std::tr1::__detail - Online Linux Manual PageSection : 3
Updated : 19 Jun 2018
Source : libstdc++
NAMEstd::tr1::__detail − Implementation details not part of the namespace std::tr1 interface.
SYNOPSIS
Classesstruct __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.
Functionstemplate<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)
Detailed DescriptionImplementation details not part of the namespace std::tr1 interface.
Function Documentation
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: __n The order of the Airy functions.
__x The argument of the Airy functions.
__i_n The output Airy function.
__k_n The output Airy function.
__ip_n The output derivative of the Airy function.
__kp_n The output derivative of the Airy function. 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 $ \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: \[ L_n(x) = \frac{e^x}{n!} \frac{d^n}{dx^n} (x^ne^{-x}) \] Parameters: __n The order of the Laguerre polynomial.
__m The degree of the Laguerre polynomial.
__x The argument of the Laguerre polynomial. Returns: The value of the associated Laguerre polynomial of order n, degree m, and argument x. 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: \[ P_l^m(x) = (1 - x^2)^{m/2}\frac{d^m}{dx^m}P_l(x) \] Parameters: l The order of the associated Legendre function. $ l >= 0 $.
m The order of the associated Legendre function. $ m <= l $.
x The argument of the associated Legendre function. $ |x| <= 1 $. 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: __n the order n of the Bernoulli number. Returns: The Bernoulli number of order n. 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: __n the order n of the Bernoulli number. Returns: The Bernoulli number of order n. 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_\nu(x) $ and $ K_\nu(x) $ and their first derivatives $ I'_\nu(x) $ and $ K'_\nu(x) $ respectively. These four functions are computed together for numerical stability. \fBParameters:\fP __nu The order of the Bessel functions.
__x The argument of the Bessel functions.
__Inu The output regular modified Bessel function.
__Knu The output irregular modified Bessel function.
__Ipnu The output derivative of the regular modified Bessel function.
__Kpnu The output derivative of the irregular modified Bessel function. 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_\nu(x) $ and Neumann $ N_\nu(x) $ functions and their first derivatives $ J'_\nu(x) $ and $ N'_\nu(x) $ respectively. These four functions are computed together for numerical stability. \fBParameters:\fP __nu The order of the Bessel functions.
__x The argument of the Bessel functions.
__Jnu The output Bessel function of the first kind.
__Nnu The output Neumann function (Bessel function of the second kind).
__Jpnu The output derivative of the Bessel function of the first kind.
__Npnu The output derivative of the Neumann function. 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 \[ B(x,y) = \frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)} \] Parameters: __x The first argument of the beta function.
__y The second argument of the beta function. Returns: The beta function. 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 \[ B(x,y) = \frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)} \] Parameters: __x The first argument of the beta function.
__y The second argument of the beta function. Returns: The beta function. 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 \[ B(x,y) = \frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)} \] Parameters: __x The first argument of the beta function.
__y The second argument of the beta function. Returns: The beta function. 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 \[ B(x,y) = \frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)} \] Parameters: __x The first argument of the beta function.
__y The second argument of the beta function. Returns: The beta function. 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: \[ \left( \right) = \frac{n!}{(n-k)! k!} \]. \fBParameters:\fP __n The first argument of the binomial coefficient.
__k The second argument of the binomial coefficient. Returns: The binomial coefficient. 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 \[ K(k) = F(k,\pi/2) = \int_0^{\pi/2}\frac{d\theta} {\sqrt{1 - k^2 sin^2\theta}} \] where $ F(k,\phi) $ is the incomplete elliptic integral of the first kind. Parameters: __k The argument of the complete elliptic function. Returns: The complete elliptic function of the first kind. 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 \[ K(k) = F(k,\pi/2) = \int_0^{\pi/2}\frac{d\theta} {\sqrt{1 - k^2sin^2\theta}} \] 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: __k The argument of the complete elliptic function. Returns: The complete elliptic function of the first kind. 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 \[ E(k,\pi/2) = \int_0^{\pi/2}\sqrt{1 - k^2 sin^2\theta} \] Parameters: __k The argument of the complete elliptic function. Returns: The complete elliptic function of the second kind. 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 \[ E(k,\pi/2) = \int_0^{\pi/2}\sqrt{1 - k^2 sin^2\theta} \] 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: __k The argument of the complete elliptic function. Returns: The complete elliptic function of the second kind. 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,\nu) = \Pi(k,\nu,\pi/2) $ using the Carlson formulation. The complete elliptic integral of the third kind is defined as \[ \Pi(k,\nu) = \int_0^{\pi/2} \frac{d\theta} {(1 - \nu \sin^2\theta)\sqrt{1 - k^2 \sin^2\theta}} \] Parameters: __k The argument of the elliptic function.
__nu The second argument of the elliptic function. Returns: The complete elliptic function of the third kind. 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 Handle b == nonpositive integer blowup - return NaN. Parameters: __a The 'numerator' parameter.
__c The 'denominator' parameter.
__x The argument of the confluent hypergeometric function. Returns: The confluent hypergeometric function. 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. \[ _1F_1(a;c;x) = \frac{\Gamma(c)}{\Gamma(a)} \sum_{n=0}^{\infty} \frac{\Gamma(a+n)}{\Gamma(c+n)} \frac{x^n}{n!} \].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: __a The 'numerator' parameter.
__c The 'denominator' parameter.
__x The argument of the confluent hypergeometric function. Returns: The confluent hypergeometric function. 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 $ \nu $: $ I_{\nu}(x) $. The regular modified cylindrical Bessel function is: \[ I_{\nu}(x) = \sum_{k=0}^{\infty} \frac{(x/2)^{\nu + 2k}}{k!\Gamma(\nu+k+1)} \] Parameters: __nu The order of the regular modified Bessel function.
__x The argument of the regular modified Bessel function. Returns: The output regular modified Bessel function. 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 $ \nu $: $ J_{\nu} $ or $ I_{\nu} $ by series expansion. The modified cylindrical Bessel function is: \[ Z_{\nu}(x) = \sum_{k=0}^{\infty} \frac{\sigma^k (x/2)^{\nu + 2k}}{k!\Gamma(\nu+k+1)} \] where $ \sigma = +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: __nu The order of the Bessel function.
__x The argument of the Bessel function.
__sgn The sign of the alternate terms -1 for the Bessel function of the first kind. +1 for the modified Bessel function of the first kind. Returns: The output Bessel function. 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 $ \nu $: $ J_{\nu}(x) $. The cylindrical Bessel function is: \[ J_{\nu}(x) = \sum_{k=0}^{\infty} \frac{(-1)^k (x/2)^{\nu + 2k}}{k!\Gamma(\nu+k+1)} \] Parameters: __nu The order of the Bessel function.
__x The argument of the Bessel function. Returns: The output Bessel function. 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_{\nu} $, $ N_{\nu} $. 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: __nu The order of the Bessel functions.
__x The argument of the Bessel functions.
__Jnu The output Bessel function of the first kind.
__Nnu The output Neumann function (Bessel function of the second kind). 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_{\nu}(x) $ of order $ \nu $. 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} $. Parameters: __nu The order of the irregular modified Bessel function.
__x The argument of the irregular modified Bessel function. Returns: The output irregular modified Bessel function. 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 $ \nu $: $ N_{\nu}(x) $. The Neumann function is defined by: \[ N_{\nu}(x) = \frac{J_{\nu}(x) \cos \nu\pi - J_{-\nu}(x)} {\sin \nu\pi} \] where for integral $ \nu = n $ a limit is taken: $ lim_{\nu \to n} $. Parameters: __nu The order of the Neumann function.
__x The argument of the Neumann function. Returns: The output Neumann function. 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,\phi) $ using the Carlson formulation. The incomplete elliptic integral of the first kind is defined as \[ F(k,\phi) = \int_0^{\phi}\frac{d\theta} {\sqrt{1 - k^2 sin^2\theta}} \] Parameters: __k The argument of the elliptic function.
__phi The integral limit argument of the elliptic function. Returns: The elliptic function of the first kind. 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,\phi) $ using the Carlson formulation. The incomplete elliptic integral of the second kind is defined as \[ E(k,\phi) = \int_0^{\phi} \sqrt{1 - k^2 sin^2\theta} \] Parameters: __k The argument of the elliptic function.
__phi The integral limit argument of the elliptic function. Returns: The elliptic function of the second kind. 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,\nu,\phi) $ using the Carlson formulation. The incomplete elliptic integral of the third kind is defined as \[ \Pi(k,\nu,\phi) = \int_0^{\phi} \frac{d\theta} {(1 - \nu \sin^2\theta) \sqrt{1 - k^2 \sin^2\theta}} \] Parameters: __k The argument of the elliptic function.
__nu The second argument of the elliptic function.
__phi The integral limit argument of the elliptic function. Returns: The elliptic function of the third kind. 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: \[ R_C(x,y) = \frac{1}{2} \int_0^\infty \frac{dt}{(t + x)^{1/2}(t + y)} \] 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: __x The first argument.
__y The second argument. Returns: The Carlson elliptic function. 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: \[ R_D(x,y,z) = \frac{3}{2} \int_0^\infty \frac{dt}{(t + x)^{1/2}(t + y)^{1/2}(t + z)^{3/2}} \] 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: __x The first of two symmetric arguments.
__y The second of two symmetric arguments.
__z The third argument. Returns: The Carlson elliptic function of the second kind. 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: \[ R_F(x,y,z) = \frac{1}{2} \int_0^\infty \frac{dt}{(t + x)^{1/2}(t + y)^{1/2}(t + z)^{1/2}} \] Parameters: __x The first of three symmetric arguments.
__y The second of three symmetric arguments.
__z The third of three symmetric arguments. Returns: The Carlson elliptic function of the first kind. 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: \[ R_J(x,y,z,p) = \frac{3}{2} \int_0^\infty \frac{dt}{(t + x)^{1/2}(t + y)^{1/2}(t + z)^{1/2}(t + p)} \] 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: __x The first of three symmetric arguments.
__y The second of three symmetric arguments.
__z The third of three symmetric arguments.
__p The fourth argument. Returns: The Carlson elliptic function of the fourth kind. 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 \[ Ei(x) = -\int_{-x}^\infty \frac{e^t}{t} dt \] Parameters: __x The argument of the exponential integral function. Returns: The exponential integral. 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 \[ E_n(x) = \int_{1}^\infty \frac{e^{-xt}}{t^n} dt \] This is something of an extension. Parameters: __n The order of the exponential integral function.
__x The argument of the exponential integral function. Returns: The exponential integral. 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 \[ E_n(x) = \int_{1}^\infty \frac{e^{-xt}}{t^n} dt \] This is something of an extension. Parameters: __n The order of the exponential integral function.
__x The argument of the exponential integral function. Returns: The exponential integral. 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 \[ E_1(x) = \int_{1}^\infty \frac{e^{-xt}}{t} dt \] Parameters: __x The argument of the exponential integral function. Returns: The exponential integral. 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 \[ E_1(x) = \int_{1}^\infty \frac{e^{-xt}}{t} dt \] Parameters: __x The argument of the exponential integral function. Returns: The exponential integral. 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 \[ E_1(x) = \int_{1}^{\infty} \frac{e^{-xt}}{t} dt \] Parameters: __x The argument of the exponential integral function. Returns: The exponential integral. 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 \[ Ei(x) = -\int_{-x}^\infty \frac{e^t}{t} dt \] Parameters: __x The argument of the exponential integral function. Returns: The exponential integral. 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 \[ Ei(x) = -\int_{-x}^\infty \frac{e^t}{t} dt \] Parameters: __x The argument of the exponential integral function. Returns: The exponential integral. 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 \[ Ei(x) = -\int_{-x}^\infty \frac{e^t}{t} dt \] Parameters: __x The argument of the exponential integral function. Returns: The exponential integral. 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 \[ E_n(x) = \int_{1}^\infty \frac{e^{-xt}}{t^n} dt \] Parameters: __n The order of the exponential integral function.
__x The argument of the exponential integral function. Returns: The exponential integral. 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 \[ E_n(x) = \int_{1}^\infty \frac{e^{-xt}}{t^n} dt \] Parameters: __n The order of the exponential integral function.
__x The argument of the exponential integral function. Returns: The exponential integral. 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 \[ E_n(x) = \int_{1}^\infty \frac{e^{-xt}}{t^n} dt \] Parameters: __n The order of the exponential integral function.
__x The argument of the exponential integral function. Returns: The exponential integral. 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 \[ E_n(x) = \int_{1}^\infty \frac{e^{-xt}}{t^n} dt \] This is something of an extension. Parameters: __n The order of the exponential integral function.
__x The argument of the exponential integral function. Returns: The exponential integral. 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) $. \fBParameters:\fP __x The argument of the gamma function. Returns: The gamma function. 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_\nu(x) $ and $ K_\nu(x) $. \[ \Gamma_1 = \frac{1}{2\mu} [\frac{1}{\Gamma(1 - \mu)} - \frac{1}{\Gamma(1 + \mu)}] \] and \[ \Gamma_2 = \frac{1}{2} [\frac{1}{\Gamma(1 - \mu)} + \frac{1}{\Gamma(1 + \mu)}] \] where $ -1/2 <= \mu <= 1/2 $ is $ \mu = \nu - N $ and $ N $. is the nearest integer to $ \nu $. The values of $ \Gamma(1 + \mu) $ and $ \Gamma(1 - \mu) $ are returned as well. The accuracy requirements on this are exquisite. Parameters: __mu The input parameter of the gamma functions.
__gam1 The output function $ \Gamma_1(\mu) $
__gam2 The output function $ \Gamma_2(\mu) $
__gampl The output function $ \Gamma(1 + \mu) $
__gammi The output function $ \Gamma(1 - \mu) $ 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 $ eta(x,s) $ for all s != 1 and x > -1. The Hurwitz zeta function is defined by: \[ \zeta(x,s) = \sum_{n=0}^{\infty} \frac{1}{(n + x)^s} \] The Riemann zeta function is a special case: \[ \zeta(s) = \zeta(1,s) \] 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 $ eta(x,s) $ for all s != 1 and x > -1. The Hurwitz zeta function is defined by: \[ \zeta(x,s) = \sum_{n=0}^{\infty} \frac{1}{(n + x)^s} \] The Riemann zeta function is a special case: \[ \zeta(s) = \zeta(1,s) \] This functions uses the double sum that converges for s != 1 and x > -1: \[ \zeta(x,s) = \frac{1}{s-1} \sum_{n=0}^{\infty} \frac{1}{n + 1} \sum_{k=0}^{n} (-1)^k \frac{n!}{(n-k)!k!} (x+k)^{-s} \] 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 \[ _2F_1(a,b;c;x) = \frac{\Gamma(c)}{\Gamma(a)\Gamma(b)} \sum_{n=0}^{\infty} \frac{\Gamma(a+n)\Gamma(b+n)}{\Gamma(c+n)} \frac{x^n}{n!} \] Parameters: __a The first 'numerator' parameter.
__a The second 'numerator' parameter.
__c The 'denominator' parameter.
__x The argument of the confluent hypergeometric function. Returns: The confluent hypergeometric function. 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 \[ _2F_1(a,b;c;x) = \frac{\Gamma(c)}{\Gamma(a)\Gamma(b)} \sum_{n=0}^{\infty} \frac{\Gamma(a+n)\Gamma(b+n)}{\Gamma(c+n)} \frac{x^n}{n!} \] The reflection formula for nonintegral $ d = c - a - b $ is: \[ _2F_1(a,b;c;x) = \frac{\Gamma(c)\Gamma(d)}{\Gamma(c-a)\Gamma(c-b)} _2F_1(a,b;1-d;1-x) + \frac{\Gamma(c)\Gamma(-d)}{\Gamma(a)\Gamma(b)} _2F_1(c-a,c-b;1+d;1-x) \] The reflection formula for integral $ m = c - a - b $ is: \[ _2F_1(a,b;a+b+m;x) = \frac{\Gamma(m)\Gamma(a+b+m)}{\Gamma(a+m)\Gamma(b+m)} \sum_{k=0}^{m-1} \frac{(m+a)_k(m+b)_k}{k!(1-m)_k} - \] 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 \[ _2F_1(a,b;c;x) = \frac{\Gamma(c)}{\Gamma(a)\Gamma(b)} \sum_{n=0}^{\infty} \frac{\Gamma(a+n)\Gamma(b+n)}{\Gamma(c+n)} \frac{x^n}{n!} \] This works and it's pretty fast. Parameters: __a The first 'numerator' parameter.
__a The second 'numerator' parameter.
__c The 'denominator' parameter.
__x The argument of the confluent hypergeometric function. Returns: The confluent hypergeometric function. 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: \[ L_n(x) = \frac{e^x}{n!} \frac{d^n}{dx^n} (x^ne^{-x}) \] Parameters: __n The order of the Laguerre polynomial.
__x The argument of the Laguerre polynomial. Returns: The value of the Laguerre polynomial of order n and argument x. 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: \[ \left( \right) = \frac{n!}{(n-k)! k!} \]. \fBParameters:\fP __n The first argument of the binomial coefficient.
__k The second argument of the binomial coefficient. Returns: The binomial coefficient. 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 \fI__log_gamma_sign\fP. \fBParameters:\fP __x The argument of the log of the gamma function. Returns: The logarithm of the gamma function. 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. \fBParameters:\fP __x The argument of the log of the gamma function. Returns: The logarithm of the gamma function. 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. \fBParameters:\fP __x The argument of the log of the gamma function. Returns: The logarithm of the gamma function. 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. \fBParameters:\fP __x The argument of the gamma function. Returns: The sign of the gamma function. 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: \[ H_n(x) = (-1)^n e^{x^2} \frac{d^n}{dx^n} e^{-x^2} \] Parameters: __n The order of the Hermite polynomial.
__x The argument of the Hermite polynomial. Returns: The value of the Hermite polynomial of order n and argument x. 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: \[ H_n(x) = (-1)^n e^{x^2} \frac{d^n}{dx^n} e^{-x^2} \] Parameters: __n The order of the Hermite polynomial.
__x The argument of the Hermite polynomial. Returns: The value of the Hermite polynomial of order n and argument x. 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 $ \alpha $: $ L_n^alpha(x) $. The associated Laguerre function 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 $ \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: \[ L_n(x) = \frac{e^x}{n!} \frac{d^n}{dx^n} (x^ne^{-x}) \] Parameters: __n The order of the Laguerre function.
__alpha The degree of the Laguerre function.
__x The argument of the Laguerre function. Returns: The value of the Laguerre function of order n, degree $ \alpha $, and argument x. 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 \[ L_n^\alpha(x) = \frac{(\alpha + 1)_n}{n!} _1F_1(-n; \alpha + 1; x) \] where $ (\alpha)_n $ is the Pochhammer symbol and $ _1F_1(a; c; x) $ is the confluent hypergeometric function. 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 $ \alpha $ for large n. Abramowitz & Stegun, 13.5.21. \fBParameters:\fP __n The order of the Laguerre function.
__alpha The degree of the Laguerre function.
__x The argument of the Laguerre function. Returns: The value of the Laguerre function of order n, degree $ \alpha $, and argument x. 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 $ \alpha $: $ L_n^\alpha(x) $ by recursion. The associated Laguerre function 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 $ \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: \[ L_n(x) = \frac{e^x}{n!} \frac{d^n}{dx^n} (x^ne^{-x}) \] Parameters: __n The order of the Laguerre function.
__alpha The degree of the Laguerre function.
__x The argument of the Laguerre function. Returns: The value of the Laguerre function of order n, degree $ \alpha $, and argument x. 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: \[ P_l(x) = \frac{1}{2^l l!}\frac{d^l}{dx^l}(x^2 - 1)^{l} \] Parameters: l The order of the Legendre polynomial. $l >= 0$.
x The argument of the Legendre polynomial. $|x| <= 1$. 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 $ \psi^{(n)}(x) $. The polygamma function is related to the Hurwitz zeta function: \[ \psi^{(n)}(x) = (-1)^{n+1} m! \zeta(m+1,x) \] 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 $ \psi(x) $ function is defined by \[ \psi(x) = \frac{\Gamma'(x)}{\Gamma(x)} \] For negative argument the reflection formula is used: \[ \psi(x) = \psi(1-x) - \pi \cot(\pi x) \]. 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 $ \psi(x) $ function is defined by \[ \psi(x) = \frac{\Gamma'(x)}{\Gamma(x)} \]. The asymptotic series is given by: \[ \psi(x) = \ln(x) - \frac{1}{2x} - \sum_{n=1}^{\infty} \frac{B_{2n}}{2 n x^{2n}} \] 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 $ \psi(x) $ function is defined by \[ \psi(x) = \frac{\Gamma'(x)}{\Gamma(x)} \]. The series is given by: \[ \psi(x) = -\gamma_E - \frac{1}{x} \sum_{k=1}^{\infty} \frac{x}{k(x + k)} \] 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 $ eta(s) $. The Riemann zeta function is defined by: \[ \zeta(s) = \sum_{k=1}^{\infty} k^{-s} for s > 1 \frac{(2\pi)^s}{pi} sin(\frac{\pi s}{2}) \Gamma (1 - s) \zeta (1 - s) for s < 1 \] For s < 1 use the reflection formula: \[ \zeta(s) = 2^s \pi^{s-1} \Gamma(1-s) \zeta(1-s) \] 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 $ eta(s) $ by an alternate series for s > 0. The Riemann zeta function is defined by: \[ \zeta(s) = \sum_{k=1}^{\infty} \frac{1}{k^{s}} for s > 1 \] For s < 1 use the reflection formula: \[ \zeta(s) = 2^s \pi^{s-1} \Gamma(1-s) \zeta(1-s) \] 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: \[ \zeta(s) = \frac{1}{1-2^{1-s}} \sum_{n=0}^{\infty} \frac{1}{2^{n+1}} \sum_{k=0}^{n} (-1)^k \frac{n!}{(n-k)!k!} (k+1)^{-s} \] Havil 2003, p. 206. The Riemann zeta function is defined by: \[ \zeta(s) = \sum_{k=1}^{\infty} \frac{1}{k^{s}} for s > 1 \] For s < 1 use the reflection formula: \[ \zeta(s) = 2^s \pi^{s-1} \Gamma(1-s) \zeta(1-s) \] 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 $ eta(s) $ using the product over prime factors. \[ \zeta(s) = \Pi_{i=1}^\infty \frac{1}{1 - p_i^{-s}} \] where $ {p_i} $ are the prime numbers. The Riemann zeta function is defined by: \[ \zeta(s) = \sum_{k=1}^{\infty} \frac{1}{k^{s}} for s > 1 \] For s < 1 use the reflection formula: \[ \zeta(s) = 2^s \pi^{s-1} \Gamma(1-s) \zeta(1-s) \] 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 $ eta(s) $ by summation for s > 1. The Riemann zeta function is defined by: \[ \zeta(s) = \sum_{k=1}^{\infty} \frac{1}{k^{s}} for s > 1 \] For s < 1 use the reflection formula: \[ \zeta(s) = 2^s \pi^{s-1} \Gamma(1-s) \zeta(1-s) \] 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: \[ j_n(x) = \left( \frac{\pi}{2x} \right) ^{1/2} J_{n+1/2}(x) \] Parameters: __n The order of the spherical Bessel function.
__x The argument of the spherical Bessel function. Returns: The output spherical Bessel function. 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: __n The order of the modified spherical Bessel function.
__x The argument of the modified spherical Bessel function.
__i_n The output regular modified spherical Bessel function.
__k_n The output irregular modified spherical Bessel function.
__ip_n The output derivative of the regular modified spherical Bessel function.
__kp_n The output derivative of the irregular modified spherical Bessel function. 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: __n The order of the spherical Bessel function.
__x The argument of the spherical Bessel function.
__j_n The output spherical Bessel function.
__n_n The output spherical Neumann function.
__jp_n The output derivative of the spherical Bessel function.
__np_n The output derivative of the spherical Neumann function. 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 \[ Y_l^m(\theta,\phi) = (-1)^m[\frac{(2l+1)}{4\pi} \frac{(l-m)!}{(l+m)!}] P_l^m(\cos\theta) \exp^{im\phi} \] 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 = Parameters: l The order of the spherical associated Legendre function. $ l >= 0 $.
m The order of the spherical associated Legendre function. $ m <= l $.
theta The radian angle argument of the spherical associated Legendre function. 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: \[ n_n(x) = \left( \frac{\pi}{2x} \right) ^{1/2} N_{n+1/2}(x) \] Parameters: __n The order of the spherical Neumann function.
__x The argument of the spherical Neumann function. Returns: The output spherical Neumann function. Definition at line 606 of file bessel_function.tcc. References __sph_bessel_jn().
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ERROR : Need New Coding : (rof_escape_sequence|91|std::tr1::__detail.3|309|\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: \[ L_n(x) = \frac{e^x}{n!} \frac{d^n}{dx^n} (x^ne^{-x}) \] |This routine returns the associated Laguerre polynomial of order n, degree m: $ L_n^m(x) $. The associated Laguerre polynomial is defined for integral $ \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: \[ L_n(x) = \frac{e^x}{n!} \frac{d^n}{dx^n} (x^ne^{-x}) \]
) (rof_escape_sequence|91|std::tr1::__detail.3|330|\[ P_l^m(x) = (1 - x^2)^{m/2}\frac{d^m}{dx^m}P_l(x) \] |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: \[ P_l^m(x) = (1 - x^2)^{m/2}\frac{d^m}{dx^m}P_l(x) \]
) (rof_nr_x|149|std::tr1::__detail.3|382|\nu(x) $ and $ K_\nu(x) $ and their first derivatives $ I'_\nu(x) $ and $ K'_\nu(x) $ respectively. These four functions are computed together for numerical stability. \fBParameters:\fP |Compute the modified Bessel functions $ I_\nu(x) $ and $ K_\nu(x) $ and their first derivatives $ I'_\nu(x) $ and $ K'_\nu(x) $ respectively. These four functions are computed together for numerical stability. \fBParameters:\fP
) (rof_nr_x|149|std::tr1::__detail.3|406|\nu(x) $ and Neumann $ N_\nu(x) $ functions and their first derivatives $ J'_\nu(x) $ and $ N'_\nu(x) $ respectively. These four functions are computed together for numerical stability. \fBParameters:\fP |Compute the Bessel $ J_\nu(x) $ and Neumann $ N_\nu(x) $ functions and their first derivatives $ J'_\nu(x) $ and $ N'_\nu(x) $ respectively. These four functions are computed together for numerical stability. \fBParameters:\fP
) (rof_escape_sequence|91|std::tr1::__detail.3|430|\[ B(x,y) = \frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)} \] |Return the beta function $ B(x,y) $. The beta function is defined by \[ B(x,y) = \frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)} \]
) (rof_escape_sequence|91|std::tr1::__detail.3|453|\[ B(x,y) = \frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)} \] |Return the beta function: $B(x,y)$. The beta function is defined by \[ B(x,y) = \frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)} \]
) (rof_escape_sequence|91|std::tr1::__detail.3|474|\[ B(x,y) = \frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)} \] |Return the beta function $B(x,y)$ using the log gamma functions. The beta function is defined by \[ B(x,y) = \frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)} \]
) (rof_escape_sequence|91|std::tr1::__detail.3|497|\[ B(x,y) = \frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)} \] |Return the beta function $B(x,y)$ using the product form. The beta function is defined by \[ B(x,y) = \frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)} \]
) (rof_escape_sequence|91|std::tr1::__detail.3|516|\[ \left( \right) = \frac{n!}{(n-k)! k!} \]. \fBParameters:\fP |Return the binomial coefficient. The binomial coefficient is given by: \[ \left( \right) = \frac{n!}{(n-k)! k!} \]. \fBParameters:\fP
) (rof_escape_sequence|91|std::tr1::__detail.3|535|\[ K(k) = F(k,\pi/2) = \int_0^{\pi/2}\frac{d\theta} {\sqrt{1 - k^2 sin^2\theta}} \] where $ F(k,\phi) $ is the incomplete elliptic integral of the first kind. |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 \[ K(k) = F(k,\pi/2) = \int_0^{\pi/2}\frac{d\theta} {\sqrt{1 - k^2 sin^2\theta}} \] where $ F(k,\phi) $ is the incomplete elliptic integral of the first kind.
) (rof_escape_sequence|91|std::tr1::__detail.3|556|\[ K(k) = F(k,\pi/2) = \int_0^{\pi/2}\frac{d\theta} {\sqrt{1 - k^2sin^2\theta}} \] |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 \[ K(k) = F(k,\pi/2) = \int_0^{\pi/2}\frac{d\theta} {\sqrt{1 - k^2sin^2\theta}} \]
) (rof_escape_sequence|91|std::tr1::__detail.3|577|\[ E(k,\pi/2) = \int_0^{\pi/2}\sqrt{1 - k^2 sin^2\theta} \] |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 \[ E(k,\pi/2) = \int_0^{\pi/2}\sqrt{1 - k^2 sin^2\theta} \]
) (rof_escape_sequence|91|std::tr1::__detail.3|598|\[ E(k,\pi/2) = \int_0^{\pi/2}\sqrt{1 - k^2 sin^2\theta} \] |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 \[ E(k,\pi/2) = \int_0^{\pi/2}\sqrt{1 - k^2 sin^2\theta} \]
) (rof_escape_sequence|91|std::tr1::__detail.3|619|\Pi(k,\nu) = \Pi(k,\nu,\pi/2) $ using the Carlson formulation. The complete elliptic integral of the third kind is defined as \[ \Pi(k,\nu) = \int_0^{\pi/2} \frac{d\theta} {(1 - \nu \sin^2\theta)\sqrt{1 - k^2 \sin^2\theta}} \] |Return the complete elliptic integral of the third kind $ \Pi(k,\nu) = \Pi(k,\nu,\pi/2) $ using the Carlson formulation. The complete elliptic integral of the third kind is defined as \[ \Pi(k,\nu) = \int_0^{\pi/2} \frac{d\theta} {(1 - \nu \sin^2\theta)\sqrt{1 - k^2 \sin^2\theta}} \]
) (rof_escape_sequence|91|std::tr1::__detail.3|677|\[ _1F_1(a;c;x) = \frac{\Gamma(c)}{\Gamma(a)} \sum_{n=0}^{\infty} \frac{\Gamma(a+n)}{\Gamma(c+n)} \frac{x^n}{n!} \].PP |This routine returns the confluent hypergeometric function by series expansion. \[ _1F_1(a;c;x) = \frac{\Gamma(c)}{\Gamma(a)} \sum_{n=0}^{\infty} \frac{\Gamma(a+n)}{\Gamma(c+n)} \frac{x^n}{n!} \].PP
) (rof_nr_x|149|std::tr1::__detail.3|703|\nu $: $ I_{\nu}(x) $. The regular modified cylindrical Bessel function is: \[ I_{\nu}(x) = \sum_{k=0}^{\infty} \frac{(x/2)^{\nu + 2k}}{k!\Gamma(\nu+k+1)} \] |Return the regular modified Bessel function of order $ \nu $: $ I_{\nu}(x) $. The regular modified cylindrical Bessel function is: \[ I_{\nu}(x) = \sum_{k=0}^{\infty} \frac{(x/2)^{\nu + 2k}}{k!\Gamma(\nu+k+1)} \]
) (rof_nr_x|149|std::tr1::__detail.3|724|\nu $: $ J_{\nu} $ or $ I_{\nu} $ by series expansion. The modified cylindrical Bessel function is: \[ Z_{\nu}(x) = \sum_{k=0}^{\infty} \frac{\sigma^k (x/2)^{\nu + 2k}}{k!\Gamma(\nu+k+1)} \] where $ \sigma = +1 $ or$ -1 $ for $ Z = I $ or $ J $ respectively. |This routine returns the cylindrical Bessel functions of order $ \nu $: $ J_{\nu} $ or $ I_{\nu} $ by series expansion. The modified cylindrical Bessel function is: \[ Z_{\nu}(x) = \sum_{k=0}^{\infty} \frac{\sigma^k (x/2)^{\nu + 2k}}{k!\Gamma(\nu+k+1)} \] where $ \sigma = +1 $ or$ -1 $ for $ Z = I $ or $ J $ respectively.
) (rof_nr_x|149|std::tr1::__detail.3|751|\nu $: $ J_{\nu}(x) $. The cylindrical Bessel function is: \[ J_{\nu}(x) = \sum_{k=0}^{\infty} \frac{(-1)^k (x/2)^{\nu + 2k}}{k!\Gamma(\nu+k+1)} \] |Return the Bessel function of order $ \nu $: $ J_{\nu}(x) $. The cylindrical Bessel function is: \[ J_{\nu}(x) = \sum_{k=0}^{\infty} \frac{(-1)^k (x/2)^{\nu + 2k}}{k!\Gamma(\nu+k+1)} \]
) (rof_nr_x|149|std::tr1::__detail.3|772|\nu} $, $ N_{\nu} $. 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 |This routine computes the asymptotic cylindrical Bessel and Neumann functions of order nu: $ J_{\nu} $, $ N_{\nu} $. 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
) (rof_nr_x|149|std::tr1::__detail.3|794|\nu}(x) $ of order $ \nu $. 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} $. |Return the irregular modified Bessel function $ K_{\nu}(x) $ of order $ \nu $. 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} $.
) (rof_nr_x|149|std::tr1::__detail.3|815|\nu $: $ N_{\nu}(x) $. The Neumann function is defined by: \[ N_{\nu}(x) = \frac{J_{\nu}(x) \cos \nu\pi - J_{-\nu}(x)} {\sin \nu\pi} \] where for integral $ \nu = n $ a limit is taken: $ lim_{\nu \to n} $. |Return the Neumann function of order $ \nu $: $ N_{\nu}(x) $. The Neumann function is defined by: \[ N_{\nu}(x) = \frac{J_{\nu}(x) \cos \nu\pi - J_{-\nu}(x)} {\sin \nu\pi} \] where for integral $ \nu = n $ a limit is taken: $ lim_{\nu \to n} $.
) (rof_escape_sequence|91|std::tr1::__detail.3|836|\phi) $ using the Carlson formulation. The incomplete elliptic integral of the first kind is defined as \[ F(k,\phi) = \int_0^{\phi}\frac{d\theta} {\sqrt{1 - k^2 sin^2\theta}} \] |Return the incomplete elliptic integral of the first kind $ F(k,\phi) $ using the Carlson formulation. The incomplete elliptic integral of the first kind is defined as \[ F(k,\phi) = \int_0^{\phi}\frac{d\theta} {\sqrt{1 - k^2 sin^2\theta}} \]
) (rof_escape_sequence|91|std::tr1::__detail.3|857|\phi) $ using the Carlson formulation. The incomplete elliptic integral of the second kind is defined as \[ E(k,\phi) = \int_0^{\phi} \sqrt{1 - k^2 sin^2\theta} \] |Return the incomplete elliptic integral of the second kind $ E(k,\phi) $ using the Carlson formulation. The incomplete elliptic integral of the second kind is defined as \[ E(k,\phi) = \int_0^{\phi} \sqrt{1 - k^2 sin^2\theta} \]
) (rof_escape_sequence|91|std::tr1::__detail.3|878|\Pi(k,\nu,\phi) $ using the Carlson formulation. The incomplete elliptic integral of the third kind is defined as \[ \Pi(k,\nu,\phi) = \int_0^{\phi} \frac{d\theta} {(1 - \nu \sin^2\theta) \sqrt{1 - k^2 \sin^2\theta}} \] |Return the incomplete elliptic integral of the third kind $ \Pi(k,\nu,\phi) $ using the Carlson formulation. The incomplete elliptic integral of the third kind is defined as \[ \Pi(k,\nu,\phi) = \int_0^{\phi} \frac{d\theta} {(1 - \nu \sin^2\theta) \sqrt{1 - k^2 \sin^2\theta}} \]
) (rof_escape_sequence|91|std::tr1::__detail.3|901|\[ R_C(x,y) = \frac{1}{2} \int_0^\infty \frac{dt}{(t + x)^{1/2}(t + y)} \] |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: \[ R_C(x,y) = \frac{1}{2} \int_0^\infty \frac{dt}{(t + x)^{1/2}(t + y)} \]
) (rof_escape_sequence|91|std::tr1::__detail.3|933|\[ R_D(x,y,z) = \frac{3}{2} \int_0^\infty \frac{dt}{(t + x)^{1/2}(t + y)^{1/2}(t + z)^{3/2}} \] |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: \[ R_D(x,y,z) = \frac{3}{2} \int_0^\infty \frac{dt}{(t + x)^{1/2}(t + y)^{1/2}(t + z)^{3/2}} \]
) (rof_escape_sequence|91|std::tr1::__detail.3|967|\[ R_F(x,y,z) = \frac{1}{2} \int_0^\infty \frac{dt}{(t + x)^{1/2}(t + y)^{1/2}(t + z)^{1/2}} \] |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: \[ R_F(x,y,z) = \frac{1}{2} \int_0^\infty \frac{dt}{(t + x)^{1/2}(t + y)^{1/2}(t + z)^{1/2}} \]
) (rof_escape_sequence|91|std::tr1::__detail.3|992|\[ R_J(x,y,z,p) = \frac{3}{2} \int_0^\infty \frac{dt}{(t + x)^{1/2}(t + y)^{1/2}(t + z)^{1/2}(t + p)} \] |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: \[ R_J(x,y,z,p) = \frac{3}{2} \int_0^\infty \frac{dt}{(t + x)^{1/2}(t + y)^{1/2}(t + z)^{1/2}(t + p)} \]
) (rof_escape_sequence|91|std::tr1::__detail.3|1028|\[ Ei(x) = -\int_{-x}^\infty \frac{e^t}{t} dt \] |Return the exponential integral $ Ei(x) $. The exponential integral is given by \[ Ei(x) = -\int_{-x}^\infty \frac{e^t}{t} dt \]
) (rof_escape_sequence|91|std::tr1::__detail.3|1047|\[ E_n(x) = \int_{1}^\infty \frac{e^{-xt}}{t^n} dt \] This is something of an extension. |Return the exponential integral $ E_n(x) $. The exponential integral is given by \[ E_n(x) = \int_{1}^\infty \frac{e^{-xt}}{t^n} dt \] This is something of an extension.
) (rof_escape_sequence|91|std::tr1::__detail.3|1068|\[ E_n(x) = \int_{1}^\infty \frac{e^{-xt}}{t^n} dt \] |Return the exponential integral $ E_n(x) $ for large argument. The exponential integral is given by \[ E_n(x) = \int_{1}^\infty \frac{e^{-xt}}{t^n} dt \]
) (rof_escape_sequence|91|std::tr1::__detail.3|1091|\[ E_1(x) = \int_{1}^\infty \frac{e^{-xt}}{t} dt \] |Return the exponential integral $ E_1(x) $. The exponential integral is given by \[ E_1(x) = \int_{1}^\infty \frac{e^{-xt}}{t} dt \]
) (rof_escape_sequence|91|std::tr1::__detail.3|1112|\[ E_1(x) = \int_{1}^\infty \frac{e^{-xt}}{t} dt \] |Return the exponential integral $ E_1(x) $ by asymptotic expansion. The exponential integral is given by \[ E_1(x) = \int_{1}^\infty \frac{e^{-xt}}{t} dt \]
) (rof_escape_sequence|91|std::tr1::__detail.3|1133|\[ E_1(x) = \int_{1}^{\infty} \frac{e^{-xt}}{t} dt \] |Return the exponential integral $ E_1(x) $ by series summation. This should be good for $ x < 1 $. The exponential integral is given by \[ E_1(x) = \int_{1}^{\infty} \frac{e^{-xt}}{t} dt \]
) (rof_escape_sequence|91|std::tr1::__detail.3|1154|\[ Ei(x) = -\int_{-x}^\infty \frac{e^t}{t} dt \] |Return the exponential integral $ Ei(x) $. The exponential integral is given by \[ Ei(x) = -\int_{-x}^\infty \frac{e^t}{t} dt \]
) (rof_escape_sequence|91|std::tr1::__detail.3|1175|\[ Ei(x) = -\int_{-x}^\infty \frac{e^t}{t} dt \] |Return the exponential integral $ Ei(x) $ by asymptotic expansion. The exponential integral is given by \[ Ei(x) = -\int_{-x}^\infty \frac{e^t}{t} dt \]
) (rof_escape_sequence|91|std::tr1::__detail.3|1196|\[ Ei(x) = -\int_{-x}^\infty \frac{e^t}{t} dt \] |Return the exponential integral $ Ei(x) $ by series summation. The exponential integral is given by \[ Ei(x) = -\int_{-x}^\infty \frac{e^t}{t} dt \]
) (rof_escape_sequence|91|std::tr1::__detail.3|1217|\[ E_n(x) = \int_{1}^\infty \frac{e^{-xt}}{t^n} dt \] |Return the exponential integral $ E_n(x) $ by continued fractions. The exponential integral is given by \[ E_n(x) = \int_{1}^\infty \frac{e^{-xt}}{t^n} dt \]
) (rof_escape_sequence|91|std::tr1::__detail.3|1240|\[ E_n(x) = \int_{1}^\infty \frac{e^{-xt}}{t^n} dt \] |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 \[ E_n(x) = \int_{1}^\infty \frac{e^{-xt}}{t^n} dt \]
) (rof_escape_sequence|91|std::tr1::__detail.3|1263|\[ E_n(x) = \int_{1}^\infty \frac{e^{-xt}}{t^n} dt \] |Return the exponential integral $ E_n(x) $ by series summation. The exponential integral is given by \[ E_n(x) = \int_{1}^\infty \frac{e^{-xt}}{t^n} dt \]
) (rof_escape_sequence|91|std::tr1::__detail.3|1284|\[ E_n(x) = \int_{1}^\infty \frac{e^{-xt}}{t^n} dt \] |Return the exponential integral $ E_n(x) $ for large order. The exponential integral is given by \[ E_n(x) = \int_{1}^\infty \frac{e^{-xt}}{t^n} dt \]
) (rof_escape_sequence|91|std::tr1::__detail.3|1307|\Gamma(x) $. \fBParameters:\fP |Return $ \Gamma(x) $. \fBParameters:\fP
) (rof_nr_x|149|std::tr1::__detail.3|1326|\nu(x) $ and $ K_\nu(x) $. \[ \Gamma_1 = \frac{1}{2\mu} [\frac{1}{\Gamma(1 - \mu)} - \frac{1}{\Gamma(1 + \mu)}] \] and \[ \Gamma_2 = \frac{1}{2} [\frac{1}{\Gamma(1 - \mu)} + \frac{1}{\Gamma(1 + \mu)}] \] where $ -1/2 <= \mu <= 1/2 $ is $ \mu = \nu - N $ and $ N $. is the nearest integer to $ \nu $. The values of $ \Gamma(1 + \mu) $ and $ \Gamma(1 - \mu) $ are returned as well. The accuracy requirements on this are exquisite. |Compute the gamma functions required by the Temme series expansions of $ N_\nu(x) $ and $ K_\nu(x) $. \[ \Gamma_1 = \frac{1}{2\mu} [\frac{1}{\Gamma(1 - \mu)} - \frac{1}{\Gamma(1 + \mu)}] \] and \[ \Gamma_2 = \frac{1}{2} [\frac{1}{\Gamma(1 - \mu)} + \frac{1}{\Gamma(1 + \mu)}] \] where $ -1/2 <= \mu <= 1/2 $ is $ \mu = \nu - N $ and $ N $. is the nearest integer to $ \nu $. The values of $ \Gamma(1 + \mu) $ and $ \Gamma(1 - \mu) $ are returned as well. The accuracy requirements on this are exquisite.
) (rof_escape_sequence|91|std::tr1::__detail.3|1332|\Gamma_1(\mu) $ |\fI__gam1\fP The output function $ \Gamma_1(\mu) $
) (rof_escape_sequence|91|std::tr1::__detail.3|1334|\Gamma_2(\mu) $ |\fI__gam2\fP The output function $ \Gamma_2(\mu) $
) (rof_escape_sequence|91|std::tr1::__detail.3|1336|\Gamma(1 + \mu) $ |\fI__gampl\fP The output function $ \Gamma(1 + \mu) $
) (rof_escape_sequence|91|std::tr1::__detail.3|1338|\Gamma(1 - \mu) $ |\fI__gammi\fP The output function $ \Gamma(1 - \mu) $
) (rof_escape_sequence|91|std::tr1::__detail.3|1350|\[ \zeta(x,s) = \sum_{n=0}^{\infty} \frac{1}{(n + x)^s} \] The Riemann zeta function is a special case: \[ \zeta(s) = \zeta(1,s) \] |Return the Hurwitz zeta function $ \zeta(x,s) $ for all s != 1 and x > -1. The Hurwitz zeta function is defined by: \[ \zeta(x,s) = \sum_{n=0}^{\infty} \frac{1}{(n + x)^s} \] The Riemann zeta function is a special case: \[ \zeta(s) = \zeta(1,s) \]
) (rof_escape_sequence|91|std::tr1::__detail.3|1359|\[ \zeta(x,s) = \sum_{n=0}^{\infty} \frac{1}{(n + x)^s} \] The Riemann zeta function is a special case: \[ \zeta(s) = \zeta(1,s) \] |Return the Hurwitz zeta function $ \zeta(x,s) $ for all s != 1 and x > -1. The Hurwitz zeta function is defined by: \[ \zeta(x,s) = \sum_{n=0}^{\infty} \frac{1}{(n + x)^s} \] The Riemann zeta function is a special case: \[ \zeta(s) = \zeta(1,s) \]
) (rof_escape_sequence|91|std::tr1::__detail.3|1361|\[ \zeta(x,s) = \frac{1}{s-1} \sum_{n=0}^{\infty} \frac{1}{n + 1} \sum_{k=0}^{n} (-1)^k \frac{n!}{(n-k)!k!} (x+k)^{-s} \] |This functions uses the double sum that converges for s != 1 and x > -1: \[ \zeta(x,s) = \frac{1}{s-1} \sum_{n=0}^{\infty} \frac{1}{n + 1} \sum_{k=0}^{n} (-1)^k \frac{n!}{(n-k)!k!} (x+k)^{-s} \]
) (rof_escape_sequence|91|std::tr1::__detail.3|1370|\[ _2F_1(a,b;c;x) = \frac{\Gamma(c)}{\Gamma(a)\Gamma(b)} \sum_{n=0}^{\infty} \frac{\Gamma(a+n)\Gamma(b+n)}{\Gamma(c+n)} \frac{x^n}{n!} \] |Return the hypogeometric function $ _2F_1(a,b;c;x) $. The hypogeometric function is defined by \[ _2F_1(a,b;c;x) = \frac{\Gamma(c)}{\Gamma(a)\Gamma(b)} \sum_{n=0}^{\infty} \frac{\Gamma(a+n)\Gamma(b+n)}{\Gamma(c+n)} \frac{x^n}{n!} \]
) (rof_escape_sequence|91|std::tr1::__detail.3|1404|\[ _2F_1(a,b;c;x) = \frac{\Gamma(c)}{\Gamma(a)\Gamma(b)} \sum_{n=0}^{\infty} \frac{\Gamma(a+n)\Gamma(b+n)}{\Gamma(c+n)} \frac{x^n}{n!} \] |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 \[ _2F_1(a,b;c;x) = \frac{\Gamma(c)}{\Gamma(a)\Gamma(b)} \sum_{n=0}^{\infty} \frac{\Gamma(a+n)\Gamma(b+n)}{\Gamma(c+n)} \frac{x^n}{n!} \]
) (rof_escape_sequence|91|std::tr1::__detail.3|1406|\[ _2F_1(a,b;c;x) = \frac{\Gamma(c)\Gamma(d)}{\Gamma(c-a)\Gamma(c-b)} _2F_1(a,b;1-d;1-x) + \frac{\Gamma(c)\Gamma(-d)}{\Gamma(a)\Gamma(b)} _2F_1(c-a,c-b;1+d;1-x) \] |The reflection formula for nonintegral $ d = c - a - b $ is: \[ _2F_1(a,b;c;x) = \frac{\Gamma(c)\Gamma(d)}{\Gamma(c-a)\Gamma(c-b)} _2F_1(a,b;1-d;1-x) + \frac{\Gamma(c)\Gamma(-d)}{\Gamma(a)\Gamma(b)} _2F_1(c-a,c-b;1+d;1-x) \]
) (rof_escape_sequence|91|std::tr1::__detail.3|1408|\[ _2F_1(a,b;a+b+m;x) = \frac{\Gamma(m)\Gamma(a+b+m)}{\Gamma(a+m)\Gamma(b+m)} \sum_{k=0}^{m-1} \frac{(m+a)_k(m+b)_k}{k!(1-m)_k} - \] |The reflection formula for integral $ m = c - a - b $ is: \[ _2F_1(a,b;a+b+m;x) = \frac{\Gamma(m)\Gamma(a+b+m)}{\Gamma(a+m)\Gamma(b+m)} \sum_{k=0}^{m-1} \frac{(m+a)_k(m+b)_k}{k!(1-m)_k} - \]
) (rof_escape_sequence|91|std::tr1::__detail.3|1417|\[ _2F_1(a,b;c;x) = \frac{\Gamma(c)}{\Gamma(a)\Gamma(b)} \sum_{n=0}^{\infty} \frac{\Gamma(a+n)\Gamma(b+n)}{\Gamma(c+n)} \frac{x^n}{n!} \] |Return the hypogeometric function $ _2F_1(a,b;c;x) $ by series expansion. The hypogeometric function is defined by \[ _2F_1(a,b;c;x) = \frac{\Gamma(c)}{\Gamma(a)\Gamma(b)} \sum_{n=0}^{\infty} \frac{\Gamma(a+n)\Gamma(b+n)}{\Gamma(c+n)} \frac{x^n}{n!} \]
) (rof_escape_sequence|91|std::tr1::__detail.3|1446|\[ L_n(x) = \frac{e^x}{n!} \frac{d^n}{dx^n} (x^ne^{-x}) \] |This routine returns the Laguerre polynomial of order n: $ L_n(x) $. The Laguerre polynomial is defined by: \[ L_n(x) = \frac{e^x}{n!} \frac{d^n}{dx^n} (x^ne^{-x}) \]
) (rof_escape_sequence|91|std::tr1::__detail.3|1465|\[ \left( \right) = \frac{n!}{(n-k)! k!} \]. \fBParameters:\fP |Return the logarithm of the binomial coefficient. The binomial coefficient is given by: \[ \left( \right) = \frac{n!}{(n-k)! k!} \]. \fBParameters:\fP
) (rof_escape_sequence|91|std::tr1::__detail.3|1484|\Gamma(x)|) $. This will return values even for $ x < 0 $. To recover the sign of $ \Gamma(x) $ for any argument use \fI__log_gamma_sign\fP. \fBParameters:\fP |Return $ log(|\Gamma(x)|) $. This will return values even for $ x < 0 $. To recover the sign of $ \Gamma(x) $ for any argument use \fI__log_gamma_sign\fP. \fBParameters:\fP
) (rof_escape_sequence|91|std::tr1::__detail.3|1503|\Gamma(x))$ by asymptotic expansion with Bernoulli number coefficients. This is like Sterling's approximation. \fBParameters:\fP |Return $log(\Gamma(x))$ by asymptotic expansion with Bernoulli number coefficients. This is like Sterling's approximation. \fBParameters:\fP
) (rof_escape_sequence|91|std::tr1::__detail.3|1520|\Gamma(x))$ by the Lanczos method. This method dominates all others on the positive axis I think. \fBParameters:\fP |Return $log(\Gamma(x))$ by the Lanczos method. This method dominates all others on the positive axis I think. \fBParameters:\fP
) (rof_escape_sequence|91|std::tr1::__detail.3|1539|\Gamma(x) $. At nonpositive integers zero is returned. \fBParameters:\fP |Return the sign of $ \Gamma(x) $. At nonpositive integers zero is returned. \fBParameters:\fP
) (rof_escape_sequence|91|std::tr1::__detail.3|1558|\[ H_n(x) = (-1)^n e^{x^2} \frac{d^n}{dx^n} e^{-x^2} \] |This routine returns the Hermite polynomial of order n: $ H_n(x) $. The Hermite polynomial is defined by: \[ H_n(x) = (-1)^n e^{x^2} \frac{d^n}{dx^n} e^{-x^2} \]
) (rof_escape_sequence|91|std::tr1::__detail.3|1579|\[ H_n(x) = (-1)^n e^{x^2} \frac{d^n}{dx^n} e^{-x^2} \] |This routine returns the Hermite polynomial of order n: $ H_n(x) $ by recursion on n. The Hermite polynomial is defined by: \[ H_n(x) = (-1)^n e^{x^2} \frac{d^n}{dx^n} e^{-x^2} \]
) (rof_escape_sequence|91|std::tr1::__detail.3|1600|\alpha $: $ L_n^alpha(x) $. The associated Laguerre function 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. |This routine returns the associated Laguerre polynomial of order n, degree $ \alpha $: $ L_n^alpha(x) $. The associated Laguerre function 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.
) (rof_escape_sequence|91|std::tr1::__detail.3|1602|\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: \[ L_n(x) = \frac{e^x}{n!} \frac{d^n}{dx^n} (x^ne^{-x}) \] |The associated Laguerre polynomial is defined for integral $ \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: \[ L_n(x) = \frac{e^x}{n!} \frac{d^n}{dx^n} (x^ne^{-x}) \]
) (rof_escape_sequence|91|std::tr1::__detail.3|1615|\alpha $, and argument x. |The value of the Laguerre function of order n, degree $ \alpha $, and argument x.
) (rof_escape_sequence|91|std::tr1::__detail.3|1625|\[ L_n^\alpha(x) = \frac{(\alpha + 1)_n}{n!} _1F_1(-n; \alpha + 1; x) \] where $ (\alpha)_n $ is the Pochhammer symbol and $ _1F_1(a; c; x) $ is the confluent hypergeometric function. |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 \[ L_n^\alpha(x) = \frac{(\alpha + 1)_n}{n!} _1F_1(-n; \alpha + 1; x) \] where $ (\alpha)_n $ is the Pochhammer symbol and $ _1F_1(a; c; x) $ is the confluent hypergeometric function.
) (rof_escape_sequence|91|std::tr1::__detail.3|1638|\alpha $ for large n. Abramowitz & Stegun, 13.5.21. \fBParameters:\fP |This routine returns the associated Laguerre polynomial of order $ n $, degree $ \alpha $ for large n. Abramowitz & Stegun, 13.5.21. \fBParameters:\fP
) (rof_escape_sequence|91|std::tr1::__detail.3|1649|\alpha $, and argument x. |The value of the Laguerre function of order n, degree $ \alpha $, and argument x.
) (rof_escape_sequence|91|std::tr1::__detail.3|1661|\alpha $: $ L_n^\alpha(x) $ by recursion. The associated Laguerre function 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. |This routine returns the associated Laguerre polynomial of order $ n $, degree $ \alpha $: $ L_n^\alpha(x) $ by recursion. The associated Laguerre function 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.
) (rof_escape_sequence|91|std::tr1::__detail.3|1663|\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: \[ L_n(x) = \frac{e^x}{n!} \frac{d^n}{dx^n} (x^ne^{-x}) \] |The associated Laguerre polynomial is defined for integral $ \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: \[ L_n(x) = \frac{e^x}{n!} \frac{d^n}{dx^n} (x^ne^{-x}) \]
) (rof_escape_sequence|91|std::tr1::__detail.3|1676|\alpha $, and argument x. |The value of the Laguerre function of order n, degree $ \alpha $, and argument x.
) (rof_escape_sequence|91|std::tr1::__detail.3|1686|\[ P_l(x) = \frac{1}{2^l l!}\frac{d^l}{dx^l}(x^2 - 1)^{l} \] |Return the Legendre polynomial by recursion on order $ l $. The Legendre function of $ l $ and $ x $, $ P_l(x) $, is defined by: \[ P_l(x) = \frac{1}{2^l l!}\frac{d^l}{dx^l}(x^2 - 1)^{l} \]
) (rof_escape_sequence|91|std::tr1::__detail.3|1702|\psi^{(n)}(x) $. The polygamma function is related to the Hurwitz zeta function: \[ \psi^{(n)}(x) = (-1)^{n+1} m! \zeta(m+1,x) \] |Return the polygamma function $ \psi^{(n)}(x) $. The polygamma function is related to the Hurwitz zeta function: \[ \psi^{(n)}(x) = (-1)^{n+1} m! \zeta(m+1,x) \]
) (rof_escape_sequence|91|std::tr1::__detail.3|1709|\psi(x) $ function is defined by \[ \psi(x) = \frac{\Gamma'(x)}{\Gamma(x)} \] For negative argument the reflection formula is used: \[ \psi(x) = \psi(1-x) - \pi \cot(\pi x) \]. |Return the digamma function. The digamma or $ \psi(x) $ function is defined by \[ \psi(x) = \frac{\Gamma'(x)}{\Gamma(x)} \] For negative argument the reflection formula is used: \[ \psi(x) = \psi(1-x) - \pi \cot(\pi x) \].
) (rof_escape_sequence|91|std::tr1::__detail.3|1718|\psi(x) $ function is defined by \[ \psi(x) = \frac{\Gamma'(x)}{\Gamma(x)} \]. The asymptotic series is given by: \[ \psi(x) = \ln(x) - \frac{1}{2x} - \sum_{n=1}^{\infty} \frac{B_{2n}}{2 n x^{2n}} \] |Return the digamma function for large argument. The digamma or $ \psi(x) $ function is defined by \[ \psi(x) = \frac{\Gamma'(x)}{\Gamma(x)} \]. The asymptotic series is given by: \[ \psi(x) = \ln(x) - \frac{1}{2x} - \sum_{n=1}^{\infty} \frac{B_{2n}}{2 n x^{2n}} \]
) (rof_escape_sequence|91|std::tr1::__detail.3|1727|\psi(x) $ function is defined by \[ \psi(x) = \frac{\Gamma'(x)}{\Gamma(x)} \]. The series is given by: \[ \psi(x) = -\gamma_E - \frac{1}{x} \sum_{k=1}^{\infty} \frac{x}{k(x + k)} \] |Return the digamma function by series expansion. The digamma or $ \psi(x) $ function is defined by \[ \psi(x) = \frac{\Gamma'(x)}{\Gamma(x)} \]. The series is given by: \[ \psi(x) = -\gamma_E - \frac{1}{x} \sum_{k=1}^{\infty} \frac{x}{k(x + k)} \]
) (rof_escape_sequence|91|std::tr1::__detail.3|1736|\[ \zeta(s) = \sum_{k=1}^{\infty} k^{-s} for s > 1 \frac{(2\pi)^s}{pi} sin(\frac{\pi s}{2}) \Gamma (1 - s) \zeta (1 - s) for s < 1 \] For s < 1 use the reflection formula: \[ \zeta(s) = 2^s \pi^{s-1} \Gamma(1-s) \zeta(1-s) \] |Return the Riemann zeta function $ \zeta(s) $. The Riemann zeta function is defined by: \[ \zeta(s) = \sum_{k=1}^{\infty} k^{-s} for s > 1 \frac{(2\pi)^s}{pi} sin(\frac{\pi s}{2}) \Gamma (1 - s) \zeta (1 - s) for s < 1 \] For s < 1 use the reflection formula: \[ \zeta(s) = 2^s \pi^{s-1} \Gamma(1-s) \zeta(1-s) \]
) (rof_escape_sequence|91|std::tr1::__detail.3|1743|\[ \zeta(s) = \sum_{k=1}^{\infty} \frac{1}{k^{s}} for s > 1 \] For s < 1 use the reflection formula: \[ \zeta(s) = 2^s \pi^{s-1} \Gamma(1-s) \zeta(1-s) \] |Evaluate the Riemann zeta function $ \zeta(s) $ by an alternate series for s > 0. The Riemann zeta function is defined by: \[ \zeta(s) = \sum_{k=1}^{\infty} \frac{1}{k^{s}} for s > 1 \] For s < 1 use the reflection formula: \[ \zeta(s) = 2^s \pi^{s-1} \Gamma(1-s) \zeta(1-s) \]
) (rof_escape_sequence|91|std::tr1::__detail.3|1750|\[ \zeta(s) = \frac{1}{1-2^{1-s}} \sum_{n=0}^{\infty} \frac{1}{2^{n+1}} \sum_{k=0}^{n} (-1)^k \frac{n!}{(n-k)!k!} (k+1)^{-s} \] Havil 2003, p. 206. |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: \[ \zeta(s) = \frac{1}{1-2^{1-s}} \sum_{n=0}^{\infty} \frac{1}{2^{n+1}} \sum_{k=0}^{n} (-1)^k \frac{n!}{(n-k)!k!} (k+1)^{-s} \] Havil 2003, p. 206.
) (rof_escape_sequence|91|std::tr1::__detail.3|1752|\[ \zeta(s) = \sum_{k=1}^{\infty} \frac{1}{k^{s}} for s > 1 \] For s < 1 use the reflection formula: \[ \zeta(s) = 2^s \pi^{s-1} \Gamma(1-s) \zeta(1-s) \] |The Riemann zeta function is defined by: \[ \zeta(s) = \sum_{k=1}^{\infty} \frac{1}{k^{s}} for s > 1 \] For s < 1 use the reflection formula: \[ \zeta(s) = 2^s \pi^{s-1} \Gamma(1-s) \zeta(1-s) \]
) (rof_escape_sequence|91|std::tr1::__detail.3|1761|\[ \zeta(s) = \Pi_{i=1}^\infty \frac{1}{1 - p_i^{-s}} \] where $ {p_i} $ are the prime numbers. The Riemann zeta function is defined by: \[ \zeta(s) = \sum_{k=1}^{\infty} \frac{1}{k^{s}} for s > 1 \] For s < 1 use the reflection formula: \[ \zeta(s) = 2^s \pi^{s-1} \Gamma(1-s) \zeta(1-s) \] |Compute the Riemann zeta function $ \zeta(s) $ using the product over prime factors. \[ \zeta(s) = \Pi_{i=1}^\infty \frac{1}{1 - p_i^{-s}} \] where $ {p_i} $ are the prime numbers. The Riemann zeta function is defined by: \[ \zeta(s) = \sum_{k=1}^{\infty} \frac{1}{k^{s}} for s > 1 \] For s < 1 use the reflection formula: \[ \zeta(s) = 2^s \pi^{s-1} \Gamma(1-s) \zeta(1-s) \]
) (rof_escape_sequence|91|std::tr1::__detail.3|1770|\[ \zeta(s) = \sum_{k=1}^{\infty} \frac{1}{k^{s}} for s > 1 \] For s < 1 use the reflection formula: \[ \zeta(s) = 2^s \pi^{s-1} \Gamma(1-s) \zeta(1-s) \] |Compute the Riemann zeta function $ \zeta(s) $ by summation for s > 1. The Riemann zeta function is defined by: \[ \zeta(s) = \sum_{k=1}^{\infty} \frac{1}{k^{s}} for s > 1 \] For s < 1 use the reflection formula: \[ \zeta(s) = 2^s \pi^{s-1} \Gamma(1-s) \zeta(1-s) \]
) (rof_escape_sequence|91|std::tr1::__detail.3|1779|\[ j_n(x) = \left( \frac{\pi}{2x} \right) ^{1/2} J_{n+1/2}(x) \] |Return the spherical Bessel function $ j_n(x) $ of order n. The spherical Bessel function is defined by: \[ j_n(x) = \left( \frac{\pi}{2x} \right) ^{1/2} J_{n+1/2}(x) \]
) (rof_escape_sequence|91|std::tr1::__detail.3|1846|\[ Y_l^m(\theta,\phi) = (-1)^m[\frac{(2l+1)}{4\pi} \frac{(l-m)!}{(l+m)!}] P_l^m(\cos\theta) \exp^{im\phi} \] is the spherical harmonic function and $ P_l^m(x) $ is the associated Legendre function. |Return the spherical associated Legendre function. The spherical associated Legendre function of $ l $, $ m $, and $ \theta $ is defined as $ Y_l^m(\theta,0) $ where \[ Y_l^m(\theta,\phi) = (-1)^m[\frac{(2l+1)}{4\pi} \frac{(l-m)!}{(l+m)!}] P_l^m(\cos\theta) \exp^{im\phi} \] is the spherical harmonic function and $ P_l^m(x) $ is the associated Legendre function.
) (rof_escape_sequence|91|std::tr1::__detail.3|1866|\[ n_n(x) = \left( \frac{\pi}{2x} \right) ^{1/2} N_{n+1/2}(x) \] |Return the spherical Neumann function $ n_n(x) $. The spherical Neumann function is defined by: \[ n_n(x) = \left( \frac{\pi}{2x} \right) ^{1/2} N_{n+1/2}(x) \]
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