Collaboration diagram for Mathematical Special Functions:

Functions
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 _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 _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

A collection of advanced mathematical special functions, defined by ISO/IEC IS 29124.

Function Documentation

◆ 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.

◆ 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.

◆ 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.