 Math::GSL::SF(3) 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)"
This routine computes the values of the regular cylindrical Bessel functions J_n($x) for n from$nmin to $nmax inclusive, returning the results in an array reference. The values are computed using recurrence relations for efficiency, and therefore may differ slightly from the exact values. "gsl_sf_bessel_Y0_e($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 $n, Y_n(x), for x>0. "gsl_sf_bessel_Yn_array($nmin, $nmax,$x)"
This routine computes the values of the irregular cylindrical Bessel functions Y_n(x) for n from $nmin to$nmax inclusive, returning the results in an array reference. The domain of the function is $x>0. The values are computed using recurrence relations for efficiency, and therefore may differ slightly from the exact values. "gsl_sf_bessel_I0_e($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($x)" "gsl_sf_bessel_In_e($n, $x,$result)"
"gsl_sf_bessel_In($n,$x)"
These routines compute the regular modified cylindrical Bessel function of order $n, I_n(x). "gsl_sf_bessel_In_array($nmin, $nmax,$x)"
This routine computes the values of the regular modified cylindrical Bessel functions I_n(x) for n from $nmin to$nmax inclusive, returning the results in an array reference. The start of the range nmin must be positive or zero. The values are computed using recurrence relations for efficiency, and therefore may differ slightly from the exact values.
"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($nmin,$nmax, $x)" This routine computes the values of the scaled regular cylindrical Bessel functions exp(-|$x|) I_n($x) for n from$nmin to $nmax inclusive, returning the results in an array reference. The start of the range nmin must be positive or zero. The values are computed using recurrence relations for efficiency, and therefore may differ slightly from the exact values. "gsl_sf_bessel_K0_e($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 $n, K_n(x), for x > 0. "gsl_sf_bessel_Kn_array($nmin, $nmax,$x)"
This routine computes the values of the Irregular Modified Cylindrical Bessel Functions K_n($x) for n from$nmin to $nmax inclusive, returning the results in an array reference. The values are computed using recurrence relations for efficiency, and therefore may differ slightly from the exact values. This function is only defined for positive$x and with throw an exception otherwise.
"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, $x)" "gsl_sf_bessel_Kn_scaled_array($nmin, $max,$x) "
This routine computes the values of the scaled irregular cylindrical Bessel functions exp(x) K_n(x) for n from nmin to nmax inclusive, storing the results in the array result_array. The start of the range nmin must be positive or zero. The domain of the function is x>0. The values are computed using recurrence relations for efficiency, and therefore may differ slightly from the exact values.
"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($lmax,$x)"
This routine computes the values of the regular spherical Bessel functions j_l(x) for l from 0 to $lmax inclusive for$lmax >= 0 and $x >= 0, returning the results in an array reference. The values are computed using recurrence relations for efficiency, and therefore may differ slightly from the exact values. "gsl_sf_bessel_jl_steed_array($lmax, $x)" This routine uses SteedXs method to compute the values of the regular spherical Bessel functions j_l(x) for l from 0 to$lmax inclusive for $lmax >= 0 and$x >= 0, storing the results in the array result_array. The Steed/Barnett algorithm is described in Comp. Phys. Comm. 21, 297 (1981). SteedXs method is more stable than the recurrence used in the other functions but is also slower.
"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($lmax,$x)"
This routine computes the values of the irregular spherical Bessel functions y_l(x) for l from 0 to $lmax inclusive for lmax >= 0, returning the results in an array reference. The values are computed using recurrence relations for efficiency, and therefore may differ slightly from the exact values. "gsl_sf_bessel_i0_scaled_e($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($lmax,$x)"
This routine computes the values of the scaled regular modified spherical Bessel functions exp(-|x|) i_l(x) for l from 0 to $lmax inclusive for$lmax >= 0, returning the results in an array reference. The values are computed using recurrence relations for efficiency, and therefore may differ slightly from the exact values.
"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($lmax,$x)"
This routine computes the values of the scaled irregular modified spherical Bessel functions exp($x) k_l($x) for l from 0 to lmax inclusive for $lmax >= 0 and$x>0, returning the results in an array reference. The values are computed using recurrence relations for efficiency, and therefore may differ slightly from the exact values.
"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 succeeded, 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 information on the functions, we refer you to the GSL offcial documentation: <http://www.gnu.org/software/gsl/manual/html_node/>

## 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 Duke'' Leto <[email protected]> and Thierry Moisan <[email protected]>

Copyright (C) 2008-2011 Jonathan Duke'' Leto and Thierry Moisan