m, float __x)

long double	std::assoc_laguerrel (unsigned int __n, unsigned int __m, long double __x)

template<typename _Tp >
__gnu_cxx::__promote< _Tp >::__type	std::assoc_legendre (unsigned int __l, unsigned int __m, _Tp __x)

float	std::assoc_legendref (unsigned int __l, unsigned int __m, float __x)

long double	std::assoc_legendrel (unsigned int __l, unsigned int __m, long double __x)

template<typename _Tpa , typename _Tpb >
__gnu_cxx::__promote_2< _Tpa, _Tpb >::__type	std::beta (_Tpa __a, _Tpb __b)

float	std::betaf (float __a, float __b)

long double	std::betal (long double __a, long double __b)

template<typename _Tp >
__gnu_cxx::__promote< _Tp >::__type	std::comp_ellint_1 (_Tp __k)

float	std::comp_ellint_1f (float __k)

long double	std::comp_ellint_1l (long double __k)

template<typename _Tp >
__gnu_cxx::__promote< _Tp >::__type	std::comp_ellint_2 (_Tp __k)

float	std::comp_ellint_2f (float __k)

long double	std::comp_ellint_2l (long double __k)

template<typename _Tp , typename _Tpn >
__gnu_cxx::__promote_2< _Tp, _Tpn >::__type	std::comp_ellint_3 (_Tp __k, _Tpn __nu)

float	std::comp_ellint_3f (float __k, float __nu)

long double	std::comp_ellint_3l (long double __k, long double __nu)

template<typename _Tpa , typename _Tpc , typename _Tp >
__gnu_cxx::__promote_3< _Tpa, _Tpc, _Tp >::__type	__gnu_cxx::conf_hyperg (_Tpa __a, _Tpc __c, _Tp __x)

float	__gnu_cxx::conf_hypergf (float __a, float __c, float __x)

long double	__gnu_cxx::conf_hypergl (long double __a, long double __c, long double __x)

template<typename _Tpnu , typename _Tp >
__gnu_cxx::__promote_2< _Tpnu, _Tp >::__type	std::cyl_bessel_i (_Tpnu __nu, _Tp __x)

float	std::cyl_bessel_if (float __nu, float __x)

long double	std::cyl_bessel_il (long double __nu, long double __x)

template<typename _Tpnu , typename _Tp >
__gnu_cxx::__promote_2< _Tpnu, _Tp >::__type	std::cyl_bessel_j (_Tpnu __nu, _Tp __x)

float	std::cyl_bessel_jf (float __nu, float __x)

long double	std::cyl_bessel_jl (long double __nu, long double __x)

template<typename _Tpnu , typename _Tp >
__gnu_cxx::__promote_2< _Tpnu, _Tp >::__type	std::cyl_bessel_k (_Tpnu __nu, _Tp __x)

float	std::cyl_bessel_kf (float __nu, float __x)

long double	std::cyl_bessel_kl (long double __nu, long double __x)

template<typename _Tpnu , typename _Tp >
__gnu_cxx::__promote_2< _Tpnu, _Tp >::__type	std::cyl_neumann (_Tpnu __nu, _Tp __x)

float	std::cyl_neumannf (float __nu, float __x)

long double	std::cyl_neumannl (long double __nu, long double __x)

template<typename _Tp , typename _Tpp >
__gnu_cxx::__promote_2< _Tp, _Tpp >::__type	std::ellint_1 (_Tp __k, _Tpp __phi)

float	std::ellint_1f (float __k, float __phi)

long double	std::ellint_1l (long double __k, long double __phi)

template<typename _Tp , typename _Tpp >
__gnu_cxx::__promote_2< _Tp, _Tpp >::__type	std::ellint_2 (_Tp __k, _Tpp __phi)

float	std::ellint_2f (float __k, float __phi)

long double	std::ellint_2l (long double __k, long double __phi)

template<typename _Tp , typename _Tpn , typename _Tpp >
__gnu_cxx::__promote_3< _Tp, _Tpn, _Tpp >::__type	std::ellint_3 (_Tp __k, _Tpn __nu, _Tpp __phi)

float	std::ellint_3f (float __k, float __nu, float __phi)

long double	std::ellint_3l (long double __k, long double __nu, long double __phi)

template<typename _Tp >
__gnu_cxx::__promote< _Tp >::__type	std::expint (_Tp __x)

float	std::expintf (float __x)

long double	std::expintl (long double __x)

template<typename _Tp >
__gnu_cxx::__promote< _Tp >::__type	std::hermite (unsigned int __n, _Tp __x)

float	std::hermitef (unsigned int __n, float __x)

long double	std::hermitel (unsigned int __n, long double __x)

template<typename _Tpa , typename _Tpb , typename _Tpc , typename _Tp >
__gnu_cxx::__promote_4< _Tpa, _Tpb, _Tpc, _Tp >::__type	__gnu_cxx::hyperg (_Tpa __a, _Tpb __b, _Tpc __c, _Tp __x)

float	__gnu_cxx::hypergf (float __a, float __b, float __c, float __x)

long double	__gnu_cxx::hypergl (long double __a, long double __b, long double __c, long double __x)

template<typename _Tp >
__gnu_cxx::__promote< _Tp >::__type	std::laguerre (unsigned int __n, _Tp __x)

float	std::laguerref (unsigned int __n, float __x)

long double	std::laguerrel (unsigned int __n, long double __x)

template<typename _Tp >
__gnu_cxx::__promote< _Tp >::__type	std::legendre (unsigned int __l, _Tp __x)

float	std::legendref (unsigned int __l, float __x)

long double	std::legendrel (unsigned int __l, long double __x)

template<typename _Tp >
__gnu_cxx::__promote< _Tp >::__type	std::riemann_zeta (_Tp __s)

float	std::riemann_zetaf (float __s)

long double	std::riemann_zetal (long double __s)

template<typename _Tp >
__gnu_cxx::__promote< _Tp >::__type	std::sph_bessel (unsigned int __n, _Tp __x)

float	std::sph_besself (unsigned int __n, float __x)

long double	std::sph_bessell (unsigned int __n, long double __x)

template<typename _Tp >
__gnu_cxx::__promote< _Tp >::__type	std::sph_legendre (unsigned int __l, unsigned int __m, _Tp __theta)

float	std::sph_legendref (unsigned int __l, unsigned int __m, float __theta)

long double	std::sph_legendrel (unsigned int __l, unsigned int __m, long double __theta)

template<typename _Tp >
__gnu_cxx::__promote< _Tp >::__type	std::sph_neumann (unsigned int __n, _Tp __x)

float	std::sph_neumannf (unsigned int __n, float __x)

long double	std::sph_neumannl (unsigned int __n, long double __x)

Detailed Description

Mathematical Special Functions

A collection of advanced mathematical special functions, defined by ISO/IEC IS 29124 and then added to ISO C++ 2017.

Introduction and History

The first significant library upgrade on the road to C++2011, TR1, included a set of 23 mathematical functions that significantly extended the standard transcendental functions inherited from C and declared in <cmath>.

Although most components from TR1 were eventually adopted for C++11 these math functions were left behind out of concern for implementability. The math functions were published as a separate international standard IS 29124 - Extensions to the C++ Library to Support Mathematical Special Functions.

For C++17 these functions were incorporated into the main standard.

assoc_laguerre - Associated Laguerre functions
assoc_legendre - Associated Legendre functions
beta - Beta functions
comp_ellint_1 - Complete elliptic functions of the first kind
comp_ellint_2 - Complete elliptic functions of the second kind
comp_ellint_3 - Complete elliptic functions of the third kind
cyl_bessel_i - Regular modified cylindrical Bessel functions
cyl_bessel_j - Cylindrical Bessel functions of the first kind
cyl_bessel_k - Irregular modified cylindrical Bessel functions
cyl_neumann - Cylindrical Neumann functions or Cylindrical Bessel functions of the second kind
ellint_1 - Incomplete elliptic functions of the first kind
ellint_2 - Incomplete elliptic functions of the second kind
ellint_3 - Incomplete elliptic functions of the third kind
expint - The exponential integral
hermite - Hermite polynomials
laguerre - Laguerre functions
legendre - Legendre polynomials
riemann_zeta - The Riemann zeta function
sph_bessel - Spherical Bessel functions
sph_legendre - Spherical Legendre functions
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 __gnu_cxx:

Argument Promotion

The arguments suppled to the non-suffixed functions will be promoted according to the following rules:

If any argument intended to be floating point is given an integral value That integral value is promoted to double.
All floating point arguments are promoted up to the largest floating point precision among them.

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.

Implementation

We strive to implement the underlying math with type generic algorithms to the greatest extent possible. In practice, the functions are thin wrappers that dispatch to function templates. Type dependence is controlled with std::numeric_limits and functions thereof.

We don't promote float to double or double to long double 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, long double should give you more dynamic range and slightly more pecision than double on many systems.

Testing

These functions have been tested against equivalent implementations from the Gnu Scientific Library, GSL and Boost and the ratio

$\frac{|f - f_{test}|}{|f_{test}|}$

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.

General Bibliography

See also: 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.; for example http://people.math.sfu.ca/~cbm/aands/; 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.; An Atlas of Functions: with Equator, the Atlas Function Calculator 2nd Edition, by Oldham, Keith B., Myland, Jan, Spanier, Jerome; Asymptotics and Special Functions by Frank W. J. Olver, Academic Press, 1974; 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; The Special Functions and Their Approximations: Volumes 1 and 2, by Yudell L. Luke, Academic Press, 1969

Function Documentation

◆ airy_ai()

template<typename _Tp >

__gnu_cxx::__promote<_Tp>::__type __gnu_cxx::airy_ai ( _Tp __x )

inline

Return the Airy function $ Ai(x) $ of real argument x.

Definition at line 1242 of file specfun.h.

◆ airy_aif()

float __gnu_cxx::airy_aif ( float __x )

inline

Return the Airy function $ Ai(x) $ of float argument x.

Definition at line 1219 of file specfun.h.

◆ airy_ail()

long double __gnu_cxx::airy_ail ( long double __x )

inline

Return the Airy function $ Ai(x) $ of long double argument x.

Definition at line 1230 of file specfun.h.

◆ airy_bi()

template<typename _Tp >

__gnu_cxx::__promote<_Tp>::__type __gnu_cxx::airy_bi ( _Tp __x )

inline

Return the Airy function $ Bi(x) $ of real argument x.

Definition at line 1277 of file specfun.h.

◆ airy_bif()

float __gnu_cxx::airy_bif ( float __x )

inline

Return the Airy function $ Bi(x) $ of float argument x.

Definition at line 1254 of file specfun.h.

◆ airy_bil()

long double __gnu_cxx::airy_bil ( long double __x )

inline

Return the Airy function $ Bi(x) $ of long double argument x.

Definition at line 1265 of file specfun.h.

◆ assoc_laguerre()

template<typename _Tp >

__gnu_cxx::__promote<_Tp>::__type std::assoc_laguerre	(	unsigned int	__n,
		unsigned int	__m,
		_Tp	__x
	)

inline

Return the associated Laguerre polynomial of nonnegative order n, nonnegative degree m and real argument x: $ L_n^m(x) $ .

The associated Laguerre function of real degree $\alpha$ , $L_n^\alpha(x)$ , is defined by

$L_n^\alpha(x) = \frac{(\alpha + 1)_n}{n!} {}_1F_1(-n; \alpha + 1; x)$

where $(\alpha)_n$ is the Pochhammer symbol and ${}_1F_1(a; c; x)$ is the confluent hypergeometric function.

The associated Laguerre polynomial is defined for integral degree $\alpha = m$ by:

$L_n^m(x) = (-1)^m \frac{d^m}{dx^m} L_{n + m}(x)$

where the Laguerre polynomial is defined by:

$L_n(x) = \frac{e^x}{n!} \frac{d^n}{dx^n} (x^ne^{-x})$

and $ x >= 0 $ .

See also: laguerre for details of the Laguerre function of degree n

Template Parameters

_Tp	The floating-point type of the argument `__x`.

Parameters

__n	The order of the Laguerre function, `__n >= 0`.
__m	The degree of the Laguerre function, `__m >= 0`.
__x	The argument of the Laguerre function, `__x >= 0`.

Exceptions

std::domain_error if __x < 0.

Definition at line 252 of file specfun.h.

◆ assoc_laguerref()

float std::assoc_laguerref	(	unsigned int	__n,
		unsigned int	__m,
		float	__x
	)

inline

Return the associated Laguerre polynomial of order n, degree m: $ L_n^m(x) $ for float argument.

See also: assoc_laguerre for more details.

Definition at line 206 of file specfun.h.

◆ assoc_laguerrel()

long double std::assoc_laguerrel	(	unsigned int	__n,
		unsigned int	__m,
		long double	__x
	)

inline

Return the associated Laguerre polynomial of order n, degree m: $ L_n^m(x) $ .

See also: assoc_laguerre for more details.

Definition at line 216 of file specfun.h.

◆ assoc_legendre()

template<typename _Tp >

__gnu_cxx::__promote<_Tp>::__type std::assoc_legendre	(	unsigned int	__l,
		unsigned int	__m,
		_Tp	__x
	)

inline

Return the associated Legendre function of degree l and order m.

The associated Legendre function is derived from the Legendre function $ P_l(x) $ by the Rodrigues formula:

$P_l^m(x) = (1 - x^2)^{m/2}\frac{d^m}{dx^m}P_l(x)$

See also: legendre for details of the Legendre function of degree l

Template Parameters

_Tp	The floating-point type of the argument `__x`.

Parameters

__l	The degree `__l >= 0`.
__m	The order `__m <= l`.
__x	The argument, `abs(__x) <= 1`.

Exceptions

std::domain_error if abs(__x) > 1.

Definition at line 298 of file specfun.h.

◆ assoc_legendref()

float std::assoc_legendref	(	unsigned int	__l,
		unsigned int	__m,
		float	__x
	)

inline

Return the associated Legendre function of degree l and order m for float argument.

See also: assoc_legendre for more details.

Definition at line 267 of file specfun.h.

◆ assoc_legendrel()

long double std::assoc_legendrel	(	unsigned int	__l,
		unsigned int	__m,
		long double	__x
	)

inline

Return the associated Legendre function of degree l and order m.

See also: assoc_legendre for more details.

Definition at line 276 of file specfun.h.

◆ beta()

template<typename _Tpa , typename _Tpb >

__gnu_cxx::__promote_2<_Tpa, _Tpb>::__type std::beta	(	_Tpa	__a,
		_Tpb	__b
	)

inline

Return the beta function, $B(a,b)$ , for real parameters a, b.

The beta function is defined by

$B(a,b) = \int_0^1 t^{a - 1} (1 - t)^{b - 1} dt = \frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)}$

where $ a > 0 $ and $ b > 0 $

Template Parameters

_Tpa	The floating-point type of the parameter `__a`.
_Tpb	The floating-point type of the parameter `__b`.

Parameters

__a	The first argument of the beta function, `__a > 0` .
__b	The second argument of the beta function, `__b > 0` .

Exceptions

std::domain_error if __a < 0 or __b < 0 .

Definition at line 343 of file specfun.h.

◆ betaf()

float std::betaf	(	float	__a,
		float	__b
	)

inline

Return the beta function, $ B(a,b) $ , for float parameters a, b.

See also: beta for more details.

Definition at line 312 of file specfun.h.

◆ betal()

long double std::betal	(	long double	__a,
		long double	__b
	)

inline

Return the beta function, $B(a,b)$ , for long double parameters a, b.

See also: beta for more details.

Definition at line 322 of file specfun.h.

◆ comp_ellint_1()

template<typename _Tp >

__gnu_cxx::__promote<_Tp>::__type std::comp_ellint_1 ( _Tp __k )

inline

Return the complete elliptic integral of the first kind $ K(k) $ for real modulus k.

The complete elliptic integral of the first kind is defined as

$K(k) = F(k,\pi/2) = \int_0^{\pi/2}\frac{d\theta} {\sqrt{1 - k^2 sin^2\theta}}$

where $F(k,\phi)$ is the incomplete elliptic integral of the first kind and the modulus $ |k| <= 1 $ .

See also: ellint_1 for details of the incomplete elliptic function of the first kind.

Template Parameters

_Tp	The floating-point type of the modulus `__k`.

Parameters

__k	The modulus, `abs(__k) <= 1`

Exceptions

std::domain_error if abs(__k) > 1 .

Definition at line 391 of file specfun.h.

◆ comp_ellint_1f()

float std::comp_ellint_1f ( float __k )

inline

Return the complete elliptic integral of the first kind $ E(k) $ for float modulus k.

See also: comp_ellint_1 for details.

Definition at line 358 of file specfun.h.

◆ comp_ellint_1l()

long double std::comp_ellint_1l ( long double __k )

inline

Return the complete elliptic integral of the first kind $ E(k) $ for long double modulus k.

See also: comp_ellint_1 for details.

Definition at line 368 of file specfun.h.

◆ comp_ellint_2()

template<typename _Tp >

__gnu_cxx::__promote<_Tp>::__type std::comp_ellint_2 ( _Tp __k )

inline

Return the complete elliptic integral of the second kind $ E(k) $ for real modulus k.

The complete elliptic integral of the second kind is defined as

$E(k) = E(k,\pi/2) = \int_0^{\pi/2}\sqrt{1 - k^2 sin^2\theta}$

where $E(k,\phi)$ is the incomplete elliptic integral of the second kind and the modulus $ |k| <= 1 $ .

See also: ellint_2 for details of the incomplete elliptic function of the second kind.

Template Parameters

_Tp	The floating-point type of the modulus `__k`.

Parameters

__k	The modulus, `abs(__k)` <= 1

Exceptions

std::domain_error if abs(__k) > 1.

Definition at line 438 of file specfun.h.

◆ comp_ellint_2f()

float std::comp_ellint_2f ( float __k )

inline

Return the complete elliptic integral of the second kind $ E(k) $ for float modulus k.

See also: comp_ellint_2 for details.

Definition at line 406 of file specfun.h.

◆ comp_ellint_2l()

long double std::comp_ellint_2l ( long double __k )

inline

Return the complete elliptic integral of the second kind $ E(k) $ for long double modulus k.

See also: comp_ellint_2 for details.

Definition at line 416 of file specfun.h.

◆ comp_ellint_3()

template<typename _Tp , typename _Tpn >

__gnu_cxx::__promote_2<_Tp, _Tpn>::__type std::comp_ellint_3	(	_Tp	__k,
		_Tpn	__nu
	)

inline

Return the complete elliptic integral of the third kind $\Pi(k,\nu) = \Pi(k,\nu,\pi/2)$ for real modulus k.

The complete elliptic integral of the third kind is defined as

$\Pi(k,\nu) = \Pi(k,\nu,\pi/2) = \int_0^{\pi/2} \frac{d\theta} {(1 - \nu \sin^2\theta)\sqrt{1 - k^2 \sin^2\theta}}$

where $\Pi(k,\nu,\phi)$ is the incomplete elliptic integral of the second kind and the modulus $ |k| <= 1 $ .

See also: ellint_3 for details of the incomplete elliptic function of the third kind.

Template Parameters

_Tp	The floating-point type of the modulus `__k`.
_Tpn	The floating-point type of the argument `__nu`.

Parameters

__k	The modulus, `abs(__k)` <= 1
__nu	The argument

Exceptions

std::domain_error if abs(__k) > 1.

Definition at line 489 of file specfun.h.

◆ comp_ellint_3f()

float std::comp_ellint_3f	(	float	__k,
		float	__nu
	)

inline

Return the complete elliptic integral of the third kind $\Pi(k,\nu)$ for float modulus k.

See also: comp_ellint_3 for details.

Definition at line 453 of file specfun.h.

◆ comp_ellint_3l()

long double std::comp_ellint_3l	(	long double	__k,
		long double	__nu
	)

inline

Return the complete elliptic integral of the third kind $\Pi(k,\nu)$ for long double modulus k.

See also: comp_ellint_3 for details.

Definition at line 463 of file specfun.h.

◆ conf_hyperg()

template<typename _Tpa , typename _Tpc , typename _Tp >

__gnu_cxx::__promote_3<_Tpa, _Tpc, _Tp>::__type __gnu_cxx::conf_hyperg	(	_Tpa	__a,
		_Tpc	__c,
		_Tp	__x
	)

inline

Return the confluent hypergeometric function ${}_1F_1(a;c;x)$ of real numeratorial parameter a, denominatorial parameter c, and argument x.

The confluent hypergeometric function is defined by

${}_1F_1(a;c;x) = \sum_{n=0}^{\infty} \frac{(a)_n x^n}{(c)_n n!}$

where the Pochhammer symbol is $ (x)_k = (x)(x+1)...(x+k-1) $ , $ (x)_0 = 1 $

Parameters

__a	The numeratorial parameter
__c	The denominatorial parameter
__x	The argument

Definition at line 1327 of file specfun.h.

◆ conf_hypergf()

float __gnu_cxx::conf_hypergf	(	float	__a,
		float	__c,
		float	__x
	)

inline

Return the confluent hypergeometric function ${}_1F_1(a;c;x)$ of float numeratorial parameter a, denominatorial parameter c, and argument x.

See also: conf_hyperg for details.

Definition at line 1295 of file specfun.h.

◆ conf_hypergl()

long double __gnu_cxx::conf_hypergl	(	long double	__a,
		long double	__c,
		long double	__x
	)

inline

Return the confluent hypergeometric function ${}_1F_1(a;c;x)$ of long double numeratorial parameter a, denominatorial parameter c, and argument x.

See also: conf_hyperg for details.

Definition at line 1306 of file specfun.h.

◆ cyl_bessel_i()

template<typename _Tpnu , typename _Tp >

__gnu_cxx::__promote_2<_Tpnu, _Tp>::__type std::cyl_bessel_i	(	_Tpnu	__nu,
		_Tp	__x
	)

inline

Return the regular modified Bessel function $I_{\nu}(x)$ for real order $\nu$ and argument $ x >= 0 $ .

The regular modified cylindrical Bessel function is:

$I_{\nu}(x) = i^{-\nu}J_\nu(ix) = \sum_{k=0}^{\infty} \frac{(x/2)^{\nu + 2k}}{k!\Gamma(\nu+k+1)}$

Template Parameters

_Tpnu	The floating-point type of the order `__nu`.
_Tp	The floating-point type of the argument `__x`.

Parameters

__nu	The order
__x	The argument, `__x >= 0`

Exceptions

std::domain_error if __x < 0 .

Definition at line 535 of file specfun.h.

◆ cyl_bessel_if()

float std::cyl_bessel_if	(	float	__nu,
		float	__x
	)

inline

Return the regular modified Bessel function $I_{\nu}(x)$ for float order $\nu$ and argument $ x >= 0 $ .

See also: cyl_bessel_i for setails.

Definition at line 504 of file specfun.h.

◆ cyl_bessel_il()

long double std::cyl_bessel_il	(	long double	__nu,
		long double	__x
	)

inline

Return the regular modified Bessel function $I_{\nu}(x)$ for long double order $\nu$ and argument $ x >= 0 $ .

See also: cyl_bessel_i for setails.

Definition at line 514 of file specfun.h.

◆ cyl_bessel_j()

template<typename _Tpnu , typename _Tp >

__gnu_cxx::__promote_2<_Tpnu, _Tp>::__type std::cyl_bessel_j	(	_Tpnu	__nu,
		_Tp	__x
	)

inline

Return the Bessel function $J_{\nu}(x)$ of real order $\nu$ and argument $ x >= 0 $ .

The cylindrical Bessel function is:

$J_{\nu}(x) = \sum_{k=0}^{\infty} \frac{(-1)^k (x/2)^{\nu + 2k}}{k!\Gamma(\nu+k+1)}$

Template Parameters

_Tpnu	The floating-point type of the order `__nu`.
_Tp	The floating-point type of the argument `__x`.

Parameters

__nu	The order
__x	The argument, `__x >= 0`

Exceptions

std::domain_error if __x < 0 .

Definition at line 581 of file specfun.h.

◆ cyl_bessel_jf()

float std::cyl_bessel_jf	(	float	__nu,
		float	__x
	)

inline

Return the Bessel function of the first kind $J_{\nu}(x)$ for float order $\nu$ and argument $ x >= 0 $ .

See also: cyl_bessel_j for setails.

Definition at line 550 of file specfun.h.

◆ cyl_bessel_jl()

long double std::cyl_bessel_jl	(	long double	__nu,
		long double	__x
	)

inline

Return the Bessel function of the first kind $J_{\nu}(x)$ for long double order $\nu$ and argument $ x >= 0 $ .

See also: cyl_bessel_j for setails.

Definition at line 560 of file specfun.h.

◆ cyl_bessel_k()

template<typename _Tpnu , typename _Tp >

__gnu_cxx::__promote_2<_Tpnu, _Tp>::__type std::cyl_bessel_k	(	_Tpnu	__nu,
		_Tp	__x
	)

inline

Return the irregular modified Bessel function $K_{\nu}(x)$ of real order $\nu$ and argument $ x $ .

The irregular modified Bessel function is defined by:

$K_{\nu}(x) = \frac{\pi}{2} \frac{I_{-\nu}(x) - I_{\nu}(x)}{\sin \nu\pi}$

where for integral $\nu = n$ a limit is taken: $lim_{\nu \to n}$ . For negative argument we have simply:

$K_{-\nu}(x) = K_{\nu}(x)$

Template Parameters

_Tpnu	The floating-point type of the order `__nu`.
_Tp	The floating-point type of the argument `__x`.

Parameters

__nu	The order
__x	The argument, `__x >= 0`

Exceptions

std::domain_error if __x < 0 .

Definition at line 633 of file specfun.h.

◆ cyl_bessel_kf()

float std::cyl_bessel_kf	(	float	__nu,
		float	__x
	)

inline

Return the irregular modified Bessel function $K_{\nu}(x)$ for float order $\nu$ and argument $ x >= 0 $ .

See also: cyl_bessel_k for setails.

Definition at line 596 of file specfun.h.

◆ cyl_bessel_kl()

long double std::cyl_bessel_kl	(	long double	__nu,
		long double	__x
	)

inline

Return the irregular modified Bessel function $K_{\nu}(x)$ for long double order $\nu$ and argument $ x >= 0 $ .

See also: cyl_bessel_k for setails.

Definition at line 606 of file specfun.h.

◆ cyl_neumann()

template<typename _Tpnu , typename _Tp >

__gnu_cxx::__promote_2<_Tpnu, _Tp>::__type std::cyl_neumann	(	_Tpnu	__nu,
		_Tp	__x
	)

inline

Return the Neumann function $N_{\nu}(x)$ of real order $\nu$ and argument $ x >= 0 $ .

The Neumann function is defined by:

$N_{\nu}(x) = \frac{J_{\nu}(x) \cos \nu\pi - J_{-\nu}(x)} {\sin \nu\pi}$

where $ x >= 0 $ and for integral order $\nu = n$ a limit is taken: $lim_{\nu \to n}$ .

Template Parameters

_Tpnu	The floating-point type of the order `__nu`.
_Tp	The floating-point type of the argument `__x`.

Parameters

__nu	The order
__x	The argument, `__x >= 0`

Exceptions

std::domain_error if __x < 0 .

Definition at line 681 of file specfun.h.

◆ cyl_neumannf()

float std::cyl_neumannf	(	float	__nu,
		float	__x
	)

inline

Return the Neumann function $N_{\nu}(x)$ of float order $\nu$ and argument $ x $ .

See also: cyl_neumann for setails.

Definition at line 648 of file specfun.h.

◆ cyl_neumannl()

long double std::cyl_neumannl	(	long double	__nu,
		long double	__x
	)

inline

Return the Neumann function $N_{\nu}(x)$ of long double order $\nu$ and argument $ x $ .

See also: cyl_neumann for setails.

Definition at line 658 of file specfun.h.

◆ ellint_1()

template<typename _Tp , typename _Tpp >

__gnu_cxx::__promote_2<_Tp, _Tpp>::__type std::ellint_1	(	_Tp	__k,
		_Tpp	__phi
	)

inline

Return the incomplete elliptic integral of the first kind $F(k,\phi)$ for real modulus $ k $ and angle $\phi$ .

The incomplete elliptic integral of the first kind is defined as

$F(k,\phi) = \int_0^{\phi}\frac{d\theta} {\sqrt{1 - k^2 sin^2\theta}}$

For $\phi= \pi/2$ this becomes the complete elliptic integral of the first kind, $ K(k) $ .

See also: comp_ellint_1.

Template Parameters

_Tp	The floating-point type of the modulus `__k`.
_Tpp	The floating-point type of the angle `__phi`.

Parameters

__k	The modulus, `abs(__k) <= 1`
__phi	The integral limit argument in radians

Exceptions

std::domain_error if abs(__k) > 1 .

Definition at line 729 of file specfun.h.

◆ ellint_1f()

float std::ellint_1f	(	float	__k,
		float	__phi
	)

inline

Return the incomplete elliptic integral of the first kind $E(k,\phi)$ for float modulus $ k $ and angle $\phi$ .

See also: ellint_1 for details.

Definition at line 696 of file specfun.h.

◆ ellint_1l()

long double std::ellint_1l	(	long double	__k,
		long double	__phi
	)

inline

Return the incomplete elliptic integral of the first kind $E(k,\phi)$ for long double modulus $ k $ and angle $\phi$ .

See also: ellint_1 for details.

Definition at line 706 of file specfun.h.

◆ ellint_2()

template<typename _Tp , typename _Tpp >

__gnu_cxx::__promote_2<_Tp, _Tpp>::__type std::ellint_2	(	_Tp	__k,
		_Tpp	__phi
	)

inline

Return the incomplete elliptic integral of the second kind $E(k,\phi)$ .

The incomplete elliptic integral of the second kind is defined as

$E(k,\phi) = \int_0^{\phi} \sqrt{1 - k^2 sin^2\theta}$

For $\phi= \pi/2$ this becomes the complete elliptic integral of the second kind, $ E(k) $ .

See also: comp_ellint_2.

Template Parameters

_Tp	The floating-point type of the modulus `__k`.
_Tpp	The floating-point type of the angle `__phi`.

Parameters

__k	The modulus, `abs(__k) <= 1`
__phi	The integral limit argument in radians

Returns: The elliptic function of the second kind.

Exceptions

std::domain_error if abs(__k) > 1 .

Definition at line 777 of file specfun.h.

◆ ellint_2f()

float std::ellint_2f	(	float	__k,
		float	__phi
	)

inline

Return the incomplete elliptic integral of the second kind $E(k,\phi)$ for float argument.

See also: ellint_2 for details.

Definition at line 744 of file specfun.h.

◆ ellint_2l()

long double std::ellint_2l	(	long double	__k,
		long double	__phi
	)

inline

Return the incomplete elliptic integral of the second kind $E(k,\phi)$ .

See also: ellint_2 for details.

Definition at line 754 of file specfun.h.

◆ ellint_3()

template<typename _Tp , typename _Tpn , typename _Tpp >

__gnu_cxx::__promote_3<_Tp, _Tpn, _Tpp>::__type std::ellint_3	(	_Tp	__k,
		_Tpn	__nu,
		_Tpp	__phi
	)

inline

Return the incomplete elliptic integral of the third kind $\Pi(k,\nu,\phi)$ .

The incomplete elliptic integral of the third kind is defined by:

$\Pi(k,\nu,\phi) = \int_0^{\phi} \frac{d\theta} {(1 - \nu \sin^2\theta) \sqrt{1 - k^2 \sin^2\theta}}$

For $\phi= \pi/2$ this becomes the complete elliptic integral of the third kind, $\Pi(k,\nu)$ .

See also: comp_ellint_3.

Template Parameters

_Tp	The floating-point type of the modulus `__k`.
_Tpn	The floating-point type of the argument `__nu`.
_Tpp	The floating-point type of the angle `__phi`.

Parameters

__k	The modulus, `abs(__k) <= 1`
__nu	The second argument
__phi	The integral limit argument in radians

Returns: The elliptic function of the third kind.

Exceptions

std::domain_error if abs(__k) > 1 .

Definition at line 830 of file specfun.h.

◆ ellint_3f()

float std::ellint_3f	(	float	__k,
		float	__nu,
		float	__phi
	)

inline

Return the incomplete elliptic integral of the third kind $\Pi(k,\nu,\phi)$ for float argument.

See also: ellint_3 for details.

Definition at line 792 of file specfun.h.

◆ ellint_3l()

long double std::ellint_3l	(	long double	__k,
		long double	__nu,
		long double	__phi
	)

inline

Return the incomplete elliptic integral of the third kind $\Pi(k,\nu,\phi)$ .

See also: ellint_3 for details.

Definition at line 802 of file specfun.h.

◆ expint()

template<typename _Tp >

__gnu_cxx::__promote<_Tp>::__type std::expint ( _Tp __x )

inline

Return the exponential integral $ Ei(x) $ for real argument x.

The exponential integral is given by

$Ei(x) = -\int_{-x}^\infty \frac{e^t}{t} dt$

Template Parameters

_Tp	The floating-point type of the argument `__x`.

Parameters

__x	The argument of the exponential integral function.

Definition at line 870 of file specfun.h.

◆ expintf()

float std::expintf ( float __x )

inline

Return the exponential integral $ Ei(x) $ for float argument x.

See also: expint for details.

Definition at line 844 of file specfun.h.

◆ expintl()

long double std::expintl ( long double __x )

inline

Return the exponential integral $ Ei(x) $ for long double argument x.

See also: expint for details.

Definition at line 854 of file specfun.h.

◆ hermite()

template<typename _Tp >

__gnu_cxx::__promote<_Tp>::__type std::hermite	(	unsigned int	__n,
		_Tp	__x
	)

inline

Return the Hermite polynomial $ H_n(x) $ of order n and real argument x.

The Hermite polynomial is defined by:

$H_n(x) = (-1)^n e^{x^2} \frac{d^n}{dx^n} e^{-x^2}$

The Hermite polynomial obeys a reflection formula:

$H_n(-x) = (-1)^n H_n(x)$

Template Parameters

_Tp	The floating-point type of the argument `__x`.

Parameters

__n	The order
__x	The argument

Definition at line 918 of file specfun.h.

◆ hermitef()

float std::hermitef	(	unsigned int	__n,
		float	__x
	)

inline

Return the Hermite polynomial $ H_n(x) $ of nonnegative order n and float argument x.

See also: hermite for details.

Definition at line 885 of file specfun.h.

◆ hermitel()

long double std::hermitel	(	unsigned int	__n,
		long double	__x
	)

inline

Return the Hermite polynomial $ H_n(x) $ of nonnegative order n and long double argument x.

See also: hermite for details.

Definition at line 895 of file specfun.h.

◆ hyperg()

template<typename _Tpa , typename _Tpb , typename _Tpc , typename _Tp >

__gnu_cxx::__promote_4<_Tpa, _Tpb, _Tpc, _Tp>::__type __gnu_cxx::hyperg	(	_Tpa	__a,
		_Tpb	__b,
		_Tpc	__c,
		_Tp	__x
	)

inline

Return the hypergeometric function ${}_2F_1(a,b;c;x)$ of real numeratorial parameters a and b, denominatorial parameter c, and argument x.

The hypergeometric function is defined by

${}_2F_1(a;c;x) = \sum_{n=0}^{\infty} \frac{(a)_n (b)_n x^n}{(c)_n n!}$

where the Pochhammer symbol is $ (x)_k = (x)(x+1)...(x+k-1) $ , $ (x)_0 = 1 $

Parameters

__a	The first numeratorial parameter
__b	The second numeratorial parameter
__c	The denominatorial parameter
__x	The argument

Definition at line 1376 of file specfun.h.

◆ hypergf()

float __gnu_cxx::hypergf	(	float	__a,
		float	__b,
		float	__c,
		float	__x
	)

inline

Return the hypergeometric function ${}_2F_1(a,b;c;x)$ of @ float numeratorial parameters a and b, denominatorial parameter c, and argument x.

See also: hyperg for details.

Definition at line 1343 of file specfun.h.

◆ hypergl()

long double __gnu_cxx::hypergl	(	long double	__a,
		long double	__b,
		long double	__c,
		long double	__x
	)

inline

Return the hypergeometric function ${}_2F_1(a,b;c;x)$ of long double numeratorial parameters a and b, denominatorial parameter c, and argument x.

See also: hyperg for details.

Definition at line 1354 of file specfun.h.

◆ laguerre()

template<typename _Tp >

__gnu_cxx::__promote<_Tp>::__type std::laguerre	(	unsigned int	__n,
		_Tp	__x
	)

inline

Returns the Laguerre polynomial $ L_n(x) $ of nonnegative degree n and real argument $ x >= 0 $ .

The Laguerre polynomial is defined by:

$L_n(x) = \frac{e^x}{n!} \frac{d^n}{dx^n} (x^ne^{-x})$

Template Parameters

_Tp	The floating-point type of the argument `__x`.

Parameters

__n	The nonnegative order
__x	The argument `__x >= 0`

Exceptions

std::domain_error if __x < 0 .

Definition at line 962 of file specfun.h.

◆ laguerref()

float std::laguerref	(	unsigned int	__n,
		float	__x
	)

inline

Returns the Laguerre polynomial $ L_n(x) $ of nonnegative degree n and float argument $ x >= 0 $ .

See also: laguerre for more details.

Definition at line 933 of file specfun.h.

◆ laguerrel()

long double std::laguerrel	(	unsigned int	__n,
		long double	__x
	)

inline

Returns the Laguerre polynomial $ L_n(x) $ of nonnegative degree n and long double argument $ x >= 0 $ .

See also: laguerre for more details.

Definition at line 943 of file specfun.h.

◆ legendre()

template<typename _Tp >

__gnu_cxx::__promote<_Tp>::__type std::legendre	(	unsigned int	__l,
		_Tp	__x
	)

inline

Return the Legendre polynomial $ P_l(x) $ of nonnegative degree $ l $ and real argument $ |x| <= 0 $ .

The Legendre function of order $ l $ and argument $ x $ , $ P_l(x) $ , is defined by:

$P_l(x) = \frac{1}{2^l l!}\frac{d^l}{dx^l}(x^2 - 1)^{l}$

Template Parameters

_Tp	The floating-point type of the argument `__x`.

Parameters

__l	The degree
__x	The argument `abs(__x)` <= 1

Exceptions

std::domain_error if abs(__x) > 1

Definition at line 1007 of file specfun.h.

◆ legendref()

float std::legendref	(	unsigned int	__l,
		float	__x
	)

inline

Return the Legendre polynomial $ P_l(x) $ of nonnegative degree $ l $ and float argument $ |x| <= 0 $ .

See also: legendre for more details.

Definition at line 977 of file specfun.h.

◆ legendrel()

long double std::legendrel	(	unsigned int	__l,
		long double	__x
	)

inline

Return the Legendre polynomial $ P_l(x) $ of nonnegative degree $ l $ and long double argument $ |x| <= 0 $ .

See also: legendre for more details.

Definition at line 987 of file specfun.h.

◆ riemann_zeta()

template<typename _Tp >

__gnu_cxx::__promote<_Tp>::__type std::riemann_zeta ( _Tp __s )

inline

Return the Riemann zeta function $\zeta(s)$ for real argument $ s $ .

The Riemann zeta function is defined by:

$\zeta(s) = \sum_{k=1}^{\infty} k^{-s} \hbox{ for } s > 1$

and

$\zeta(s) = \frac{1}{1-2^{1-s}}\sum_{k=1}^{\infty}(-1)^{k-1}k^{-s} \hbox{ for } 0 <= s <= 1$

For s < 1 use the reflection formula:

$\zeta(s) = 2^s \pi^{s-1} \sin(\frac{\pi s}{2}) \Gamma(1-s) \zeta(1-s)$

Template Parameters

_Tp	The floating-point type of the argument `__s`.

Parameters

__s	The argument `s != 1`

Definition at line 1058 of file specfun.h.

◆ riemann_zetaf()

float std::riemann_zetaf ( float __s )

inline

Return the Riemann zeta function $\zeta(s)$ for float argument $ s $ .

See also: riemann_zeta for more details.

Definition at line 1022 of file specfun.h.

◆ riemann_zetal()

long double std::riemann_zetal ( long double __s )

inline

Return the Riemann zeta function $\zeta(s)$ for long double argument $ s $ .

See also: riemann_zeta for more details.

Definition at line 1032 of file specfun.h.

◆ sph_bessel()

template<typename _Tp >

__gnu_cxx::__promote<_Tp>::__type std::sph_bessel	(	unsigned int	__n,
		_Tp	__x
	)

inline

Return the spherical Bessel function $ j_n(x) $ of nonnegative order n and real argument $ x >= 0 $ .

The spherical Bessel function is defined by:

$j_n(x) = \left(\frac{\pi}{2x} \right) ^{1/2} J_{n+1/2}(x)$

Template Parameters

_Tp	The floating-point type of the argument `__x`.

Parameters

__n	The integral order `n >= 0`
__x	The real argument `x >= 0`

Exceptions

std::domain_error if __x < 0 .

Definition at line 1102 of file specfun.h.

◆ sph_besself()

float std::sph_besself	(	unsigned int	__n,
		float	__x
	)

inline

Return the spherical Bessel function $ j_n(x) $ of nonnegative order n and float argument $ x >= 0 $ .

See also: sph_bessel for more details.

Definition at line 1073 of file specfun.h.

◆ sph_bessell()

long double std::sph_bessell	(	unsigned int	__n,
		long double	__x
	)

inline

Return the spherical Bessel function $ j_n(x) $ of nonnegative order n and long double argument $ x >= 0 $ .

See also: sph_bessel for more details.

Definition at line 1083 of file specfun.h.

◆ sph_legendre()

template<typename _Tp >

__gnu_cxx::__promote<_Tp>::__type std::sph_legendre	(	unsigned int	__l,
		unsigned int	__m,
		_Tp	__theta
	)

inline

Return the spherical Legendre function of nonnegative integral degree l and order m and real angle $\theta$ in radians.

The spherical Legendre function is defined by

$Y_l^m(\theta,\phi) = (-1)^m[\frac{(2l+1)}{4\pi} \frac{(l-m)!}{(l+m)!}] P_l^m(\cos\theta) \exp^{im\phi}$

Template Parameters

_Tp	The floating-point type of the angle `__theta`.

Parameters

__l	The order `__l >= 0`
__m	The degree `__m >= 0` and `__m <= __l`
__theta	The radian polar angle argument

Definition at line 1149 of file specfun.h.

◆ sph_legendref()

float std::sph_legendref	(	unsigned int	__l,
		unsigned int	__m,
		float	__theta
	)

inline

Return the spherical Legendre function of nonnegative integral degree l and order m and float angle $\theta$ in radians.

See also: sph_legendre for details.

Definition at line 1117 of file specfun.h.

◆ sph_legendrel()

long double std::sph_legendrel	(	unsigned int	__l,
		unsigned int	__m,
		long double	__theta
	)

inline

Return the spherical Legendre function of nonnegative integral degree l and order m and long double angle $\theta$ in radians.

See also: sph_legendre for details.

Definition at line 1128 of file specfun.h.

◆ sph_neumann()

template<typename _Tp >

__gnu_cxx::__promote<_Tp>::__type std::sph_neumann	(	unsigned int	__n,
		_Tp	__x
	)

inline

Return the spherical Neumann function of integral order $ n >= 0 $ and real argument $ x >= 0 $ .

The spherical Neumann function is defined by

$n_n(x) = \left(\frac{\pi}{2x} \right) ^{1/2} N_{n+1/2}(x)$

Template Parameters

_Tp	The floating-point type of the argument `__x`.

Parameters

__n	The integral order `n >= 0`
__x	The real argument `__x >= 0`

Exceptions

std::domain_error if __x < 0 .

Definition at line 1193 of file specfun.h.

◆ sph_neumannf()

float std::sph_neumannf	(	unsigned int	__n,
		float	__x
	)

inline

Return the spherical Neumann function of integral order $ n >= 0 $ and float argument $ x >= 0 $ .

See also: sph_neumann for details.

Definition at line 1164 of file specfun.h.

◆ sph_neumannl()

long double std::sph_neumannl	(	unsigned int	__n,
		long double	__x
	)

inline

Return the spherical Neumann function of integral order $ n >= 0 $ and long double $ x >= 0 $ .

See also: sph_neumann for details.

Definition at line 1174 of file specfun.h.

Functions

Detailed Description

Mathematical Special Functions

Introduction and History

Contents

Argument Promotion

NaN Arguments

Implementation

Testing

General Bibliography

Function Documentation

◆ airy_ai()

◆ airy_aif()

◆ airy_ail()

◆ airy_bi()

◆ airy_bif()

◆ airy_bil()

◆ assoc_laguerre()

◆ assoc_laguerref()

◆ assoc_laguerrel()

◆ assoc_legendre()

◆ assoc_legendref()

◆ assoc_legendrel()

◆ beta()

◆ betaf()

◆ betal()

◆ comp_ellint_1()

◆ comp_ellint_1f()

◆ comp_ellint_1l()

◆ comp_ellint_2()

◆ comp_ellint_2f()

◆ comp_ellint_2l()

◆ comp_ellint_3()

◆ comp_ellint_3f()

◆ comp_ellint_3l()

◆ conf_hyperg()

◆ conf_hypergf()

◆ conf_hypergl()

◆ cyl_bessel_i()

◆ cyl_bessel_if()

◆ cyl_bessel_il()

◆ cyl_bessel_j()

◆ cyl_bessel_jf()

◆ cyl_bessel_jl()

◆ cyl_bessel_k()

◆ cyl_bessel_kf()

◆ cyl_bessel_kl()

◆ cyl_neumann()

◆ cyl_neumannf()

◆ cyl_neumannl()

◆ ellint_1()

◆ ellint_1f()

◆ ellint_1l()

◆ ellint_2()

◆ ellint_2f()

◆ ellint_2l()

◆ ellint_3()

◆ ellint_3f()

◆ ellint_3l()

◆ expint()

◆ expintf()

◆ expintl()

◆ hermite()

◆ hermitef()

◆ hermitel()

◆ hyperg()

◆ hypergf()

◆ hypergl()

◆ laguerre()

◆ laguerref()

◆ laguerrel()

◆ legendre()

◆ legendref()

◆ legendrel()

◆ riemann_zeta()

◆ riemann_zetaf()

◆ riemann_zetal()

◆ sph_bessel()

◆ sph_besself()

◆ sph_bessell()