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 >
|
inline |
Return the associated Laguerre polynomial of nonnegative order n, nonnegative degree m and real argument x: 
The associated Laguerre function of real degree 
![\[
L_n^\alpha(x) = \frac{(\alpha + 1)_n}{n!}
{}_1F_1(-n; \alpha + 1; x)
\]](form_44_dark.png)
where 
The associated Laguerre polynomial is defined for integral degree 
![\[
L_n^m(x) = (-1)^m \frac{d^m}{dx^m} L_{n + m}(x)
\]](form_48_dark.png)
where the Laguerre polynomial is defined by:
![\[
L_n(x) = \frac{e^x}{n!} \frac{d^n}{dx^n} (x^ne^{-x})
\]](form_49_dark.png)
and 
- 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.
◆ assoc_laguerref()
|
inline |
◆ assoc_laguerrel()
|
inline |
◆ assoc_legendre()
template<typename _Tp >
|
inline |
Return the associated Legendre function of degree l and order m.
The associated Legendre function is derived from the Legendre function 
![\[
P_l^m(x) = (1 - x^2)^{m/2}\frac{d^m}{dx^m}P_l(x)
\]](form_52_dark.png)
- 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.
◆ assoc_legendref()
|
inline |
◆ assoc_legendrel()
|
inline |
◆ beta()
Return the beta function, 
a, b.
The beta function is defined by
![\[
B(a,b) = \int_0^1 t^{a - 1} (1 - t)^{b - 1} dt
= \frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)}
\]](form_55_dark.png)
where 
- 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 < 0or__b < 0.
◆ betaf()
|
inline |
◆ betal()
|
inline |
◆ comp_ellint_1()
template<typename _Tp >
|
inline |
Return the complete elliptic integral of the first kind 
k.
The complete elliptic integral of the first kind is defined as
![\[
K(k) = F(k,\pi/2) = \int_0^{\pi/2}\frac{d\theta}
{\sqrt{1 - k^2 sin^2\theta}}
\]](form_60_dark.png)
where 
- 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.
◆ comp_ellint_1f()
|
inline |
◆ comp_ellint_1l()
|
inline |
◆ comp_ellint_2()
template<typename _Tp >
|
inline |
Return the complete elliptic integral of the second kind 
k.
The complete elliptic integral of the second kind is defined as
![\[
E(k) = E(k,\pi/2) = \int_0^{\pi/2}\sqrt{1 - k^2 sin^2\theta}
\]](form_63_dark.png)
where 
- 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.
◆ comp_ellint_2f()
|
inline |
◆ comp_ellint_2l()
|
inline |
◆ comp_ellint_3()
Return the complete elliptic integral of the third kind 
k.
The complete elliptic integral of the third kind is defined as
![\[
\Pi(k,\nu) = \Pi(k,\nu,\pi/2) = \int_0^{\pi/2}
\frac{d\theta}
{(1 - \nu \sin^2\theta)\sqrt{1 - k^2 \sin^2\theta}}
\]](form_67_dark.png)
where 
- 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.
◆ comp_ellint_3f()
|
inline |
◆ comp_ellint_3l()
|
inline |
◆ cyl_bessel_i()
Return the regular modified Bessel function 
The regular modified cylindrical Bessel function is:
![\[
I_{\nu}(x) = i^{-\nu}J_\nu(ix) = \sum_{k=0}^{\infty}
\frac{(x/2)^{\nu + 2k}}{k!\Gamma(\nu+k+1)}
\]](form_71_dark.png)
- 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.
◆ cyl_bessel_if()
|
inline |
◆ cyl_bessel_il()
|
inline |
◆ cyl_bessel_j()
Return the Bessel function 
The cylindrical Bessel function is:
![\[
J_{\nu}(x) = \sum_{k=0}^{\infty}
\frac{(-1)^k (x/2)^{\nu + 2k}}{k!\Gamma(\nu+k+1)}
\]](form_73_dark.png)
- 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.
◆ cyl_bessel_jf()
|
inline |
◆ cyl_bessel_jl()
|
inline |
◆ cyl_bessel_k()
Return the irregular modified Bessel function 
The irregular modified Bessel function is defined by:
![\[
K_{\nu}(x) = \frac{\pi}{2}
\frac{I_{-\nu}(x) - I_{\nu}(x)}{\sin \nu\pi}
\]](form_76_dark.png)
where for integral 
![\[
K_{-\nu}(x) = K_{\nu}(x)
\]](form_79_dark.png)
- 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.
◆ cyl_bessel_kf()
|
inline |
◆ cyl_bessel_kl()
|
inline |
◆ cyl_neumann()
Return the Neumann function 
The Neumann function is defined by:
![\[
N_{\nu}(x) = \frac{J_{\nu}(x) \cos \nu\pi - J_{-\nu}(x)}
{\sin \nu\pi}
\]](form_81_dark.png)
where
- 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.
◆ cyl_neumannf()
|
inline |
◆ cyl_neumannl()
|
inline |
◆ ellint_1()
Return the incomplete elliptic integral of the first kind real modulus 
The incomplete elliptic integral of the first kind is defined as
![\[
F(k,\phi) = \int_0^{\phi}\frac{d\theta}
{\sqrt{1 - k^2 sin^2\theta}}
\]](form_84_dark.png)
For 
- 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.
◆ ellint_1f()
|
inline |
◆ ellint_1l()
|
inline |
◆ ellint_2()
Return the incomplete elliptic integral of the second kind
The incomplete elliptic integral of the second kind is defined as
![\[
E(k,\phi) = \int_0^{\phi} \sqrt{1 - k^2 sin^2\theta}
\]](form_86_dark.png)
For
- 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.
◆ ellint_2f()
|
inline |
◆ ellint_2l()
|
inline |
◆ ellint_3()
|
inline |
Return the incomplete elliptic integral of the third kind
The incomplete elliptic integral of the third kind is defined by:
![\[
\Pi(k,\nu,\phi) = \int_0^{\phi}
\frac{d\theta}
{(1 - \nu \sin^2\theta)
\sqrt{1 - k^2 \sin^2\theta}}
\]](form_87_dark.png)
For 
- 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.
◆ ellint_3f()
|
inline |
◆ ellint_3l()
|
inline |
◆ expint()
template<typename _Tp >
|
inline |
![\[
Ei(x) = -\int_{-x}^\infty \frac{e^t}{t} dt
\]](form_89_dark.png)
◆ expintf()
|
inline |
◆ expintl()
|
inline |
◆ hermite()
template<typename _Tp >
|
inline |
Return the Hermite polynomial 
real argument x.
The Hermite polynomial is defined by:
![\[
H_n(x) = (-1)^n e^{x^2} \frac{d^n}{dx^n} e^{-x^2}
\]](form_91_dark.png)
The Hermite polynomial obeys a reflection formula:
![\[
H_n(-x) = (-1)^n H_n(x)
\]](form_92_dark.png)
- Template Parameters
-
_Tp The floating-point type of the argument __x.
- Parameters
-
__n The order __x The argument
◆ hermitef()
|
inline |
◆ hermitel()
|
inline |
◆ laguerre()
template<typename _Tp >
|
inline |
Returns the Laguerre polynomial 
n and real argument
The Laguerre polynomial is defined by:
- 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.
◆ laguerref()
|
inline |
◆ laguerrel()
|
inline |
◆ legendre()
template<typename _Tp >
|
inline |
Return the Legendre polynomial 
The Legendre function of order
![\[
P_l(x) = \frac{1}{2^l l!}\frac{d^l}{dx^l}(x^2 - 1)^{l}
\]](form_96_dark.png)
- 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
◆ legendref()
|
inline |
◆ legendrel()
|
inline |
◆ riemann_zeta()
template<typename _Tp >
|
inline |
![\[
\zeta(s) = \sum_{k=1}^{\infty} k^{-s} \hbox{ for } s > 1
\]](form_100_dark.png)
![\[
\zeta(s) = \frac{1}{1-2^{1-s}}\sum_{k=1}^{\infty}(-1)^{k-1}k^{-s}
\hbox{ for } 0 <= s <= 1
\]](form_101_dark.png)
![\[
\zeta(s) = 2^s \pi^{s-1} \sin(\frac{\pi s}{2}) \Gamma(1-s) \zeta(1-s)
\]](form_102_dark.png)
◆ riemann_zetaf()
|
inline |
◆ riemann_zetal()
|
inline |
◆ sph_bessel()
template<typename _Tp >
|
inline |
Return the spherical Bessel function 
The spherical Bessel function is defined by:
![\[
j_n(x) = \left(\frac{\pi}{2x} \right) ^{1/2} J_{n+1/2}(x)
\]](form_104_dark.png)
- 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.
◆ sph_besself()
|
inline |
◆ sph_bessell()
|
inline |
◆ sph_legendre()
template<typename _Tp >
|
inline |
Return the spherical Legendre function of nonnegative integral degree l and order m and real angle 
The spherical Legendre function is defined by
![\[
Y_l^m(\theta,\phi) = (-1)^m[\frac{(2l+1)}{4\pi}
\frac{(l-m)!}{(l+m)!}]
P_l^m(\cos\theta) \exp^{im\phi}
\]](form_106_dark.png)
- Template Parameters
-
_Tp The floating-point type of the angle __theta.
- Parameters
-
__l The order __l >= 0__m The degree __m >= 0and__m <= __l__theta The radian polar angle argument
◆ sph_legendref()
|
inline |
◆ sph_legendrel()
|
inline |
◆ sph_neumann()
template<typename _Tp >
|
inline |
Return the spherical Neumann function of integral order 
The spherical Neumann function is defined by
![\[
n_n(x) = \left(\frac{\pi}{2x} \right) ^{1/2} N_{n+1/2}(x)
\]](form_108_dark.png)
- 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.
◆ sph_neumannf()
|
inline |
