Math::GSL::SF(3)

NAME
Math::GSL::SF - Special Functions

SYNOPSIS
    use Math::GSL::SF qw /:all/;

DESCRIPTION
This module contains a data structure named gsl_sf_result. To create a new one use

    $r = Math::GSL::SF::gsl_sf_result_struct->new;

You can then access the elements of the structure in this way :

    my $val   = $r->{val};

    my $error = $r->{err};

Here is a list of all included functions:

"gsl_sf_airy_Ai_e($x, $mode)"
"gsl_sf_airy_Ai($x, $mode, $result)"
    - These routines compute the Airy function Ai($x) with an accuracy specified by $mode. $mode should be $GSL_PREC_DOUBLE, $GSL_PREC_SINGLE or $GSL_PREC_APPROX. $result is a gsl_sf_result structure.

"gsl_sf_airy_Bi_e($x, $mode, $result)"
"gsl_sf_airy_Bi($x, $mode)"
    - These routines compute the Airy function Bi($x) with an accuracy specified by $mode. $mode should be $GSL_PREC_DOUBLE, $GSL_PREC_SINGLE or $GSL_PREC_APPROX. $result is a gsl_sf_result structure.

"gsl_sf_airy_Ai_scaled_e($x, $mode, $result)"
"gsl_sf_airy_Ai_scaled($x, $mode)"
    - These routines compute a scaled version of the Airy function S_A($x) Ai($x). For $x>0 the scaling factor S_A($x) is \exp(+(2/3) $x**(3/2)), and is 1 for $x<0. $result is a gsl_sf_result structure.

"gsl_sf_airy_Bi_scaled_e($x, $mode, $result)"
"gsl_sf_airy_Bi_scaled($x, $mode)"
    - These routines compute a scaled version of the Airy function S_B($x) Bi($x). For $x>0 the scaling factor S_B($x) is exp(-(2/3) $x**(3/2)), and is 1 for $x<0. $result is a gsl_sf_result structure.

"gsl_sf_airy_Ai_deriv_e($x, $mode, $result)"
"gsl_sf_airy_Ai_deriv($x, $mode)"
    - These routines compute the Airy function derivative Ai'($x) with an accuracy specified by $mode. $result is a gsl_sf_result structure.

"gsl_sf_airy_Bi_deriv_e($x, $mode, $result)"
"gsl_sf_airy_Bi_deriv($x, $mode)"
    -These routines compute the Airy function derivative Bi'($x) with an accuracy specified by $mode. $result is a gsl_sf_result structure.

"gsl_sf_airy_Ai_deriv_scaled_e($x, $mode, $result)"
"gsl_sf_airy_Ai_deriv_scaled($x, $mode)"
    -These routines compute the scaled Airy function derivative S_A(x) Ai'(x). For x>0 the scaling factor S_A(x) is \exp(+(2/3) x^(3/2)), and is 1 for x<0. $result is a gsl_sf_result structure.

"gsl_sf_airy_Bi_deriv_scaled_e($x, $mode, $result)"
"gsl_sf_airy_Bi_deriv_scaled($x, $mode)"
    -These routines compute the scaled Airy function derivative S_B(x) Bi'(x). For x>0 the scaling factor S_B(x) is exp(-(2/3) x^(3/2)), and is 1 for x<0. $result is a gsl_sf_result structure.

"gsl_sf_airy_zero_Ai_e($s, $result)"
"gsl_sf_airy_zero_Ai($s)"
    -These routines compute the location of the s-th zero of the Airy function Ai($x). $result is a gsl_sf_result structure.

"gsl_sf_airy_zero_Bi_e($s, $result)"
"gsl_sf_airy_zero_Bi($s)"
    -These routines compute the location of the s-th zero of the Airy function Bi($x). $result is a gsl_sf_result structure.

"gsl_sf_airy_zero_Ai_deriv_e($s, $result)"
"gsl_sf_airy_zero_Ai_deriv($s)"
    -These routines compute the location of the s-th zero of the Airy function derivative Ai'(x). $result is a gsl_sf_result structure.

"gsl_sf_airy_zero_Bi_deriv_e($s, $result)"
"gsl_sf_airy_zero_Bi_deriv($s)"
    - These routines compute the location of the s-th zero of the Airy function derivative Bi'(x). $result is a gsl_sf_result structure.

"gsl_sf_bessel_J0_e($x, $result)"
"gsl_sf_bessel_J0($x)"
    -These routines compute the regular cylindrical Bessel function of zeroth order, J_0($x). $result is a gsl_sf_result structure.

"gsl_sf_bessel_J1_e($x, $result)"
"gsl_sf_bessel_J1($x)"
    - These routines compute the regular cylindrical Bessel function of first order, J_1($x). $result is a gsl_sf_result structure.

"gsl_sf_bessel_Jn_e($n, $x, $result)"
"gsl_sf_bessel_Jn($n, $x)"
    -These routines compute the regular cylindrical Bessel function of order n, J_n($x).

"gsl_sf_bessel_Jn_array($nmin, $nmax, $x, $result_array)" - This routine computes the values of the regular cylindrical Bessel functions J_n($x) for n from $nmin to $nmax inclusive, storing the results in the array $result_array. The values are computed using recurrence relations for efficiency, and therefore may differ slightly from the exact values.
"gsl_sf_bessel_Y0_e($x, $result)"
"gsl_sf_bessel_Y0($x)"
    - These routines compute the irregular spherical Bessel function of zeroth order, y_0(x) = -\cos(x)/x.

"gsl_sf_bessel_Y1_e($x, $result)"
"gsl_sf_bessel_Y1($x)"
    -These routines compute the irregular spherical Bessel function of first order, y_1(x) = -(\cos(x)/x + \sin(x))/x.

"gsl_sf_bessel_Yn_e"($n, $x, $result)
"gsl_sf_bessel_Yn($n, $x)"
    -These routines compute the irregular cylindrical Bessel function of order $n, Y_n(x), for x>0.

"gsl_sf_bessel_Yn_array"
    -

"gsl_sf_bessel_I0_e($x, $result)"
"gsl_sf_bessel_I0($x)"
    -These routines compute the regular modified cylindrical Bessel function of zeroth order, I_0(x).

"gsl_sf_bessel_I1_e($x, $result)"
"gsl_sf_bessel_I1($x)"
    -These routines compute the regular modified cylindrical Bessel function of first order, I_1(x).

"gsl_sf_bessel_In_e($n, $x, $result)"
"gsl_sf_bessel_In($n, $x)"
    -These routines compute the regular modified cylindrical Bessel function of order $n, I_n(x).

"gsl_sf_bessel_In_array"
    -

"gsl_sf_bessel_I0_scaled_e($x, $result)"
"gsl_sf_bessel_I0_scaled($x)"
    -These routines compute the scaled regular modified cylindrical Bessel function of zeroth order \exp(-|x|) I_0(x).

"gsl_sf_bessel_I1_scaled_e($x, $result)"
"gsl_sf_bessel_I1_scaled($x)"
    -These routines compute the scaled regular modified cylindrical Bessel function of first order \exp(-|x|) I_1(x).

"gsl_sf_bessel_In_scaled_e($n, $x, $result)"
"gsl_sf_bessel_In_scaled($n, $x)"
    -These routines compute the scaled regular modified cylindrical Bessel function of order $n, \exp(-|x|) I_n(x)

"gsl_sf_bessel_In_scaled_array"
    -

"gsl_sf_bessel_K0_e($x, $result)"
"gsl_sf_bessel_K0($x)"
    -These routines compute the irregular modified cylindrical Bessel function of zeroth order, K_0(x), for x > 0.

"gsl_sf_bessel_K1_e($x, $result)"
"gsl_sf_bessel_K1($x)"
    -These routines compute the irregular modified cylindrical Bessel function of first order, K_1(x), for x > 0.

"gsl_sf_bessel_Kn_e($n, $x, $result)"
"gsl_sf_bessel_Kn($n, $x)"
    -These routines compute the irregular modified cylindrical Bessel function of order $n, K_n(x), for x > 0.

"gsl_sf_bessel_Kn_array"
    -

"gsl_sf_bessel_K0_scaled_e($x, $result)"
"gsl_sf_bessel_K0_scaled($x)"
    -These routines compute the scaled irregular modified cylindrical Bessel function of zeroth order \exp(x) K_0(x) for x>0.

"gsl_sf_bessel_K1_scaled_e($x, $result)"
"gsl_sf_bessel_K1_scaled($x)"
    -

"gsl_sf_bessel_Kn_scaled_e($n, $x, $result)"
"gsl_sf_bessel_Kn_scaled($n, $x)"
    -

"gsl_sf_bessel_Kn_scaled_array "
    -

"gsl_sf_bessel_j0_e($x, $result)"
"gsl_sf_bessel_j0($x)"
    -

"gsl_sf_bessel_j1_e($x, $result)"
"gsl_sf_bessel_j1($x)"
    -

"gsl_sf_bessel_j2_e($x, $result)"
"gsl_sf_bessel_j2($x)"
    -

"gsl_sf_bessel_jl_e($l, $x, $result)"
"gsl_sf_bessel_jl($l, $x)"
    -

"gsl_sf_bessel_jl_array"
    -

"gsl_sf_bessel_jl_steed_array"
    -

"gsl_sf_bessel_y0_e($x, $result)"
"gsl_sf_bessel_y0($x)"
    -

"gsl_sf_bessel_y1_e($x, $result)"
"gsl_sf_bessel_y1($x)"
    -

"gsl_sf_bessel_y2_e($x, $result)"
"gsl_sf_bessel_y2($x)"
    -

"gsl_sf_bessel_yl_e($l, $x, $result)"
"gsl_sf_bessel_yl($l, $x)"
    -

"gsl_sf_bessel_yl_array"
    -

"gsl_sf_bessel_i0_scaled_e($x, $result)"
"gsl_sf_bessel_i0_scaled($x)"
    -

"gsl_sf_bessel_i1_scaled_e($x, $result)"
"gsl_sf_bessel_i1_scaled($x)"
    -

"gsl_sf_bessel_i2_scaled_e($x, $result)"
"gsl_sf_bessel_i2_scaled($x)"
    -

"gsl_sf_bessel_il_scaled_e($l, $x, $result)"
"gsl_sf_bessel_il_scaled($x)"
    -

"gsl_sf_bessel_il_scaled_array"
    -

"gsl_sf_bessel_k0_scaled_e($x, $result)"
"gsl_sf_bessel_k0_scale($x)"
    -

"gsl_sf_bessel_k1_scaled_e($x, $result)"
"gsl_sf_bessel_k1_scaled($x)"
    -

"gsl_sf_bessel_k2_scaled_e($x, $result) "
"gsl_sf_bessel_k2_scaled($x)"
    -

"gsl_sf_bessel_kl_scaled_e($l, $x, $result)"
"gsl_sf_bessel_kl_scaled($l, $x)"
    -

"gsl_sf_bessel_kl_scaled_array"
    -

"gsl_sf_bessel_Jnu_e($nu, $x, $result)"
"gsl_sf_bessel_Jnu($nu, $x)"
    -

"gsl_sf_bessel_sequence_Jnu_e "
    -

"gsl_sf_bessel_Ynu_e($nu, $x, $result)"
"gsl_sf_bessel_Ynu($nu, $x)"
    -

"gsl_sf_bessel_Inu_scaled_e($nu, $x, $result)"
"gsl_sf_bessel_Inu_scaled($nu, $x)"
    -

"gsl_sf_bessel_Inu_e($nu, $x, $result)"
"gsl_sf_bessel_Inu($nu, $x)"
    -

"gsl_sf_bessel_Knu_scaled_e($nu, $x, $result)"
"gsl_sf_bessel_Knu_scaled($nu, $x)"
    -

"gsl_sf_bessel_Knu_e($nu, $x, $result)"
"gsl_sf_bessel_Knu($nu, $x)"
    -

"gsl_sf_bessel_lnKnu_e($nu, $x, $result)"
"gsl_sf_bessel_lnKnu($nu, $x)"
    -

"gsl_sf_bessel_zero_J0_e($s, $result)"
"gsl_sf_bessel_zero_J0($s)"
    -

"gsl_sf_bessel_zero_J1_e($s, $result)"
"gsl_sf_bessel_zero_J1($s)"
    -

"gsl_sf_bessel_zero_Jnu_e($nu, $s, $result)"
"gsl_sf_bessel_zero_Jnu($nu, $s)"
    -

"gsl_sf_clausen_e($x, $result)"
"gsl_sf_clausen($x)"
    -

"gsl_sf_hydrogenicR_1_e($Z, $r, $result)"
"gsl_sf_hydrogenicR_1($Z, $r)"
    -

"gsl_sf_hydrogenicR_e($n, $l, $Z, $r, $result)"
"gsl_sf_hydrogenicR($n, $l, $Z, $r)"
    -

"gsl_sf_coulomb_wave_FG_e($eta, $x, $L_F, $k, $F, gsl_sf_result * Fp, gsl_sf_result * G, $Gp)" - This function computes the Coulomb wave functions F_L(\eta,x), G_{L-k}(\eta,x) and their derivatives F'_L(\eta,x), G'_{L-k}(\eta,x) with respect to $x. The parameters are restricted to L, L-k > -1/2, x > 0 and integer $k. Note that L itself is not restricted to being an integer. The results are stored in the parameters $F, $G for the function values and $Fp, $Gp for the derivative values. $F, $G, $Fp, $Gp are all gsl_result structs. If an overflow occurs, $GSL_EOVRFLW is returned and scaling exponents are returned as second and third values.
"gsl_sf_coulomb_wave_F_array " -
"gsl_sf_coulomb_wave_FG_array" -
"gsl_sf_coulomb_wave_FGp_array" -
"gsl_sf_coulomb_wave_sphF_array" -
"gsl_sf_coulomb_CL_e($L, $eta, $result)" - This function computes the Coulomb wave function normalization constant C_L($eta) for $L > -1.
"gsl_sf_coulomb_CL_arrayi" -
"gsl_sf_coupling_3j_e($two_ja, $two_jb, $two_jc, $two_ma, $two_mb, $two_mc, $result)"
"gsl_sf_coupling_3j($two_ja, $two_jb, $two_jc, $two_ma, $two_mb, $two_mc)"
    - These routines compute the Wigner 3-j coefficient,
       (ja jb jc
        ma mb mc)
     where the arguments are given in half-integer units, ja = $two_ja/2, ma = $two_ma/2, etc.

"gsl_sf_coupling_6j_e($two_ja, $two_jb, $two_jc, $two_jd, $two_je, $two_jf, $result)"
"gsl_sf_coupling_6j($two_ja, $two_jb, $two_jc, $two_jd, $two_je, $two_jf)"
    - These routines compute the Wigner 6-j coefficient,
       {ja jb jc
        jd je jf}
     where the arguments are given in half-integer units, ja = $two_ja/2, ma = $two_ma/2, etc.

"gsl_sf_coupling_RacahW_e"
"gsl_sf_coupling_RacahW"
    -

"gsl_sf_coupling_9j_e($two_ja, $two_jb, $two_jc, $two_jd, $two_je, $two_jf, $two_jg, $two_jh, $two_ji, $result)"
"gsl_sf_coupling_9j($two_ja, $two_jb, $two_jc, $two_jd, $two_je, $two_jf, $two_jg, $two_jh, $two_ji)"
    -These routines compute the Wigner 9-j coefficient,

              {ja jb jc
               jd je jf
               jg jh ji}
     where the arguments are given in half-integer units, ja = two_ja/2, ma = two_ma/2, etc.

"gsl_sf_dawson_e($x, $result)"
"gsl_sf_dawson($x)"
    -These routines compute the value of Dawson's integral for $x.

"gsl_sf_debye_1_e($x, $result)"
"gsl_sf_debye_1($x)"
    -These routines compute the first-order Debye function D_1(x) = (1/x) \int_0^x dt (t/(e^t - 1)).

"gsl_sf_debye_2_e($x, $result)"
"gsl_sf_debye_2($x)"
    -These routines compute the second-order Debye function D_2(x) = (2/x^2) \int_0^x dt (t^2/(e^t - 1)).

"gsl_sf_debye_3_e($x, $result)"
"gsl_sf_debye_3($x)"
    -These routines compute the third-order Debye function D_3(x) = (3/x^3) \int_0^x dt (t^3/(e^t - 1)).

"gsl_sf_debye_4_e($x, $result)"
"gsl_sf_debye_4($x)"
    -These routines compute the fourth-order Debye function D_4(x) = (4/x^4) \int_0^x dt (t^4/(e^t - 1)).

"gsl_sf_debye_5_e($x, $result)"
"gsl_sf_debye_5($x)"
    -These routines compute the fifth-order Debye function D_5(x) = (5/x^5) \int_0^x dt (t^5/(e^t - 1)).

"gsl_sf_debye_6_e($x, $result)"
"gsl_sf_debye_6($x)"
    -These routines compute the sixth-order Debye function D_6(x) = (6/x^6) \int_0^x dt (t^6/(e^t - 1)).

"gsl_sf_dilog_e ($x, $result)"
"gsl_sf_dilog($x)"
    - These routines compute the dilogarithm for a real argument. In Lewin's notation this is Li_2(x), the real part of the dilogarithm of a real x. It is defined by the integral representation Li_2(x) = - \Re \int_0^x ds \log(1-s) / s. Note that \Im(Li_2(x)) = 0 for x <= 1, and -\pi\log(x) for x > 1. Note that Abramowitz & Stegun refer to the Spence integral S(x)=Li_2(1-x) as the dilogarithm rather than Li_2(x).

"gsl_sf_complex_dilog_xy_e" -
"gsl_sf_complex_dilog_e($r, $theta, $result_re, $result_im)" - This function computes the full complex-valued dilogarithm for the complex argument z = r \exp(i \theta). The real and imaginary parts of the result are returned in the $result_re and $result_im gsl_result structs.
"gsl_sf_complex_spence_xy_e" -
"gsl_sf_multiply"
"gsl_sf_multiply_e($x, $y, $result)" - This function multiplies $x and $y storing the product and its associated error in $result.
"gsl_sf_multiply_err_e($x, $dx, $y, $dy, $result)" - This function multiplies $x and $y with associated absolute errors $dx and $dy. The product xy +/- xy \sqrt((dx/x)^2 +(dy/y)^2) is stored in $result.
    -

"gsl_sf_ellint_Kcomp_e($k, $mode, $result)"
"gsl_sf_ellint_Kcomp($k, $mode)"
    -These routines compute the complete elliptic integral K($k) to the accuracy specified by the mode variable mode. Note that Abramowitz & Stegun define this function in terms of the parameter m = k^2.

"gsl_sf_ellint_Ecomp_e($k, $mode, $result)"
"gsl_sf_ellint_Ecomp($k, $mode)"
    -

"gsl_sf_ellint_Pcomp_e($k, $n, $mode, $result)"
"gsl_sf_ellint_Pcomp($k, $n, $mode)"
    -

"gsl_sf_ellint_Dcomp_e"
"gsl_sf_ellint_Dcomp "
    -

"gsl_sf_ellint_F_e($phi, $k, $mode, $result)"
"gsl_sf_ellint_F($phi, $k, $mode)"
    -These routines compute the incomplete elliptic integral F($phi,$k) to the accuracy specified by the mode variable mode. Note that Abramowitz & Stegun define this function in terms of the parameter m = k^2.

"gsl_sf_ellint_E_e($phi, $k, $mode, $result)"
"gsl_sf_ellint_E($phi, $k, $mode)"
    -These routines compute the incomplete elliptic integral E($phi,$k) to the accuracy specified by the mode variable mode. Note that Abramowitz & Stegun define this function in terms of the parameter m = k^2.

"gsl_sf_ellint_P_e($phi, $k, $n, $mode, $result)"
"gsl_sf_ellint_P($phi, $k, $n, $mode)"
    -These routines compute the incomplete elliptic integral \Pi(\phi,k,n) to the accuracy specified by the mode variable mode. Note that Abramowitz & Stegun define this function in terms of the parameters m = k^2 and \sin^2(\alpha) = k^2, with the change of sign n \to -n.

"gsl_sf_ellint_D_e($phi, $k, $n, $mode, $result)"
"gsl_sf_ellint_D($phi, $k, $n, $mode)"
    -These functions compute the incomplete elliptic integral D(\phi,k) which is defined through the Carlson form RD(x,y,z) by the following relation, D(\phi,k,n) = (1/3)(\sin(\phi))^3 RD (1-\sin^2(\phi), 1-k^2 \sin^2(\phi), 1). The argument $n is not used and will be removed in a future release.

"gsl_sf_ellint_RC_e($x, $y, $mode, $result)"
"gsl_sf_ellint_RC($x, $y, $mode)"
    - These routines compute the incomplete elliptic integral RC($x,$y) to the accuracy specified by the mode variable $mode.

"gsl_sf_ellint_RD_e($x, $y, $z, $mode, $result)"
"gsl_sf_ellint_RD($x, $y, $z, $mode)"
    - These routines compute the incomplete elliptic integral RD($x,$y,$z) to the accuracy specified by the mode variable $mode.

"gsl_sf_ellint_RF_e($x, $y, $z, $mode, $result)"
"gsl_sf_ellint_RF($x, $y, $z, $mode)"
    - These routines compute the incomplete elliptic integral RF($x,$y,$z) to the accuracy specified by the mode variable $mode.

"gsl_sf_ellint_RJ_e($x, $y, $z, $p, $mode, $result)"
"gsl_sf_ellint_RJ($x, $y, $z, $p, $mode)"
    - These routines compute the incomplete elliptic integral RJ($x,$y,$z,$p) to the accuracy specified by the mode variable $mode.

"gsl_sf_elljac_e($u, $m)" - This function computes the Jacobian elliptic functions sn(u|m), cn(u|m), dn(u|m) by descending Landen transformations. The function returns 0 if the operation succeded, 1 otherwise and then returns the result of sn, cn and dn in this order.
"gsl_sf_erfc_e($x, $result)"
"gsl_sf_erfc($x)"
    -These routines compute the complementary error function erfc(x) = 1 - erf(x) = (2/\sqrt(\pi)) \int_x^\infty \exp(-t^2).

"gsl_sf_log_erfc_e($x, $result)"
"gsl_sf_log_erfc($x)"
    -These routines compute the logarithm of the complementary error function \log(\erfc(x)).

"gsl_sf_erf_e($x, $result)"
"gsl_sf_erf($x)"
    -These routines compute the error function erf(x), where erf(x) = (2/\sqrt(\pi)) \int_0^x dt \exp(-t^2).

"gsl_sf_erf_Z_e($x, $result)"
"gsl_sf_erf_Z($x)"
    -These routines compute the Gaussian probability density function Z(x) = (1/\sqrt{2\pi}) \exp(-x^2/2).

"gsl_sf_erf_Q_e($x, $result)"
"gsl_sf_erf_Q($x)"
    - These routines compute the upper tail of the Gaussian probability function Q(x) = (1/\sqrt{2\pi}) \int_x^\infty dt \exp(-t^2/2). The hazard function for the normal distribution, also known as the inverse Mill's ratio, is defined as, h(x) = Z(x)/Q(x) = \sqrt{2/\pi} \exp(-x^2 / 2) / \erfc(x/\sqrt 2) It decreases rapidly as x approaches -\infty and asymptotes to h(x) \sim x as x approaches +\infty.

"gsl_sf_hazard_e($x, $result)"
"gsl_sf_hazard($x)"
    - These routines compute the hazard function for the normal distribution.

"gsl_sf_exp_e($x, $result)"
"gsl_sf_exp($x)"
    - These routines provide an exponential function \exp(x) using GSL semantics and error checking.

"gsl_sf_exp_e10_e" -
"gsl_sf_exp_mult_e "
"gsl_sf_exp_mult"
    -

"gsl_sf_exp_mult_e10_e" -
"gsl_sf_expm1_e($x, $result)"
"gsl_sf_expm1($x)"
    -These routines compute the quantity \exp(x)-1 using an algorithm that is accurate for small x.

"gsl_sf_exprel_e($x, $result)"
"gsl_sf_exprel($x)"
    -These routines compute the quantity (\exp(x)-1)/x using an algorithm that is accurate for small x. For small x the algorithm is based on the expansion (\exp(x)-1)/x = 1 + x/2 + x^2/(2*3) + x^3/(2*3*4) + \dots.

"gsl_sf_exprel_2_e($x, $result)"
"gsl_sf_exprel_2($x)"
    -These routines compute the quantity 2(\exp(x)-1-x)/x^2 using an algorithm that is accurate for small x. For small x the algorithm is based on the expansion 2(\exp(x)-1-x)/x^2 = 1 + x/3 + x^2/(3*4) + x^3/(3*4*5) + \dots.

"gsl_sf_exprel_n_e($x, $result)"
"gsl_sf_exprel_n($x)"
    -These routines compute the N-relative exponential, which is the n-th generalization of the functions gsl_sf_exprel and gsl_sf_exprel2. The N-relative exponential is given by,
     exprel_N(x) = N!/x^N (\exp(x) - \sum_{k=0}^{N-1} x^k/k!)
      = 1 + x/(N+1) + x^2/((N+1)(N+2)) + ...
      = 1F1 (1,1+N,x)

"gsl_sf_exp_err_e($x, $dx, $result)" - This function exponentiates $x with an associated absolute error $dx.
"gsl_sf_exp_err_e10_e" -
"gsl_sf_exp_mult_err_e($x, $dx, $y, $dy, $result)" -
"gsl_sf_exp_mult_err_e10_e" -
"gsl_sf_expint_E1_e($x, $result)"
"gsl_sf_expint_E1($x)"
    -These routines compute the exponential integral E_1(x), E_1(x) := \Re \int_1^\infty dt \exp(-xt)/t.

"gsl_sf_expint_E2_e($x, $result)"
"gsl_sf_expint_E2($x)"
    -These routines compute the second-order exponential integral E_2(x),
      E_2(x) := \Re \int_1^\infty dt \exp(-xt)/t^2.

"gsl_sf_expint_En_e($n, $x, $result)"
"gsl_sf_expint_En($n, $x)"
    -These routines compute the exponential integral E_n(x) of order n,
      E_n(x) := \Re \int_1^\infty dt \exp(-xt)/t^n.

"gsl_sf_expint_E1_scaled_e "
"gsl_sf_expint_E1_scaled"
    -

"gsl_sf_expint_E2_scaled_e"
"gsl_sf_expint_E2_scaled "
    -

"gsl_sf_expint_En_scaled_e"
"gsl_sf_expint_En_scaled"
    -

"gsl_sf_expint_Ei_e($x, $result)"
"gsl_sf_expint_Ei($x)"
    -These routines compute the exponential integral Ei(x), Ei(x) := - PV(\int_{-x}^\infty dt \exp(-t)/t) where PV denotes the principal value of the integral.

"gsl_sf_expint_Ei_scaled_e"
"gsl_sf_expint_Ei_scaled "
    -

"gsl_sf_Shi_e($x, $result)"
"gsl_sf_Shi($x)"
    -These routines compute the integral Shi(x) = \int_0^x dt \sinh(t)/t.

"gsl_sf_Chi_e($x, $result)"
"gsl_sf_Chi($x)"
    -These routines compute the integral Chi(x) := \Re[ \gamma_E + \log(x) + \int_0^x dt (\cosh[t]-1)/t] , where \gamma_E is the Euler constant (available as $M_EULER from the Math::GSL::Const module).

"gsl_sf_expint_3_e($x, $result)"
"gsl_sf_expint_3($x)"
    -These routines compute the third-order exponential integral Ei_3(x) = \int_0^xdt \exp(-t^3) for x >= 0.

"gsl_sf_Si_e($x, $result)"
"gsl_sf_Si($x)"
    -These routines compute the Sine integral Si(x) = \int_0^x dt \sin(t)/t.

"gsl_sf_Ci_e($x, $result)"
"gsl_sf_Ci($x)"
    -These routines compute the Cosine integral Ci(x) = -\int_x^\infty dt \cos(t)/t for x > 0.

"gsl_sf_fermi_dirac_m1_e($x, $result)"
"gsl_sf_fermi_dirac_m1($x)"
    -These routines compute the complete Fermi-Dirac integral with an index of -1. This integral is given by F_{-1}(x) = e^x / (1 + e^x).

"gsl_sf_fermi_dirac_0_e($x, $result)"
"gsl_sf_fermi_dirac_0($x)"
    -These routines compute the complete Fermi-Dirac integral with an index of 0. This integral is given by F_0(x) = \ln(1 + e^x).

"gsl_sf_fermi_dirac_1_e($x, $result)"
"gsl_sf_fermi_dirac_1($x)"
    -These routines compute the complete Fermi-Dirac integral with an index of 1, F_1(x) = \int_0^\infty dt (t /(\exp(t-x)+1)).

"gsl_sf_fermi_dirac_2_e($x, $result)"
"gsl_sf_fermi_dirac_2($x)"
    -These routines compute the complete Fermi-Dirac integral with an index of 2, F_2(x) = (1/2) \int_0^\infty dt (t^2 /(\exp(t-x)+1)).

"gsl_sf_fermi_dirac_int_e($j, $x, $result)"
"gsl_sf_fermi_dirac_int($j, $x)"
    -These routines compute the complete Fermi-Dirac integral with an integer index of j, F_j(x) = (1/\Gamma(j+1)) \int_0^\infty dt (t^j /(\exp(t-x)+1)).

"gsl_sf_fermi_dirac_mhalf_e($x, $result)"
"gsl_sf_fermi_dirac_mhalf($x)"
    -These routines compute the complete Fermi-Dirac integral F_{-1/2}(x).

"gsl_sf_fermi_dirac_half_e($x, $result)"
"gsl_sf_fermi_dirac_half($x)"
    -These routines compute the complete Fermi-Dirac integral F_{1/2}(x).

"gsl_sf_fermi_dirac_3half_e($x, $result)"
"gsl_sf_fermi_dirac_3half($x)"
    -These routines compute the complete Fermi-Dirac integral F_{3/2}(x).

"gsl_sf_fermi_dirac_inc_0_e($x, $b, $result)"
"gsl_sf_fermi_dirac_inc_0($x, $b, $result)"
    -These routines compute the incomplete Fermi-Dirac integral with an index of zero, F_0(x,b) = \ln(1 + e^{b-x}) - (b-x).

"gsl_sf_legendre_Pl_e($l, $x, $result)"
"gsl_sf_legendre_Pl($l, $x)"
    -These functions evaluate the Legendre polynomial P_l(x) for a specific value of l, x subject to l >= 0, |x| <= 1

"gsl_sf_legendre_Pl_array"
"gsl_sf_legendre_Pl_deriv_array"
    -

"gsl_sf_legendre_P1_e($x, $result)"
"gsl_sf_legendre_P2_e($x, $result)"
"gsl_sf_legendre_P3_e($x, $result)"
"gsl_sf_legendre_P1($x)"
"gsl_sf_legendre_P2($x)"
"gsl_sf_legendre_P3($x)"
    -These functions evaluate the Legendre polynomials P_l(x) using explicit representations for l=1, 2, 3.

"gsl_sf_legendre_Q0_e($x, $result)"
"gsl_sf_legendre_Q0($x)"
    -These routines compute the Legendre function Q_0(x) for x > -1, x != 1.

"gsl_sf_legendre_Q1_e($x, $result)"
"gsl_sf_legendre_Q1($x)"
    -These routines compute the Legendre function Q_1(x) for x > -1, x != 1.

"gsl_sf_legendre_Ql_e($l, $x, $result)"
"gsl_sf_legendre_Ql($l, $x)"
    -These routines compute the Legendre function Q_l(x) for x > -1, x != 1 and l >= 0.

"gsl_sf_legendre_Plm_e($l, $m, $x, $result)"
"gsl_sf_legendre_Plm($l, $m, $x)"
    -These routines compute the associated Legendre polynomial P_l^m(x) for m >= 0, l >= m, |x| <= 1.

"gsl_sf_legendre_Plm_array"
"gsl_sf_legendre_Plm_deriv_array "
    -

"gsl_sf_legendre_sphPlm_e($l, $m, $x, $result)"
"gsl_sf_legendre_sphPlm($l, $m, $x)"
    -These routines compute the normalized associated Legendre polynomial $\sqrt{(2l+1)/(4\pi)} \sqrt{(l-m)!/(l+m)!} P_l^m(x)$ suitable for use in spherical harmonics. The parameters must satisfy m >= 0, l >= m, |x| <= 1. Theses routines avoid the overflows that occur for the standard normalization of P_l^m(x).

"gsl_sf_legendre_sphPlm_array "
"gsl_sf_legendre_sphPlm_deriv_array"
    -

"gsl_sf_legendre_array_size" -
"gsl_sf_lngamma_e($x, $result)"
"gsl_sf_lngamma($x)"
    -These routines compute the logarithm of the Gamma function, \log(\Gamma(x)), subject to x not being a negative integer or zero. For x<0 the real part of \log(\Gamma(x)) is returned, which is equivalent to \log(|\Gamma(x)|). The function is computed using the real Lanczos method.

"gsl_sf_lngamma_sgn_e($x, $result_lg)" - This routine returns the sign of the gamma function and the logarithm of its magnitude into this order, subject to $x not being a negative integer or zero. The function is computed using the real Lanczos method. The value of the gamma function can be reconstructed using the relation \Gamma(x) = sgn * \exp(resultlg).
"gsl_sf_gamma_e "
"gsl_sf_gamma"
    -

"gsl_sf_gammastar_e"
"gsl_sf_gammastar "
    -

"gsl_sf_gammainv_e"
"gsl_sf_gammainv"
    -

"gsl_sf_lngamma_complex_e "
    -

"gsl_sf_gamma_inc_Q_e"
"gsl_sf_gamma_inc_Q"
    -

"gsl_sf_gamma_inc_P_e "
"gsl_sf_gamma_inc_P"
    -

"gsl_sf_gamma_inc_e"
"gsl_sf_gamma_inc "
    -

"gsl_sf_taylorcoeff_e"
"gsl_sf_taylorcoeff"
    -

"gsl_sf_fact_e "
"gsl_sf_fact"
    -

"gsl_sf_doublefact_e"
"gsl_sf_doublefact "
    -

"gsl_sf_lnfact_e"
"gsl_sf_lnfact"
    -

"gsl_sf_lndoublefact_e "
"gsl_sf_lndoublefact"
    -

"gsl_sf_lnchoose_e"
"gsl_sf_lnchoose "
    -

"gsl_sf_choose_e"
"gsl_sf_choose"
    -

"gsl_sf_lnpoch_e "
"gsl_sf_lnpoch"
    -

"gsl_sf_lnpoch_sgn_e"
    -

"gsl_sf_poch_e "
"gsl_sf_poch"
    -

"gsl_sf_pochrel_e"
"gsl_sf_pochrel "
    -

"gsl_sf_lnbeta_e"
"gsl_sf_lnbeta"
    -

"gsl_sf_lnbeta_sgn_e "
    -

"gsl_sf_beta_e"
"gsl_sf_beta"
    -

"gsl_sf_beta_inc_e "
"gsl_sf_beta_inc"
    -

"gsl_sf_gegenpoly_1_e"
"gsl_sf_gegenpoly_2_e "
"gsl_sf_gegenpoly_3_e"
"gsl_sf_gegenpoly_1"
"gsl_sf_gegenpoly_2 "
"gsl_sf_gegenpoly_3"
    -

"gsl_sf_gegenpoly_n_e"
"gsl_sf_gegenpoly_n "
    -

"gsl_sf_gegenpoly_array"
"gsl_sf_hyperg_0F1_e"
"gsl_sf_hyperg_0F1 "
    -

"gsl_sf_hyperg_1F1_int_e"
"gsl_sf_hyperg_1F1_int"
    -

"gsl_sf_hyperg_1F1_e "
"gsl_sf_hyperg_1F1"
    -

"gsl_sf_hyperg_U_int_e"
"gsl_sf_hyperg_U_int "
    -

"gsl_sf_hyperg_U_int_e10_e"
    -

"gsl_sf_hyperg_U_e"
"gsl_sf_hyperg_U "
    -

"gsl_sf_hyperg_U_e10_e"
    -

"gsl_sf_hyperg_2F1_e"
"gsl_sf_hyperg_2F1 "
    -

"gsl_sf_hyperg_2F1_conj_e"
"gsl_sf_hyperg_2F1_conj"
    -

"gsl_sf_hyperg_2F1_renorm_e "
"gsl_sf_hyperg_2F1_renorm"
    -

"gsl_sf_hyperg_2F1_conj_renorm_e"
"gsl_sf_hyperg_2F1_conj_renorm "
    -

"gsl_sf_hyperg_2F0_e"
"gsl_sf_hyperg_2F0"
    -

"gsl_sf_laguerre_1_e "
"gsl_sf_laguerre_2_e"
"gsl_sf_laguerre_3_e"
"gsl_sf_laguerre_1 "
"gsl_sf_laguerre_2"
"gsl_sf_laguerre_3"
    -

"gsl_sf_laguerre_n_e "
"gsl_sf_laguerre_n"
    -

"gsl_sf_lambert_W0_e"
"gsl_sf_lambert_W0 "
    -

"gsl_sf_lambert_Wm1_e"
"gsl_sf_lambert_Wm1"
    -

"gsl_sf_conicalP_half_e "
"gsl_sf_conicalP_half"
    -

"gsl_sf_conicalP_mhalf_e"
"gsl_sf_conicalP_mhalf "
    -

"gsl_sf_conicalP_0_e"
"gsl_sf_conicalP_0"
    -

"gsl_sf_conicalP_1_e "
"gsl_sf_conicalP_1"
    -

"gsl_sf_conicalP_sph_reg_e"
"gsl_sf_conicalP_sph_reg "
    -

"gsl_sf_conicalP_cyl_reg_e"
"gsl_sf_conicalP_cyl_reg"
    -

"gsl_sf_legendre_H3d_0_e "
"gsl_sf_legendre_H3d_0"
    -

"gsl_sf_legendre_H3d_1_e"
"gsl_sf_legendre_H3d_1 "
    -

"gsl_sf_legendre_H3d_e"
"gsl_sf_legendre_H3d"
    -

"gsl_sf_legendre_H3d_array "
    -

"gsl_sf_log_e"
"gsl_sf_log"
    -

"gsl_sf_log_abs_e "
"gsl_sf_log_abs"
    -

"gsl_sf_complex_log_e"
    -

"gsl_sf_log_1plusx_e "
"gsl_sf_log_1plusx"
    -

"gsl_sf_log_1plusx_mx_e"
"gsl_sf_log_1plusx_mx "
    -

"gsl_sf_mathieu_a_array"
"gsl_sf_mathieu_b_array"
    -

"gsl_sf_mathieu_a "
"gsl_sf_mathieu_b"
    -

"gsl_sf_mathieu_a_coeff"
"gsl_sf_mathieu_b_coeff "
    -

"gsl_sf_mathieu_alloc"
    -

"gsl_sf_mathieu_free"
    -

"gsl_sf_mathieu_ce "
"gsl_sf_mathieu_se"
    -

"gsl_sf_mathieu_ce_array"
"gsl_sf_mathieu_se_array "
    -

"gsl_sf_mathieu_Mc"
"gsl_sf_mathieu_Ms"
    -

"gsl_sf_mathieu_Mc_array "
"gsl_sf_mathieu_Ms_array"
    -

"gsl_sf_pow_int_e"
"gsl_sf_pow_int "
    -

"gsl_sf_psi_int_e"
"gsl_sf_psi_int"
    -

"gsl_sf_psi_e "
"gsl_sf_psi"
    -

"gsl_sf_psi_1piy_e"
"gsl_sf_psi_1piy "
    -

"gsl_sf_complex_psi_e"
    -

"gsl_sf_psi_1_int_e"
"gsl_sf_psi_1_int "
    -

"gsl_sf_psi_1_e "
"gsl_sf_psi_1"
    -

"gsl_sf_psi_n_e "
"gsl_sf_psi_n"
    -

"gsl_sf_result_smash_e"
    -

"gsl_sf_synchrotron_1_e "
"gsl_sf_synchrotron_1"
    -

"gsl_sf_synchrotron_2_e"
"gsl_sf_synchrotron_2 "
    -

"gsl_sf_transport_2_e"
"gsl_sf_transport_2"
    -

"gsl_sf_transport_3_e "
"gsl_sf_transport_3"
    -

"gsl_sf_transport_4_e"
"gsl_sf_transport_4 "
    -

"gsl_sf_transport_5_e"
"gsl_sf_transport_5"
    -

"gsl_sf_sin_e "
"gsl_sf_sin"
    -

"gsl_sf_cos_e"
"gsl_sf_cos "
    -

"gsl_sf_hypot_e"
"gsl_sf_hypot"
    -

"gsl_sf_complex_sin_e "
    -

"gsl_sf_complex_cos_e"
    -

"gsl_sf_complex_logsin_e"
    -

"gsl_sf_sinc_e "
"gsl_sf_sinc"
    -

"gsl_sf_lnsinh_e"
"gsl_sf_lnsinh "
    -

"gsl_sf_lncosh_e"
"gsl_sf_lncosh"
    -

"gsl_sf_polar_to_rect "
    -

"gsl_sf_rect_to_polar"
    -

"gsl_sf_sin_err_e"
"gsl_sf_cos_err_e "
    -

"gsl_sf_angle_restrict_symm_e"
"gsl_sf_angle_restrict_symm"
    -

"gsl_sf_angle_restrict_pos_e "
"gsl_sf_angle_restrict_pos"
    -

"gsl_sf_angle_restrict_symm_err_e"
"gsl_sf_angle_restrict_pos_err_e "
    "gsl_sf_atanint_e"
    "gsl_sf_atanint"
        -These routines compute the Arctangent integral, which is defined as AtanInt(x) = \int_0^x dt \arctan(t)/t.

"gsl_sf_zeta_int_e "
"gsl_sf_zeta_int"
"gsl_sf_zeta_e gsl_sf_zeta "
"gsl_sf_zetam1_e"
"gsl_sf_zetam1"
"gsl_sf_zetam1_int_e "
"gsl_sf_zetam1_int"
"gsl_sf_hzeta_e"
"gsl_sf_hzeta "
"gsl_sf_eta_int_e"
"gsl_sf_eta_int"
"gsl_sf_eta_e"
"gsl_sf_eta "

This module also contains the following constants used as mode in various of those functions :

* GSL_PREC_DOUBLE - Double-precision, a relative accuracy of approximately 2 * 10^-16.
* GSL_PREC_SINGLE - Single-precision, a relative accuracy of approximately 10^-7.
* GSL_PREC_APPROX - Approximate values, a relative accuracy of approximately 5 * 10^-4.

 You can import the functions that you want to use by giving a space separated
 list to Math::GSL::SF when you use the package.  You can also write
 use Math::GSL::SF qw/:all/
 to use all avaible functions of the module. Note that
 the tag names begin with a colon.  Other tags are also available, here is a
 complete list of all tags for this module :

"airy"
"bessel"
"clausen"
"hydrogenic"
"coulumb"
"coupling"
"dawson"
"debye"
"dilog"
"factorial"
"misc"
"elliptic"
"error"
"hypergeometric"
"laguerre"
"legendre"
"gamma"
"transport"
"trig"
"zeta"
"eta"
"vars"

For more informations on the functions, we refer you to the GSL offcial documentation: <http://www.gnu.org/software/gsl/manual/html_node/>

Tip : search on google: site:http://www.gnu.org/software/gsl/manual/html_node/name_of_the_function_you_want

EXAMPLES
This example computes the dilogarithm of 1/10 :

    use Math::GSL::SF qw/dilog/;
    my $x = gsl_sf_dilog(0.1);
    print "gsl_sf_dilog(0.1) = $x\n";

An example using Math::GSL::SF and gnuplot is in the examples/sf folder of the source code.

AUTHORS
Jonathan Leto <jonathan@leto.net> and Thierry Moisan <thierry.moisan@gmail.com>

COPYRIGHT AND LICENSE
Copyright (C) 2008-2009 Jonathan Leto and Thierry Moisan

This program is free software; you can redistribute it and/or modify it under the same terms as Perl itself.