Classes¶

struct __floating_point_constant
A class to encapsulate type dependent floating point constants. Not everything will be able to be expressed as type logic. struct __numeric_constants
A structure for numeric constants.

Functions¶

template<typename _Tp > void __airy (const _Tp __x, _Tp &__Ai, _Tp &__Bi, _Tp &__Aip, _Tp &__Bip)
template<typename _Tp > _Tp __assoc_laguerre (const unsigned int __n, const unsigned int __m, const _Tp __x)
template<typename _Tp > _Tp __assoc_legendre_p (const unsigned int __l, const unsigned int __m, const _Tp __x)
template<typename _Tp > _Tp __bernoulli (const int __n)
template<typename _Tp > _Tp __bernoulli_series (unsigned int __n)
template<typename _Tp > void __bessel_ik (const _Tp __nu, const _Tp __x, _Tp &__Inu, _Tp &__Knu, _Tp &__Ipnu, _Tp &__Kpnu)
template<typename _Tp > void __bessel_jn (const _Tp __nu, const _Tp __x, _Tp &__Jnu, _Tp &__Nnu, _Tp &__Jpnu, _Tp &__Npnu)
template<typename _Tp > _Tp __beta (_Tp __x, _Tp __y)
template<typename _Tp > _Tp __beta_gamma (_Tp __x, _Tp __y)
template<typename _Tp > _Tp __beta_lgamma (_Tp __x, _Tp __y)
template<typename _Tp > _Tp __beta_product (_Tp __x, _Tp __y)
template<typename _Tp > _Tp __bincoef (const unsigned int __n, const unsigned int __k)
template<typename _Tp > _Tp __comp_ellint_1 (const _Tp __k)
template<typename _Tp > _Tp __comp_ellint_1_series (const _Tp __k)
template<typename _Tp > _Tp __comp_ellint_2 (const _Tp __k)
template<typename _Tp > _Tp __comp_ellint_2_series (const _Tp __k)
template<typename _Tp > _Tp __comp_ellint_3 (const _Tp __k, const _Tp __nu)
template<typename _Tp > _Tp __conf_hyperg (const _Tp __a, const _Tp __c, const _Tp __x)
template<typename _Tp > _Tp __conf_hyperg_luke (const _Tp __a, const _Tp __c, const _Tp __xin)
template<typename _Tp > _Tp __conf_hyperg_series (const _Tp __a, const _Tp __c, const _Tp __x)
template<typename _Tp > _Tp __cyl_bessel_i (const _Tp __nu, const _Tp __x)
template<typename _Tp > _Tp __cyl_bessel_ij_series (const _Tp __nu, const _Tp __x, const _Tp __sgn, const unsigned int __max_iter)
template<typename _Tp > _Tp __cyl_bessel_j (const _Tp __nu, const _Tp __x)
template<typename _Tp > void __cyl_bessel_jn_asymp (const _Tp __nu, const _Tp __x, _Tp &__Jnu, _Tp &__Nnu)
template<typename _Tp > _Tp __cyl_bessel_k (const _Tp __nu, const _Tp __x)
template<typename _Tp > _Tp __cyl_neumann_n (const _Tp __nu, const _Tp __x)
template<typename _Tp > _Tp __ellint_1 (const _Tp __k, const _Tp __phi)
template<typename _Tp > _Tp __ellint_2 (const _Tp __k, const _Tp __phi)
template<typename _Tp > _Tp __ellint_3 (const _Tp __k, const _Tp __nu, const _Tp __phi)
template<typename _Tp > _Tp __ellint_rc (const _Tp __x, const _Tp __y)
template<typename _Tp > _Tp __ellint_rd (const _Tp __x, const _Tp __y, const _Tp __z)
template<typename _Tp > _Tp __ellint_rf (const _Tp __x, const _Tp __y, const _Tp __z)
template<typename _Tp > _Tp __ellint_rj (const _Tp __x, const _Tp __y, const _Tp __z, const _Tp __p)
template<typename _Tp > _Tp __expint (const _Tp __x)
template<typename _Tp > _Tp __expint (const unsigned int __n, const _Tp __x)
template<typename _Tp > _Tp __expint_asymp (const unsigned int __n, const _Tp __x)
template<typename _Tp > _Tp __expint_E1 (const _Tp __x)
template<typename _Tp > _Tp __expint_E1_asymp (const _Tp __x)
template<typename _Tp > _Tp __expint_E1_series (const _Tp __x)
template<typename _Tp > _Tp __expint_Ei (const _Tp __x)
template<typename _Tp > _Tp __expint_Ei_asymp (const _Tp __x)
template<typename _Tp > _Tp __expint_Ei_series (const _Tp __x)
template<typename _Tp > _Tp __expint_En_cont_frac (const unsigned int __n, const _Tp __x)
template<typename _Tp > _Tp __expint_En_recursion (const unsigned int __n, const _Tp __x)
template<typename _Tp > _Tp __expint_En_series (const unsigned int __n, const _Tp __x)
template<typename _Tp > _Tp __expint_large_n (const unsigned int __n, const _Tp __x)
template<typename _Tp > _Tp __gamma (const _Tp __x)
template<typename _Tp > void __gamma_temme (const _Tp __mu, _Tp &__gam1, _Tp &__gam2, _Tp &__gampl, _Tp &__gammi)
template<typename _Tp > _Tp __hurwitz_zeta (const _Tp __a, const _Tp __s)
template<typename _Tp > _Tp __hurwitz_zeta_glob (const _Tp __a, const _Tp __s)
template<typename _Tp > _Tp __hyperg (const _Tp __a, const _Tp __b, const _Tp __c, const _Tp __x)
template<typename _Tp > _Tp __hyperg_luke (const _Tp __a, const _Tp __b, const _Tp __c, const _Tp __xin)
template<typename _Tp > _Tp __hyperg_reflect (const _Tp __a, const _Tp __b, const _Tp __c, const _Tp __x)
template<typename _Tp > _Tp __hyperg_series (const _Tp __a, const _Tp __b, const _Tp __c, const _Tp __x)
template<typename _Tp > bool __isnan (const _Tp __x)
template<> bool __isnan< float > (const float __x)
template<> bool __isnan< long double > (const long double __x)
template<typename _Tp > _Tp __laguerre (const unsigned int __n, const _Tp __x)
template<typename _Tp > _Tp __log_bincoef (const unsigned int __n, const unsigned int __k)
template<typename _Tp > _Tp __log_gamma (const _Tp __x)
template<typename _Tp > _Tp __log_gamma_bernoulli (const _Tp __x)
template<typename _Tp > _Tp __log_gamma_lanczos (const _Tp __x)
template<typename _Tp > _Tp __log_gamma_sign (const _Tp __x)
template<typename _Tp > _Tp __poly_hermite (const unsigned int __n, const _Tp __x)
template<typename _Tp > _Tp __poly_hermite_recursion (const unsigned int __n, const _Tp __x)
template<typename _Tpa , typename _Tp > _Tp __poly_laguerre (const unsigned int __n, const _Tpa __alpha1, const _Tp __x)
template<typename _Tpa , typename _Tp > _Tp __poly_laguerre_hyperg (const unsigned int __n, const _Tpa __alpha1, const _Tp __x)
template<typename _Tpa , typename _Tp > _Tp __poly_laguerre_large_n (const unsigned __n, const _Tpa __alpha1, const _Tp __x)
template<typename _Tpa , typename _Tp > _Tp __poly_laguerre_recursion (const unsigned int __n, const _Tpa __alpha1, const _Tp __x)
template<typename _Tp > _Tp __poly_legendre_p (const unsigned int __l, const _Tp __x)
template<typename _Tp > _Tp __psi (const unsigned int __n, const _Tp __x)
template<typename _Tp > _Tp __psi (const _Tp __x)
template<typename _Tp > _Tp __psi_asymp (const _Tp __x)
template<typename _Tp > _Tp __psi_series (const _Tp __x)
template<typename _Tp > _Tp __riemann_zeta (const _Tp __s)
template<typename _Tp > _Tp __riemann_zeta_alt (const _Tp __s)
template<typename _Tp > _Tp __riemann_zeta_glob (const _Tp __s)
template<typename _Tp > _Tp __riemann_zeta_product (const _Tp __s)
template<typename _Tp > _Tp __riemann_zeta_sum (const _Tp __s)
template<typename _Tp > _Tp __sph_bessel (const unsigned int __n, const _Tp __x)
template<typename _Tp > void __sph_bessel_ik (const unsigned int __n, const _Tp __x, _Tp &__i_n, _Tp &__k_n, _Tp &__ip_n, _Tp &__kp_n)
template<typename _Tp > void __sph_bessel_jn (const unsigned int __n, const _Tp __x, _Tp &__j_n, _Tp &__n_n, _Tp &__jp_n, _Tp &__np_n)
template<typename _Tp > _Tp __sph_legendre (const unsigned int __l, const unsigned int __m, const _Tp __theta)
template<typename _Tp > _Tp __sph_neumann (const unsigned int __n, const _Tp __x)

template<typename _Tp > void std::tr1::__detail::__airy (const _Tp __x, _Tp & __Ai, _Tp & __Bi, _Tp & __Aip, _Tp & __Bip) [inline]¶

Compute the Airy functions $ Ai(x) $ and $ Bi(x) $ and their first derivatives $ Ai'(x) $ and $ Bi(x) $ respectively. Parameters:

Definition at line 370 of file modified_bessel_func.tcc.

References __bessel_ik(), __bessel_jn(), std::tr1::__detail::__numeric_constants< _Tp >::__pi(), std::tr1::__detail::__numeric_constants< _Tp >::__sqrt3(), std::abs(), and std::sqrt().

template<typename _Tp > _Tp std::tr1::__detail::__assoc_laguerre (const unsigned int __n, const unsigned int __m, const _Tp __x) [inline]¶

This routine returns the associated Laguerre polynomial of order n, degree m: $ L_n^m(x) $. The associated Laguerre polynomial is defined for integral $ lpha = m $ by: L_n^m(x) = (-1)^m ac{d^m}{dx^m} L_{n + m}(x) ] where the Laguerre polynomial is defined by: L_n(x) = ac{e^x}{n!} ac{d^n}{dx^n} (x^ne^{-x}) ]

Parameters:

Returns:

Definition at line 297 of file poly_laguerre.tcc.

template<typename _Tp > _Tp std::tr1::__detail::__assoc_legendre_p (const unsigned int __l, const unsigned int __m, const _Tp __x) [inline]¶

Return the associated Legendre function by recursion on $ l $. 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}ac{d^m}{dx^m}P_l(x) ]

Parameters:

Definition at line 133 of file legendre_function.tcc.

References __poly_legendre_p(), and std::sqrt().

template<typename _Tp > _Tp std::tr1::__detail::__bernoulli (const int __n) [inline]¶

This returns Bernoulli number $B_n$. Parameters:

Returns:

Definition at line 133 of file gamma.tcc.

template<typename _Tp > _Tp std::tr1::__detail::__bernoulli_series (unsigned int __n) [inline]¶

This returns Bernoulli numbers from a table or by summation for larger values. Recursion is unstable.

Parameters:

Returns:

Definition at line 70 of file gamma.tcc.

References std::pow().

template<typename _Tp > void std::tr1::__detail::__bessel_ik (const _Tp __nu, const _Tp __x, _Tp & __Inu, _Tp & __Knu, _Tp & __Ipnu, _Tp & __Kpnu) [inline]¶

Compute the modified Bessel functions $ I_0(x) $ and $ K_0(x) $ and their first derivatives $ I'_0(x) $ and $ K'_0(x) $ respectively. These four functions are computed together for numerical stability. Parameters:

Definition at line 81 of file modified_bessel_func.tcc.

References __gamma_temme(), std::tr1::__detail::__numeric_constants< _Tp >::__pi(), std::abs(), std::cosh(), std::exp(), std::log(), std::sin(), std::sinh(), and std::sqrt().

Referenced by __airy(), __cyl_bessel_i(), __cyl_bessel_k(), and __sph_bessel_ik().

template<typename _Tp > void std::tr1::__detail::__bessel_jn (const _Tp __nu, const _Tp __x, _Tp & __Jnu, _Tp & __Nnu, _Tp & __Jpnu, _Tp & __Npnu) [inline]¶

Compute the Bessel $ J_0(x) $ and Neumann $ N_0(x) $ functions and their first derivatives $ J'_0(x) $ and $ N'_0(x) $ respectively. These four functions are computed together for numerical stability. Parameters:

Definition at line 128 of file bessel_function.tcc.

References __gamma_temme(), std::tr1::__detail::__numeric_constants< _Tp >::__pi(), std::abs(), std::cosh(), std::exp(), std::log(), std::max(), std::sin(), std::sinh(), and std::sqrt().

Referenced by __airy(), __cyl_bessel_j(), __cyl_neumann_n(), and __sph_bessel_jn().

template<typename _Tp > _Tp std::tr1::__detail::__beta (_Tp __x, _Tp __y) [inline]¶

Return the beta function $ B(x,y) $. The beta function is defined by B(x,y) = ac{Gamma(x)Gamma(y)}{Gamma(x+y)} ]

Parameters:

Returns:

Definition at line 185 of file beta_function.tcc.

References __beta_lgamma().

Referenced by __ellint_rj().

template<typename _Tp > _Tp std::tr1::__detail::__beta_gamma (_Tp __x, _Tp __y) [inline]¶

Return the beta function: $B(x,y)$. The beta function is defined by B(x,y) = ac{Gamma(x)Gamma(y)}{Gamma(x+y)} ]

Parameters:

Returns:

Definition at line 75 of file beta_function.tcc.

References __gamma().

template<typename _Tp > _Tp std::tr1::__detail::__beta_lgamma (_Tp __x, _Tp __y) [inline]¶

Return the beta function $B(x,y)$ using the log gamma functions. The beta function is defined by B(x,y) = ac{Gamma(x)Gamma(y)}{Gamma(x+y)} ]

Parameters:

Returns:

Definition at line 123 of file beta_function.tcc.

References __log_gamma(), and std::exp().

Referenced by __beta().

template<typename _Tp > _Tp std::tr1::__detail::__beta_product (_Tp __x, _Tp __y) [inline]¶

Return the beta function $B(x,y)$ using the product form. The beta function is defined by B(x,y) = ac{Gamma(x)Gamma(y)}{Gamma(x+y)} ]

Parameters:

Returns:

Definition at line 154 of file beta_function.tcc.

template<typename _Tp > _Tp std::tr1::__detail::__bincoef (const unsigned int __n, const unsigned int __k) [inline]¶

Return the binomial coefficient. The binomial coefficient is given by: ac{n!}{(n-k)! k!} ]. Parameters:

Returns:

Definition at line 310 of file gamma.tcc.

References std::exp(), and std::log().

template<typename _Tp > _Tp std::tr1::__detail::__comp_ellint_1 (const _Tp __k) [inline]¶

Return the complete elliptic integral of the first kind $ K(k) $ using the Carlson formulation. The complete elliptic integral of the first kind is defined as K(k) = F(k,i/2)
= int_0^{i/2}ac{dheta}
{rt{1 - k^2 sin^2heta}} ] where $ F(k,hi)
$ is the incomplete elliptic integral of the first kind.

Parameters:

Returns:

Definition at line 191 of file ell_integral.tcc.

References __ellint_rf(), and std::abs().

Referenced by __ellint_1().

template<typename _Tp > _Tp std::tr1::__detail::__comp_ellint_1_series (const _Tp __k) [inline]¶

Return the complete elliptic integral of the first kind $ K(k) $ by series expansion. The complete elliptic integral of the first kind is defined as K(k) = F(k,i/2)
= int_0^{i/2}ac{dheta}
{rt{1 - k^2sin^2heta}} ]

This routine is not bad as long as |k| is somewhat smaller than 1 but is not is good as the Carlson elliptic integral formulation.

Parameters:

Returns:

Definition at line 153 of file ell_integral.tcc.

References std::tr1::__detail::__numeric_constants< _Tp >::__pi_2().

template<typename _Tp > _Tp std::tr1::__detail::__comp_ellint_2 (const _Tp __k) [inline]¶

Return the complete elliptic integral of the second kind $ E(k) $ using the Carlson formulation. The complete elliptic integral of the second kind is defined as E(k,i/2)
= int_0^{i/2}rt{1
- k^2 sin^2heta} ]

Parameters:

Returns:

Definition at line 402 of file ell_integral.tcc.

References __ellint_rd(), __ellint_rf(), and std::abs().

Referenced by __ellint_2().

template<typename _Tp > _Tp std::tr1::__detail::__comp_ellint_2_series (const _Tp __k) [inline]¶

Return the complete elliptic integral of the second kind $ E(k) $ by series expansion. The complete elliptic integral of the second kind is defined as E(k,i/2)
= int_0^{i/2}rt{1
- k^2 sin^2heta} ]

This routine is not bad as long as |k| is somewhat smaller than 1 but is not is good as the Carlson elliptic integral formulation.

Parameters:

Returns:

Definition at line 266 of file ell_integral.tcc.

References std::tr1::__detail::__numeric_constants< _Tp >::__pi_2().

template<typename _Tp > _Tp std::tr1::__detail::__comp_ellint_3 (const _Tp __k, const _Tp __nu) [inline]¶

Return the complete elliptic integral of the third kind $ Pi(k,0) = Pi(k,0,i/2)
$ using the Carlson formulation. The complete elliptic integral of the third kind is defined as Pi(k,0) = int_0^{i/2}
ac{dheta} {(1 - 0 n^2heta)rt{1 - k^2 n^2heta}} ]

Parameters:

Returns:

Definition at line 670 of file ell_integral.tcc.

References __ellint_rf(), __ellint_rj(), and std::abs().

Referenced by __ellint_3().

template<typename _Tp > _Tp std::tr1::__detail::__conf_hyperg (const _Tp __a, const _Tp __c, const _Tp __x) [inline]¶

Return the confluent hypogeometric function $ _1F_1(a;c;x) $. Todo

Parameters:

Returns:

Definition at line 222 of file hypergeometric.tcc.

References __conf_hyperg_luke(), __conf_hyperg_series(), and std::exp().

template<typename _Tp > _Tp std::tr1::__detail::__conf_hyperg_luke (const _Tp __a, const _Tp __c, const _Tp __xin) [inline]¶

Return the hypogeometric function $ _2F_1(a,b;c;x) $ by an iterative procedure described in Luke, Algorithms for the Computation of Mathematical Functions. Like the case of the 2F1 rational approximations, these are probably guaranteed to converge for x < 0, barring gross numerical instability in the pre-asymptotic regime.

Definition at line 115 of file hypergeometric.tcc.

References std::abs(), and std::pow().

Referenced by __conf_hyperg().

template<typename _Tp > _Tp std::tr1::__detail::__conf_hyperg_series (const _Tp __a, const _Tp __c, const _Tp __x) [inline]¶

This routine returns the confluent hypergeometric function by series expansion. _1F_1(a;c;x) = ac{Gamma(c)}{Gamma(a)} m_{n=0}^{infty} ac{Gamma(a+n)}{Gamma(c+n)} ac{x^n}{n!} ].PP If a and b are integers and a < 0 and either b > 0 or b < a then the series is a polynomial with a finite number of terms. If b is an integer and b <= 0 the confluent hypergeometric function is undefined.

Parameters:

Returns:

Definition at line 78 of file hypergeometric.tcc.

References std::abs().

Referenced by __conf_hyperg().

template<typename _Tp > _Tp std::tr1::__detail::__cyl_bessel_i (const _Tp __nu, const _Tp __x) [inline]¶

Return the regular modified Bessel function of order $ 0 $: $ I_{0}(x) $. The regular modified cylindrical Bessel function is: I_{0}(x) = m_{k=0}^{infty} ac{(x/2)^{0 + 2k}}{k!Gamma(0+k+1)} ]

Parameters:

Returns:

Definition at line 265 of file modified_bessel_func.tcc.

References __bessel_ik(), and __cyl_bessel_ij_series().

template<typename _Tp > _Tp std::tr1::__detail::__cyl_bessel_ij_series (const _Tp __nu, const _Tp __x, const _Tp __sgn, const unsigned int __max_iter) [inline]¶

This routine returns the cylindrical Bessel functions of order $ 0 $: $ J_{0} $ or $ I_{0} $ by series expansion. The modified cylindrical Bessel function is: Z_{0}(x) = m_{k=0}^{infty} ac{gma^k (x/2)^{0 + 2k}}{k!Gamma(0+k+1)} ] where $ gma = +1 $ or$ -1 $ for $ Z = I $ or $ J $ respectively.

See Abramowitz & Stegun, 9.1.10 Abramowitz & Stegun, 9.6.7 (1) Handbook of Mathematical Functions, ed. Milton Abramowitz and Irene A. Stegun, Dover Publications, Equation 9.1.10 p. 360 and Equation 9.6.10 p. 375

Parameters:

Returns:

Definition at line 410 of file bessel_function.tcc.

References __log_gamma(), std::abs(), std::exp(), and std::log().

Referenced by __cyl_bessel_i(), and __cyl_bessel_j().

template<typename _Tp > _Tp std::tr1::__detail::__cyl_bessel_j (const _Tp __nu, const _Tp __x) [inline]¶

Return the Bessel function of order $ 0 $: $ J_{0}(x) $. The cylindrical Bessel function is: J_{0}(x) = m_{k=0}^{infty} ac{(-1)^k (x/2)^{0 + 2k}}{k!Gamma(0+k+1)} ]

Parameters:

Returns:

Definition at line 454 of file bessel_function.tcc.

References __bessel_jn(), __cyl_bessel_ij_series(), and __cyl_bessel_jn_asymp().

template<typename _Tp > void std::tr1::__detail::__cyl_bessel_jn_asymp (const _Tp __nu, const _Tp __x, _Tp & __Jnu, _Tp & __Nnu) [inline]¶

This routine computes the asymptotic cylindrical Bessel and Neumann functions of order nu: $ J_{0} $, $ N_{0} $. References: (1) Handbook of Mathematical Functions, ed. Milton Abramowitz and Irene A. Stegun, Dover Publications, Section 9 p. 364, Equations 9.2.5-9.2.10

Parameters:

Definition at line 353 of file bessel_function.tcc.

References std::cos(), std::sin(), and std::sqrt().

Referenced by __cyl_bessel_j(), and __cyl_neumann_n().

template<typename _Tp > _Tp std::tr1::__detail::__cyl_bessel_k (const _Tp __nu, const _Tp __x) [inline]¶

Return the irregular modified Bessel function $ K_{0}(x) $ of order $ 0 $. The irregular modified Bessel function is defined by: K_{0}(x) = ac{i}{2}
ac{I_{-0}(x) - I_{0}(x)}{n 0i}
] where for integral $ 0 = n $ a limit is taken: $ lim_{0 o n} $.

Parameters:

Returns:

Definition at line 301 of file modified_bessel_func.tcc.

References __bessel_ik().

template<typename _Tp > _Tp std::tr1::__detail::__cyl_neumann_n (const _Tp __nu, const _Tp __x) [inline]¶

Return the Neumann function of order $ 0 $: $ N_{0}(x) $. The Neumann function is defined by: N_{0}(x) = ac{J_{0}(x) os 0i
- J_{-0}(x)} {n 0i}
] where for integral $ 0 = n $ a limit is taken: $ lim_{0 o n} $.

Parameters:

Returns:

Definition at line 496 of file bessel_function.tcc.

References __bessel_jn(), and __cyl_bessel_jn_asymp().

template<typename _Tp > _Tp std::tr1::__detail::__ellint_1 (const _Tp __k, const _Tp __phi) [inline]¶

Return the incomplete elliptic integral of the first kind $ F(k,hi)
$ using the Carlson formulation. The incomplete elliptic integral of the first kind is defined as F(k,hi)
= int_0^{hi}ac{dheta}
{rt{1 - k^2 sin^2heta}} ]

Parameters:

Returns:

Definition at line 219 of file ell_integral.tcc.

References __comp_ellint_1(), __ellint_rf(), std::tr1::__detail::__numeric_constants< _Tp >::__pi(), std::abs(), std::cos(), and std::sin().

template<typename _Tp > _Tp std::tr1::__detail::__ellint_2 (const _Tp __k, const _Tp __phi) [inline]¶

Return the incomplete elliptic integral of the second kind $ E(k,hi)
$ using the Carlson formulation. The incomplete elliptic integral of the second kind is defined as E(k,hi)
= int_0^{hi}
rt{1 - k^2 sin^2heta} ]

Parameters:

Returns:

Definition at line 436 of file ell_integral.tcc.

References __comp_ellint_2(), __ellint_rd(), __ellint_rf(), std::tr1::__detail::__numeric_constants< _Tp >::__pi(), std::abs(), std::cos(), and std::sin().

template<typename _Tp > _Tp std::tr1::__detail::__ellint_3 (const _Tp __k, const _Tp __nu, const _Tp __phi) [inline]¶

Return the incomplete elliptic integral of the third kind $ Pi(k,0,hi)
$ using the Carlson formulation. The incomplete elliptic integral of the third kind is defined as Pi(k,0,hi)
= int_0^{hi}
ac{dheta} {(1 - 0 n^2heta) rt{1 - k^2 n^2heta}} ]

Parameters:

Returns:

Definition at line 710 of file ell_integral.tcc.

References __comp_ellint_3(), __ellint_rf(), __ellint_rj(), std::tr1::__detail::__numeric_constants< _Tp >::__pi(), std::abs(), std::cos(), and std::sin().

template<typename _Tp > _Tp std::tr1::__detail::__ellint_rc (const _Tp __x, const _Tp __y) [inline]¶

Return the Carlson elliptic function $ R_C(x,y) = R_F(x,y,y) $ where $ R_F(x,y,z) $ is the Carlson elliptic function of the first kind. The Carlson elliptic function is defined by: R_C(x,y) = ac{1}{2} int_0^infty ac{dt}{(t + x)^{1/2}(t + y)} ]

Based on Carlson's algorithms:

• B. C. Carlson Numer. Math. 33, 1 (1979)
• B. C. Carlson, Special Functions of Applied Mathematics (1977)
• Numerical Recipes in C, 2nd ed, pp. 261-269, by Press, Teukolsky, Vetterling, Flannery (1992)

Parameters:

Returns:

Definition at line 495 of file ell_integral.tcc.

References std::abs(), std::max(), std::min(), std::pow(), and std::sqrt().

Referenced by __ellint_rj().

template<typename _Tp > _Tp std::tr1::__detail::__ellint_rd (const _Tp __x, const _Tp __y, const _Tp __z) [inline]¶

Return the Carlson elliptic function of the second kind $ R_D(x,y,z) = R_J(x,y,z,z) $ where $ R_J(x,y,z,p) $ is the Carlson elliptic function of the third kind. The Carlson elliptic function of the second kind is defined by: R_D(x,y,z) = ac{3}{2} int_0^infty ac{dt}{(t + x)^{1/2}(t + y)^{1/2}(t + z)^{3/2}} ]

Based on Carlson's algorithms:

• B. C. Carlson Numer. Math. 33, 1 (1979)
• B. C. Carlson, Special Functions of Applied Mathematics (1977)
• Numerical Recipes in C, 2nd ed, pp. 261-269, by Press, Teukolsky, Vetterling, Flannery (1992)

Parameters:

Returns:

Definition at line 314 of file ell_integral.tcc.

References std::abs(), std::max(), std::min(), std::pow(), and std::sqrt().

Referenced by __comp_ellint_2(), and __ellint_2().

template<typename _Tp > _Tp std::tr1::__detail::__ellint_rf (const _Tp __x, const _Tp __y, const _Tp __z) [inline]¶

Return the Carlson elliptic function $ R_F(x,y,z) $ of the first kind. The Carlson elliptic function of the first kind is defined by: R_F(x,y,z) = ac{1}{2} int_0^infty ac{dt}{(t + x)^{1/2}(t + y)^{1/2}(t + z)^{1/2}} ]

Parameters:

Returns:

Definition at line 74 of file ell_integral.tcc.

References std::abs(), std::max(), std::min(), std::pow(), and std::sqrt().

Referenced by __comp_ellint_1(), __comp_ellint_2(), __comp_ellint_3(), __ellint_1(), __ellint_2(), and __ellint_3().

template<typename _Tp > _Tp std::tr1::__detail::__ellint_rj (const _Tp __x, const _Tp __y, const _Tp __z, const _Tp __p) [inline]¶

Return the Carlson elliptic function $ R_J(x,y,z,p) $ of the third kind. The Carlson elliptic function of the third kind is defined by: R_J(x,y,z,p) = ac{3}{2} int_0^infty ac{dt}{(t + x)^{1/2}(t + y)^{1/2}(t + z)^{1/2}(t + p)} ]

Based on Carlson's algorithms:

• B. C. Carlson Numer. Math. 33, 1 (1979)
• B. C. Carlson, Special Functions of Applied Mathematics (1977)
• Numerical Recipes in C, 2nd ed, pp. 261-269, by Press, Teukolsky, Vetterling, Flannery (1992)

Parameters:

Returns:

Definition at line 566 of file ell_integral.tcc.

References __beta(), __ellint_rc(), std::abs(), std::max(), std::min(), std::pow(), and std::sqrt().

Referenced by __comp_ellint_3(), and __ellint_3().

template<typename _Tp > _Tp std::tr1::__detail::__expint (const _Tp __x) [inline]¶

Return the exponential integral $ Ei(x) $. The exponential integral is given by Ei(x) = -int_{-x}^infty ac{e^t}{t} dt ]

Parameters:

Returns:

Definition at line 512 of file exp_integral.tcc.

References __expint_Ei().

template<typename _Tp > _Tp std::tr1::__detail::__expint (const unsigned int __n, const _Tp __x) [inline]¶

Return the exponential integral $ E_n(x) $. The exponential integral is given by E_n(x) = int_{1}^infty ac{e^{-xt}}{t^n} dt ] This is something of an extension.

Parameters:

Returns:

Definition at line 472 of file exp_integral.tcc.

References __expint_E1(), __expint_En_recursion(), and std::exp().

template<typename _Tp > _Tp std::tr1::__detail::__expint_asymp (const unsigned int __n, const _Tp __x) [inline]¶

Return the exponential integral $ E_n(x) $ for large argument. The exponential integral is given by E_n(x) = int_{1}^infty ac{e^{-xt}}{t^n} dt ]

This is something of an extension.

Parameters:

Returns:

Definition at line 404 of file exp_integral.tcc.

References std::abs(), and std::exp().

template<typename _Tp > _Tp std::tr1::__detail::__expint_E1 (const _Tp __x) [inline]¶

Return the exponential integral $ E_1(x) $. The exponential integral is given by E_1(x) = int_{1}^infty ac{e^{-xt}}{t} dt ]

Parameters:

Returns:

Definition at line 374 of file exp_integral.tcc.

References __expint_E1_asymp(), __expint_E1_series(), __expint_Ei(), and __expint_En_cont_frac().

Referenced by __expint(), __expint_Ei(), and __expint_En_recursion().

template<typename _Tp > _Tp std::tr1::__detail::__expint_E1_asymp (const _Tp __x) [inline]¶

Return the exponential integral $ E_1(x) $ by asymptotic expansion. The exponential integral is given by E_1(x) = int_{1}^infty ac{e^{-xt}}{t} dt ]

Parameters:

Returns:

Definition at line 114 of file exp_integral.tcc.

References std::abs(), and std::exp().

Referenced by __expint_E1().

template<typename _Tp > _Tp std::tr1::__detail::__expint_E1_series (const _Tp __x) [inline]¶

Return the exponential integral $ E_1(x) $ by series summation. This should be good for $ x < 1 $. The exponential integral is given by E_1(x) = int_{1}^{infty} ac{e^{-xt}}{t} dt ]

Parameters:

Returns:

Definition at line 77 of file exp_integral.tcc.

References std::tr1::__detail::__numeric_constants< _Tp >::__gamma_e(), std::abs(), and std::log().

Referenced by __expint_E1().

template<typename _Tp > _Tp std::tr1::__detail::__expint_Ei (const _Tp __x) [inline]¶

Return the exponential integral $ Ei(x) $. The exponential integral is given by Ei(x) = -int_{-x}^infty ac{e^t}{t} dt ]

Parameters:

Returns:

Definition at line 350 of file exp_integral.tcc.

References __expint_E1(), __expint_Ei_asymp(), __expint_Ei_series(), and std::log().

Referenced by __expint(), and __expint_E1().

template<typename _Tp > _Tp std::tr1::__detail::__expint_Ei_asymp (const _Tp __x) [inline]¶

Return the exponential integral $ Ei(x) $ by asymptotic expansion. The exponential integral is given by Ei(x) = -int_{-x}^infty ac{e^t}{t} dt ]

Parameters:

Returns:

Definition at line 317 of file exp_integral.tcc.

References std::exp().

Referenced by __expint_Ei().

template<typename _Tp > _Tp std::tr1::__detail::__expint_Ei_series (const _Tp __x) [inline]¶

Return the exponential integral $ Ei(x) $ by series summation. The exponential integral is given by Ei(x) = -int_{-x}^infty ac{e^t}{t} dt ]

Parameters:

Returns:

Definition at line 286 of file exp_integral.tcc.

References std::tr1::__detail::__numeric_constants< _Tp >::__gamma_e(), and std::log().

Referenced by __expint_Ei().

template<typename _Tp > _Tp std::tr1::__detail::__expint_En_cont_frac (const unsigned int __n, const _Tp __x) [inline]¶

Return the exponential integral $ E_n(x) $ by continued fractions. The exponential integral is given by E_n(x) = int_{1}^infty ac{e^{-xt}}{t^n} dt ]

Parameters:

Returns:

Definition at line 197 of file exp_integral.tcc.

References std::abs(), std::exp(), and std::min().

Referenced by __expint_E1().

template<typename _Tp > _Tp std::tr1::__detail::__expint_En_recursion (const unsigned int __n, const _Tp __x) [inline]¶

Return the exponential integral $ E_n(x) $ by recursion. Use upward recursion for $ x < n $ and downward recursion (Miller's algorithm) otherwise. The exponential integral is given by E_n(x) = int_{1}^infty ac{e^{-xt}}{t^n} dt ]

Parameters:

Returns:

Definition at line 242 of file exp_integral.tcc.

References __expint_E1(), and std::exp().

Referenced by __expint().

template<typename _Tp > _Tp std::tr1::__detail::__expint_En_series (const unsigned int __n, const _Tp __x) [inline]¶

Return the exponential integral $ E_n(x) $ by series summation. The exponential integral is given by E_n(x) = int_{1}^infty ac{e^{-xt}}{t^n} dt ]

Parameters:

Returns:

Definition at line 151 of file exp_integral.tcc.

References std::tr1::__detail::__numeric_constants< _Tp >::__gamma_e(), __psi(), std::abs(), and std::log().

template<typename _Tp > _Tp std::tr1::__detail::__expint_large_n (const unsigned int __n, const _Tp __x) [inline]¶

Return the exponential integral $ E_n(x) $ for large order. The exponential integral is given by E_n(x) = int_{1}^infty ac{e^{-xt}}{t^n} dt ]

This is something of an extension.

Parameters:

Returns:

Definition at line 438 of file exp_integral.tcc.

References std::abs(), and std::exp().

template<typename _Tp > _Tp std::tr1::__detail::__gamma (const _Tp __x) [inline]¶

Return $ Gamma(x) $. Parameters:

Returns:

Definition at line 333 of file gamma.tcc.

References __log_gamma(), and std::exp().

Referenced by __beta_gamma(), and __gamma_temme().

template<typename _Tp > void std::tr1::__detail::__gamma_temme (const _Tp __mu, _Tp & __gam1, _Tp & __gam2, _Tp & __gampl, _Tp & __gammi) [inline]¶

Compute the gamma functions required by the Temme series expansions of $ N_0(x) $ and $ K_0(x) $. Gamma_1 = ac{1}{2} [ac{1}{Gamma(1 - )} - ac{1}{Gamma(1 + )}] ] and Gamma_2 = ac{1}{2} [ac{1}{Gamma(1 - )} + ac{1}{Gamma(1 + )}] ] where $ -1/2 <= <= 1/2 $ is $ = 0 - N $ and $ N $. is the nearest integer to $ 0 $. The values of $ Gamma(1 + ) $ and $ Gamma(1 - ) $ are returned as well. The accuracy requirements on this are exquisite.

Parameters:

Definition at line 90 of file bessel_function.tcc.

References __gamma(), and std::abs().

Referenced by __bessel_ik(), and __bessel_jn().

template<typename _Tp > _Tp std::tr1::__detail::__hurwitz_zeta (const _Tp __a, const _Tp __s) [inline]¶

Return the Hurwitz zeta function $ ta(x,s) $ for all s != 1 and x > -1. The Hurwitz zeta function is defined by: ta(x,s) = m_{n=0}^{infty} ac{1}{(n + x)^s} ] The Riemann zeta function is a special case: ta(s) = ta(1,s) ]

Definition at line 426 of file riemann_zeta.tcc.

References __hurwitz_zeta_glob().

Referenced by __psi().

template<typename _Tp > _Tp std::tr1::__detail::__hurwitz_zeta_glob (const _Tp __a, const _Tp __s) [inline]¶

Return the Hurwitz zeta function $ ta(x,s) $ for all s != 1 and x > -1. The Hurwitz zeta function is defined by: ta(x,s) = m_{n=0}^{infty} ac{1}{(n + x)^s} ] The Riemann zeta function is a special case: ta(s) = ta(1,s) ]

This functions uses the double sum that converges for s != 1 and x > -1: ta(x,s) = ac{1}{s-1} m_{n=0}^{infty} ac{1}{n + 1} m_{k=0}^{n} (-1)^k ac{n!}{(n-k)!k!} (x+k)^{-s} ]

Definition at line 361 of file riemann_zeta.tcc.

References __log_gamma(), std::abs(), std::exp(), std::log(), and std::pow().

Referenced by __hurwitz_zeta().

template<typename _Tp > _Tp std::tr1::__detail::__hyperg (const _Tp __a, const _Tp __b, const _Tp __c, const _Tp __x) [inline]¶

Return the hypogeometric function $ _2F_1(a,b;c;x) $. The hypogeometric function is defined by _2F_1(a,b;c;x) = ac{Gamma(c)}{Gamma(a)Gamma(b)} m_{n=0}^{infty} ac{Gamma(a+n)Gamma(b+n)}{Gamma(c+n)} ac{x^n}{n!} ]

Parameters:

Returns:

Definition at line 724 of file hypergeometric.tcc.

References __hyperg_luke(), __hyperg_reflect(), __hyperg_series(), std::abs(), and std::pow().

template<typename _Tp > _Tp std::tr1::__detail::__hyperg_luke (const _Tp __a, const _Tp __b, const _Tp __c, const _Tp __xin) [inline]¶

Return the hypogeometric function $ _2F_1(a,b;c;x) $ by an iterative procedure described in Luke, Algorithms for the Computation of Mathematical Functions.

Definition at line 300 of file hypergeometric.tcc.

References std::abs(), and std::pow().

Referenced by __hyperg().

template<typename _Tp > _Tp std::tr1::__detail::__hyperg_reflect (const _Tp __a, const _Tp __b, const _Tp __c, const _Tp __x) [inline]¶

Return the hypogeometric function $ _2F_1(a,b;c;x) $ by the reflection formulae in Abramowitz & Stegun formula 15.3.6 for d = c - a - b not integral and formula 15.3.11 for d = c - a - b integral. This assumes a, b, c != negative integer. The hypogeometric function is defined by _2F_1(a,b;c;x) = ac{Gamma(c)}{Gamma(a)Gamma(b)} m_{n=0}^{infty} ac{Gamma(a+n)Gamma(b+n)}{Gamma(c+n)} ac{x^n}{n!} ]

The reflection formula for nonintegral $ d = c - a - b $ is: _2F_1(a,b;c;x) = ac{Gamma(c)Gamma(d)}{Gamma(c-a)Gamma(c-b)} _2F_1(a,b;1-d;1-x) + ac{Gamma(c)Gamma(-d)}{Gamma(a)Gamma(b)} _2F_1(c-a,c-b;1+d;1-x) ]

The reflection formula for integral $ m = c - a - b $ is: _2F_1(a,b;a+b+m;x) = ac{Gamma(m)Gamma(a+b+m)}{Gamma(a+m)Gamma(b+m)} m_{k=0}^{m-1} ac{(m+a)_k(m+b)_k}{k!(1-m)_k} - ]

Definition at line 434 of file hypergeometric.tcc.

References std::tr1::__detail::__numeric_constants< _Tp >::__gamma_e(), __hyperg_series(), __log_gamma(), __log_gamma_sign(), __psi(), std::abs(), std::exp(), and std::log().

Referenced by __hyperg().

template<typename _Tp > _Tp std::tr1::__detail::__hyperg_series (const _Tp __a, const _Tp __b, const _Tp __c, const _Tp __x) [inline]¶

Return the hypogeometric function $ _2F_1(a,b;c;x) $ by series expansion. The hypogeometric function is defined by _2F_1(a,b;c;x) = ac{Gamma(c)}{Gamma(a)Gamma(b)} m_{n=0}^{infty} ac{Gamma(a+n)Gamma(b+n)}{Gamma(c+n)} ac{x^n}{n!} ]

This works and it's pretty fast.

Parameters:

Returns:

Definition at line 266 of file hypergeometric.tcc.

References std::abs().

Referenced by __hyperg(), and __hyperg_reflect().

template<typename _Tp > _Tp std::tr1::__detail::__laguerre (const unsigned int __n, const _Tp __x) [inline]¶

This routine returns the Laguerre polynomial of order n: $ L_n(x) $. The Laguerre polynomial is defined by: L_n(x) = ac{e^x}{n!} ac{d^n}{dx^n} (x^ne^{-x}) ]

Parameters:

Returns:

Definition at line 320 of file poly_laguerre.tcc.

template<typename _Tp > _Tp std::tr1::__detail::__log_bincoef (const unsigned int __n, const unsigned int __k) [inline]¶

Return the logarithm of the binomial coefficient. The binomial coefficient is given by: ac{n!}{(n-k)! k!} ]. Parameters:

Returns:

Definition at line 279 of file gamma.tcc.

References __log_gamma(), and std::log().

template<typename _Tp > _Tp std::tr1::__detail::__log_gamma (const _Tp __x) [inline]¶

Return $ log(|Gamma(x)|) $. This will return values even for $ x < 0 $. To recover the sign of $ Gamma(x) $ for any argument use __log_gamma_sign. Parameters:

Returns:

Definition at line 221 of file gamma.tcc.

References std::tr1::__detail::__numeric_constants< _Tp >::__lnpi(), __log_gamma_lanczos(), std::abs(), std::log(), and std::sin().

Referenced by __beta_lgamma(), __cyl_bessel_ij_series(), __gamma(), __hurwitz_zeta_glob(), __hyperg_reflect(), __log_bincoef(), __poly_laguerre_large_n(), __psi(), __riemann_zeta(), __riemann_zeta_glob(), and __sph_legendre().

template<typename _Tp > _Tp std::tr1::__detail::__log_gamma_bernoulli (const _Tp __x) [inline]¶

Return $log(Gamma(x))$ by asymptotic expansion with Bernoulli number coefficients. This is like Sterling's approximation. Parameters:

Returns:

Definition at line 149 of file gamma.tcc.

References std::__lg(), and std::log().

template<typename _Tp > _Tp std::tr1::__detail::__log_gamma_lanczos (const _Tp __x) [inline]¶

Return $log(Gamma(x))$ by the Lanczos method. This method dominates all others on the positive axis I think. Parameters:

Returns:

Definition at line 177 of file gamma.tcc.

References std::tr1::__detail::__numeric_constants< _Tp >::__euler(), and std::log().

Referenced by __log_gamma().

template<typename _Tp > _Tp std::tr1::__detail::__log_gamma_sign (const _Tp __x) [inline]¶

Return the sign of $ Gamma(x) $. At nonpositive integers zero is returned. Parameters:

Returns:

Definition at line 248 of file gamma.tcc.

References std::sin().

Referenced by __hyperg_reflect().

template<typename _Tp > _Tp std::tr1::__detail::__poly_hermite (const unsigned int __n, const _Tp __x) [inline]¶

This routine returns the Hermite polynomial of order n: $ H_n(x) $. The Hermite polynomial is defined by: H_n(x) = (-1)^n e^{x^2} ac{d^n}{dx^n} e^{-x^2} ]

Parameters:

Returns:

Definition at line 112 of file poly_hermite.tcc.

References __poly_hermite_recursion().

template<typename _Tp > _Tp std::tr1::__detail::__poly_hermite_recursion (const unsigned int __n, const _Tp __x) [inline]¶

This routine returns the Hermite polynomial of order n: $ H_n(x) $ by recursion on n. The Hermite polynomial is defined by: H_n(x) = (-1)^n e^{x^2} ac{d^n}{dx^n} e^{-x^2} ]

Parameters:

Returns:

Definition at line 70 of file poly_hermite.tcc.

Referenced by __poly_hermite().

template<typename _Tpa , typename _Tp > _Tp std::tr1::__detail::__poly_laguerre (const unsigned int __n, const _Tpa __alpha1, const _Tp __x) [inline]¶

This routine returns the associated Laguerre polynomial of order n, degree $ lpha $: $ L_n^alpha(x) $. The associated Laguerre function is defined by L_n^lpha(x) = ac{(lpha + 1)_n}{n!} _1F_1(-n; lpha + 1; x) ] where $ (lpha)_n $ is the Pochhammer symbol and $ _1F_1(a; c; x) $ is the confluent hypergeometric function.

The associated Laguerre polynomial is defined for integral $ lpha = m $ by: L_n^m(x) = (-1)^m ac{d^m}{dx^m} L_{n + m}(x) ] where the Laguerre polynomial is defined by: L_n(x) = ac{e^x}{n!} ac{d^n}{dx^n} (x^ne^{-x}) ]

Parameters:

Returns:

Definition at line 244 of file poly_laguerre.tcc.

References __poly_laguerre_hyperg(), __poly_laguerre_large_n(), and __poly_laguerre_recursion().

template<typename _Tpa , typename _Tp > _Tp std::tr1::__detail::__poly_laguerre_hyperg (const unsigned int __n, const _Tpa __alpha1, const _Tp __x) [inline]¶

Evaluate the polynomial based on the confluent hypergeometric function in a safe way, with no restriction on the arguments. The associated Laguerre function is defined by L_n^lpha(x) = ac{(lpha + 1)_n}{n!} _1F_1(-n; lpha + 1; x) ] where $ (lpha)_n $ is the Pochhammer symbol and $ _1F_1(a; c; x) $ is the confluent hypergeometric function.

This function assumes x != 0.

This is from the GNU Scientific Library.

Definition at line 127 of file poly_laguerre.tcc.

References std::abs().

Referenced by __poly_laguerre().

template<typename _Tpa , typename _Tp > _Tp std::tr1::__detail::__poly_laguerre_large_n (const unsigned __n, const _Tpa __alpha1, const _Tp __x) [inline]¶

This routine returns the associated Laguerre polynomial of order $ n $, degree $ lpha $ for large n. Abramowitz & Stegun, 13.5.21. Parameters:

Returns:

This is from the GNU Scientific Library.

Definition at line 72 of file poly_laguerre.tcc.

References __log_gamma(), std::tr1::__detail::__numeric_constants< _Tp >::__pi_2(), std::exp(), std::log(), std::sin(), and std::sqrt().

Referenced by __poly_laguerre().

template<typename _Tpa , typename _Tp > _Tp std::tr1::__detail::__poly_laguerre_recursion (const unsigned int __n, const _Tpa __alpha1, const _Tp __x) [inline]¶

This routine returns the associated Laguerre polynomial of order $ n $, degree $ lpha $: $ L_n^lpha(x) $ by recursion. The associated Laguerre function is defined by L_n^lpha(x) = ac{(lpha + 1)_n}{n!} _1F_1(-n; lpha + 1; x) ] where $ (lpha)_n $ is the Pochhammer symbol and $ _1F_1(a; c; x) $ is the confluent hypergeometric function.

The associated Laguerre polynomial is defined for integral $ lpha = m $ by: L_n^m(x) = (-1)^m ac{d^m}{dx^m} L_{n + m}(x) ] where the Laguerre polynomial is defined by: L_n(x) = ac{e^x}{n!} ac{d^n}{dx^n} (x^ne^{-x}) ]

Parameters:

Returns:

Definition at line 184 of file poly_laguerre.tcc.

Referenced by __poly_laguerre().

template<typename _Tp > _Tp std::tr1::__detail::__poly_legendre_p (const unsigned int __l, const _Tp __x) [inline]¶

Return the Legendre polynomial by recursion on order $ l $. The Legendre function of $ l $ and $ x $, $ P_l(x) $, is defined by: P_l(x) = ac{1}{2^l l!}ac{d^l}{dx^l}(x^2 - 1)^{l} ]

Parameters:

Definition at line 76 of file legendre_function.tcc.

Referenced by __assoc_legendre_p(), and __sph_legendre().

template<typename _Tp > _Tp std::tr1::__detail::__psi (const unsigned int __n, const _Tp __x) [inline]¶

Return the polygamma function $ si^{(n)}(x)
$. The polygamma function is related to the Hurwitz zeta function: si^{(n)}(x)
= (-1)^{n+1} m! ta(m+1,x) ]

Definition at line 444 of file gamma.tcc.

References __hurwitz_zeta(), __log_gamma(), __psi(), and std::exp().

template<typename _Tp > _Tp std::tr1::__detail::__psi (const _Tp __x) [inline]¶

Return the digamma function. The digamma or $ si(x)
$ function is defined by si(x)
= ac{Gamma'(x)}{Gamma(x)} ] For negative argument the reflection formula is used: si(x)
= si(1-x)
- i
ot(i
x) ].

Definition at line 415 of file gamma.tcc.

References std::tr1::__detail::__numeric_constants< _Tp >::__pi(), __psi_asymp(), __psi_series(), std::abs(), std::cos(), and std::sin().

Referenced by __expint_En_series(), __hyperg_reflect(), and __psi().

template<typename _Tp > _Tp std::tr1::__detail::__psi_asymp (const _Tp __x) [inline]¶

Return the digamma function for large argument. The digamma or $ si(x)
$ function is defined by si(x)
= ac{Gamma'(x)}{Gamma(x)} ]. The asymptotic series is given by: si(x)
= ac{1}{2x} - m_{n=1}^{infty} ac{B_{2n}}{2 n x^{2n}} ]

Definition at line 384 of file gamma.tcc.

References std::abs(), and std::log().

Referenced by __psi().

template<typename _Tp > _Tp std::tr1::__detail::__psi_series (const _Tp __x) [inline]¶

Return the digamma function by series expansion. The digamma or $ si(x)
$ function is defined by si(x)
= ac{Gamma'(x)}{Gamma(x)} ]. The series is given by: si(x)
= -mma_E - ac{1}{x} m_{k=1}^{infty} ac{x}{k(x + k)} ]

Definition at line 354 of file gamma.tcc.

References std::tr1::__detail::__numeric_constants< _Tp >::__gamma_e(), and std::abs().

Referenced by __psi().

template<typename _Tp > _Tp std::tr1::__detail::__riemann_zeta (const _Tp __s) [inline]¶

Return the Riemann zeta function $ ta(s) $. The Riemann zeta function is defined by: ta(s) = m_{k=1}^{infty} k^{-s} for s > 1 ac{(2i)^s}{pi}
sin(ac{i
s}{2}) Gamma (1 - s) ta (1 - s) for s < 1 ] For s < 1 use the reflection formula: ta(s) = 2^s i^{s-1}
Gamma(1-s) ta(1-s) ]

Definition at line 289 of file riemann_zeta.tcc.

References __log_gamma(), std::tr1::__detail::__numeric_constants< _Tp >::__pi(), __riemann_zeta_glob(), __riemann_zeta_product(), __riemann_zeta_sum(), std::exp(), std::pow(), and std::sin().

template<typename _Tp > _Tp std::tr1::__detail::__riemann_zeta_alt (const _Tp __s) [inline]¶

Evaluate the Riemann zeta function $ ta(s) $ by an alternate series for s > 0. The Riemann zeta function is defined by: ta(s) = m_{k=1}^{infty} ac{1}{k^{s}} for s > 1 ] For s < 1 use the reflection formula: ta(s) = 2^s i^{s-1}
Gamma(1-s) ta(1-s) ]

Definition at line 111 of file riemann_zeta.tcc.

References std::abs(), and std::pow().

template<typename _Tp > _Tp std::tr1::__detail::__riemann_zeta_glob (const _Tp __s) [inline]¶

Evaluate the Riemann zeta function by series for all s != 1. Convergence is great until largish negative numbers. Then the convergence of the > 0 sum gets better. The series is: ta(s) = ac{1}{1-2^{1-s}} m_{n=0}^{infty} ac{1}{2^{n+1}} m_{k=0}^{n} (-1)^k ac{n!}{(n-k)!k!} (k+1)^{-s} ] Havil 2003, p. 206.

The Riemann zeta function is defined by: ta(s) = m_{k=1}^{infty} ac{1}{k^{s}} for s > 1 ] For s < 1 use the reflection formula: ta(s) = 2^s i^{s-1}
Gamma(1-s) ta(1-s) ]

Definition at line 153 of file riemann_zeta.tcc.

References __log_gamma(), std::tr1::__detail::__numeric_constants< _Tp >::__pi(), std::abs(), std::exp(), std::log(), std::pow(), and std::sin().

Referenced by __riemann_zeta().

template<typename _Tp > _Tp std::tr1::__detail::__riemann_zeta_product (const _Tp __s) [inline]¶

Compute the Riemann zeta function $ ta(s) $ using the product over prime factors. ta(s) = Pi_{i=1}^infty ac{1}{1 - p_i^{-s}} ] where $ {p_i} $ are the prime numbers. The Riemann zeta function is defined by: ta(s) = m_{k=1}^{infty} ac{1}{k^{s}} for s > 1 ] For s < 1 use the reflection formula: ta(s) = 2^s i^{s-1}
Gamma(1-s) ta(1-s) ]

Definition at line 248 of file riemann_zeta.tcc.

References std::pow().

Referenced by __riemann_zeta().

template<typename _Tp > _Tp std::tr1::__detail::__riemann_zeta_sum (const _Tp __s) [inline]¶

Compute the Riemann zeta function $ ta(s) $ by summation for s > 1. The Riemann zeta function is defined by: ta(s) = m_{k=1}^{infty} ac{1}{k^{s}} for s > 1 ] For s < 1 use the reflection formula: ta(s) = 2^s i^{s-1}
Gamma(1-s) ta(1-s) ]

Definition at line 74 of file riemann_zeta.tcc.

References std::pow().

Referenced by __riemann_zeta().

template<typename _Tp > _Tp std::tr1::__detail::__sph_bessel (const unsigned int __n, const _Tp __x) [inline]¶

Return the spherical Bessel function $ j_n(x) $ of order n. The spherical Bessel function is defined by: j_n(x) = ac{i}{2x}
ight) ^{1/2} J_{n+1/2}(x) ]

Parameters:

Returns:

Definition at line 568 of file bessel_function.tcc.

References __sph_bessel_jn().

template<typename _Tp > void std::tr1::__detail::__sph_bessel_ik (const unsigned int __n, const _Tp __x, _Tp & __i_n, _Tp & __k_n, _Tp & __ip_n, _Tp & __kp_n) [inline]¶

Compute the spherical modified Bessel functions $ i_n(x) $ and $ k_n(x) $ and their first derivatives $ i'_n(x) $ and $ k'_n(x) $ respectively. Parameters:

Definition at line 335 of file modified_bessel_func.tcc.

References __bessel_ik(), std::tr1::__detail::__numeric_constants< _Tp >::__sqrtpio2(), and std::sqrt().

template<typename _Tp > void std::tr1::__detail::__sph_bessel_jn (const unsigned int __n, const _Tp __x, _Tp & __j_n, _Tp & __n_n, _Tp & __jp_n, _Tp & __np_n) [inline]¶

Compute the spherical Bessel $ j_n(x) $ and Neumann $ n_n(x) $ functions and their first derivatives $ j'_n(x) $ and $ n'_n(x) $ respectively. Parameters:

Definition at line 533 of file bessel_function.tcc.

References __bessel_jn(), std::tr1::__detail::__numeric_constants< _Tp >::__sqrtpio2(), and std::sqrt().

Referenced by __sph_bessel(), and __sph_neumann().

template<typename _Tp > _Tp std::tr1::__detail::__sph_legendre (const unsigned int __l, const unsigned int __m, const _Tp __theta) [inline]¶

Return the spherical associated Legendre function. The spherical associated Legendre function of $ l $, $ m $, and $ heta $ is defined as $ Y_l^m(heta,0) $ where Y_l^m(heta,hi)
= (-1)^m[ac{(2l+1)}{4i}
ac{(l-m)!}{(l+m)!}] P_l^m(osheta) \xp^{imhi}
] is the spherical harmonic function and $ P_l^m(x) $ is the associated Legendre function.

This function differs from the associated Legendre function by argument ($x = os(heta)$) and by a normalization factor but this factor is rather large for large $ l $ and $ m $ and so this function is stable for larger differences of $ l $ and $ m $.

Parameters:

Definition at line 213 of file legendre_function.tcc.

References std::tr1::__detail::__numeric_constants< _Tp >::__lnpi(), __log_gamma(), __poly_legendre_p(), std::cos(), std::exp(), std::log(), and std::sqrt().

template<typename _Tp > _Tp std::tr1::__detail::__sph_neumann (const unsigned int __n, const _Tp __x) [inline]¶

Return the spherical Neumann function $ n_n(x) $. The spherical Neumann function is defined by: n_n(x) = ac{i}{2x}
ight) ^{1/2} N_{n+1/2}(x) ]

Parameters:

Returns:

Definition at line 606 of file bessel_function.tcc.

References __sph_bessel_jn().