Math::GSL::SF(3pm)

📦

version 0.49 · 6 versions

💖 Math::GSL::SF 💖

welcome to the Math::GSL::SF page, straight out of the User Contributed Perl Documentation part of the manual, bestie 🫶 this is where the Special Functions live and there are so many of them. buckle up, hydrate, we're doing this. 💅

✨ NAME ✨

Math::GSL::SF — Special Functions. that's the name, that's the whole vibe, iconic. 👑

🦋 SYNOPSIS 🦋

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

🌸 DESCRIPTION 🌸

okay so first things first: this module ships a data structure called gsl_sf_result, and if you wanna manifest a brand new one you literally just go:

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

and then you can reach into its little guts like this:

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

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

slay. that's val and err, right there, no cap.

now here's the full list of every single function included 🌈 — yes, all of them, we are NOT skipping anybody:

"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. and $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)"

same energy but for the Airy function Bi($x), 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 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 it's 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)"

scaled Airy again, this time S_B($x) Bi($x). for $x>0 the scaling factor S_B($x) is exp(-(2/3) $x**(3/2)), and it's 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 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)"

the Airy function derivative Bi'($x), 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)"

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 it's 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)"

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 it's 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)

same deal, but 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)

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)

the location of the s-th zero of the Airy function derivative Bi'(x). $result is a gsl_sf_result structure.

and now we slide into Bessel territory 🧚

"gsl_sf_bessel_J0_e($x, $result)"
gsl_sf_bessel_J0($x)

these 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)

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)"

the regular cylindrical Bessel function of order n, J_n($x). 🔥

"gsl_sf_bessel_Jn_array($nmin, $nmax, $x)"

this one computes the values of the regular cylindrical Bessel functions J_n($x) for n from $nmin to $nmax inclusive, and hands the results back in an array reference. heads up though: the values are computed using recurrence relations for efficiency, so they may differ slightly from the exact values. 👀

"gsl_sf_bessel_Y0_e($x, $result)"
gsl_sf_bessel_Y0($x)

these 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)

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 compute the irregular cylindrical Bessel function of order $n, Y_n(x), for x>0. 💜

"gsl_sf_bessel_Yn_array($nmin, $nmax, $x)"

values of the irregular cylindrical Bessel functions Y_n(x) for n from $nmin to $nmax inclusive, returned in an array reference. the domain of this function is $x>0. and again, recurrence relations for efficiency, so they may differ slightly from the exact values.

"gsl_sf_bessel_I0_e($x, $result)"
gsl_sf_bessel_I0($x)

the regular modified cylindrical Bessel function of zeroth order, I_0(x). ✨

"gsl_sf_bessel_I1_e($x, $result)"
gsl_sf_bessel_I1($x)

(no further deets given, she's mysterious 💀)

"gsl_sf_bessel_In_e($n, $x, $result)"
"gsl_sf_bessel_In($n, $x)"

the regular modified cylindrical Bessel function of order $n, I_n(x).

"gsl_sf_bessel_In_array($nmin, $nmax, $x)"

computes the values of the regular modified cylindrical Bessel functions I_n(x) for n from $nmin to $nmax inclusive, results in an array reference. important: the start of the range nmin must be positive or zero. and yeah — recurrence relations for efficiency, so they may differ slightly from the exact values. 🫶

"gsl_sf_bessel_I0_scaled_e($x, $result)"
gsl_sf_bessel_I0_scaled($x)

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)

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)"

the scaled regular modified cylindrical Bessel function of order $n, \exp(-|x|) I_n(x).

"gsl_sf_bessel_In_scaled_array($nmin, $nmax, $x)"

computes the values of the scaled regular cylindrical Bessel functions exp(-|$x|) I_n($x) for n from $nmin to $nmax inclusive, returned in an array reference. the start of the range nmin must be positive or zero. and, as ever, recurrence relations for efficiency, so they may differ slightly from the exact values.

"gsl_sf_bessel_K0_e($x, $result)"
gsl_sf_bessel_K0($x)

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)

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)"

the irregular modified cylindrical Bessel function of order $n, K_n(x), for x > 0.

"gsl_sf_bessel_Kn_array($nmin, $nmax, $x)"

computes the values of the Irregular Modified Cylindrical Bessel Functions K_n($x) for n from $nmin to $nmax inclusive, results in an array reference. recurrence relations for efficiency, so slight differences from the exact values are possible. also this function is ONLY defined for positive $x — anything else and it will throw an exception, so don't play yourself. 🚨

"gsl_sf_bessel_K0_scaled_e($x, $result)"
gsl_sf_bessel_K0_scaled($x)

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)

(no extra tea in the docs for this one 💀)

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

(also glossed over in the source, we move 🫶)

"gsl_sf_bessel_Kn_scaled_array($nmin, $max, $x) "

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. values are computed using recurrence relations for efficiency, so they may differ slightly from the exact values. 🌈

lowercase era, let's go 🦋

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

(no description in the original, just vibes)

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

(still no description)

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

(you guessed it, no description)

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

(no description here either, she's keeping it cute and quiet)

"gsl_sf_bessel_jl_array($lmax, $x)"

this one 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. recurrence relations for efficiency, so they may differ slightly from the exact values. ✨

"gsl_sf_bessel_jl_steed_array($lmax, $x)"

this one uses Steed's 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). Steed's method is more stable than the recurrence used in the other functions, but it's also slower. pick your fighter. ⚔️

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

(no description given)

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

(no description given)

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

(no description given)

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

(no description given)

"gsl_sf_bessel_yl_array($lmax, $x)"

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. recurrence relations for efficiency, so they may differ slightly from the exact values. 🎀

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

(and no description for this last one either — mysterious to the very end 💀)

okay bestie, continuing the DESCRIPTION section like we never left 💅 here's the rest of the function menu, and it's giving "I know a guy" energy. no cap, every single one is here. periodt.

gsl_sf_bessel_i1_scaled_e($x, $result) and gsl_sf_bessel_i1_scaled($x) — no description in the source, just floating here. ✨
gsl_sf_bessel_i2_scaled_e($x, $result) and gsl_sf_bessel_i2_scaled($x) — again, no description. 💅
gsl_sf_bessel_il_scaled_e($l, $x, $result) and gsl_sf_bessel_il_scaled($x) — no description, keep scrolling.
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, $result) and gsl_sf_bessel_k0_scale($x) — no description. (and yes, it's _scale not _scaled, we respect the source 🙈)
gsl_sf_bessel_k1_scaled_e($x, $result) and gsl_sf_bessel_k1_scaled($x) — no description.
gsl_sf_bessel_k2_scaled_e($x, $result) and gsl_sf_bessel_k2_scaled($x) — no description.
gsl_sf_bessel_kl_scaled_e($l, $x, $result) and gsl_sf_bessel_kl_scaled($l, $x) — no description.
gsl_sf_bessel_kl_scaled_array($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, $result) and gsl_sf_bessel_Jnu($nu, $x) — no description.
gsl_sf_bessel_sequence_Jnu_e — just a name, no description.
gsl_sf_bessel_Ynu_e($nu, $x, $result) and gsl_sf_bessel_Ynu($nu, $x) — no description.
gsl_sf_bessel_Inu_scaled_e($nu, $x, $result) and gsl_sf_bessel_Inu_scaled($nu, $x) — no description.
gsl_sf_bessel_Inu_e($nu, $x, $result) and gsl_sf_bessel_Inu($nu, $x) — no description.
gsl_sf_bessel_Knu_scaled_e($nu, $x, $result) and gsl_sf_bessel_Knu_scaled($nu, $x) — no description.
gsl_sf_bessel_Knu_e($nu, $x, $result) and gsl_sf_bessel_Knu($nu, $x) — no description.
gsl_sf_bessel_lnKnu_e($nu, $x, $result) and gsl_sf_bessel_lnKnu($nu, $x) — no description.
gsl_sf_bessel_zero_J0_e($s, $result) and gsl_sf_bessel_zero_J0($s) — no description.
gsl_sf_bessel_zero_J1_e($s, $result) and gsl_sf_bessel_zero_J1($s) — no description.
gsl_sf_bessel_zero_Jnu_e($nu, $s, $result) and gsl_sf_bessel_zero_Jnu($nu, $s) — no description.
gsl_sf_clausen_e($x, $result) and gsl_sf_clausen($x) — no description.
gsl_sf_hydrogenicR_1_e($Z, $r, $result) and gsl_sf_hydrogenicR_1($Z, $r) — no description.
gsl_sf_hydrogenicR_e($n, $l, $Z, $r, $result) and gsl_sf_hydrogenicR($n, $l, $Z, $r) — no description.
gsl_sf_coulomb_wave_FG_e($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, $GSL_EOVRFLW is returned and scaling exponents are returned as second and third values. 🌀
gsl_sf_coulomb_wave_F_array — just a name, no description.
gsl_sf_coulomb_wave_FG_array — just a name, no description.
gsl_sf_coulomb_wave_FGp_array — just a name, no description.
gsl_sf_coulomb_wave_sphF_array — just a name, no description.
gsl_sf_coulomb_CL_e($L, $eta, $result) — this function computes the Coulomb wave function normalization constant C_L($eta) for $L > -1. ✨
gsl_sf_coulomb_CL_arrayi — just a name, no description. (yes, arrayi, not our typo 💀)
gsl_sf_coupling_3j_e($two_ja, $two_jb, $two_jc, $two_ma, $two_mb, $two_mc, $result) and gsl_sf_coupling_3j($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, $result) and gsl_sf_coupling_6j($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 and gsl_sf_coupling_RacahW — no description.
gsl_sf_coupling_9j_e($two_ja, $two_jb, $two_jc, $two_jd, $two_je, $two_jf, $two_jg, $two_jh, $two_ji, $result) and gsl_sf_coupling_9j($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. (note: no $ on two_ja here, we keep it real 💅)

gsl_sf_dawson_e($x, $result) and gsl_sf_dawson($x) — these routines compute the value of Dawson's integral for $x. 🦋
gsl_sf_debye_1_e($x, $result) and gsl_sf_debye_1($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, $result) and gsl_sf_debye_2($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, $result) and gsl_sf_debye_3($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, $result) and gsl_sf_debye_4($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, $result) and gsl_sf_debye_5($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, $result) and gsl_sf_debye_6($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, $result) and gsl_sf_dilog($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 — just a name, no description.
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 $result_im gsl_result structs. 💖
gsl_sf_complex_spence_xy_e — just a name, no description.
gsl_sf_multiply — just a name, no description.
gsl_sf_multiply_e($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, $result) and gsl_sf_ellint_Kcomp($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, $result) and gsl_sf_ellint_Ecomp($k, $mode) — no description.
gsl_sf_ellint_Pcomp_e($k, $n, $mode, $result) and gsl_sf_ellint_Pcomp($k, $n, $mode) — no description.
gsl_sf_ellint_Dcomp_e and gsl_sf_ellint_Dcomp — no description.
gsl_sf_ellint_F_e($phi, $k, $mode, $result) and gsl_sf_ellint_F($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, $result) and gsl_sf_ellint_E($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, $result) and gsl_sf_ellint_P($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, $result) and gsl_sf_ellint_D($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, $result) and gsl_sf_ellint_RC($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, $result) and gsl_sf_ellint_RD($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, $result) and gsl_sf_ellint_RF($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, $result) and gsl_sf_ellint_RJ($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, $result) and gsl_sf_erfc($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, $result) and gsl_sf_log_erfc($x) — these routines compute the logarithm of the complementary error function \log(\erfc(x)). 😭
gsl_sf_erf_e($x, $result) and gsl_sf_erf($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, $result) and gsl_sf_erf_Z($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 give you 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, aka 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. It's giving dramatic exits and chill entrances, honestly. 💫
gsl_sf_hazard_e($x, $result), gsl_sf_hazard($x) — these compute the hazard function for the normal distribution. That's it, that's the whole job.
gsl_sf_exp_e($x, $result), gsl_sf_exp($x) — an exponential function \exp(x) with GSL semantics and error checking baked in. Standard exp but with ✨standards✨.
gsl_sf_exp_e10_e — the page lists this one and moves on, no blurb attached.
gsl_sf_exp_mult_e, gsl_sf_exp_mult — listed together, no description on this page.
gsl_sf_exp_mult_e10_e — also listed with no blurb. iykyk.
gsl_sf_expm1_e($x, $result), gsl_sf_expm1($x) — computes \exp(x)-1 using an algorithm that's accurate for small x. Because subtracting 1 from something tiny is where naive code literally falls apart.
gsl_sf_exprel_e($x, $result), gsl_sf_exprel($x) — computes (\exp(x)-1)/x with an algorithm 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. No division-by-basically-zero shenanigans. 💅
gsl_sf_exprel_2_e($x, $result), gsl_sf_exprel_2($x) — computes 2(\exp(x)-1-x)/x^2, again with a small-x-accurate algorithm. For small x it uses 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) — computes the N-relative exponential, the n-th generalization of the functions gsl_sf_exprel and gsl_sf_exprel2. It's defined as:

 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) — exponentiates $x with an associated absolute error $dx. Errors are part of the whole personality here.
gsl_sf_exp_err_e10_e — listed, no blurb.
gsl_sf_exp_mult_err_e($x, $dx, $y, $dy, $result) — listed, no blurb.
gsl_sf_exp_mult_err_e10_e — listed, no blurb.

gsl_sf_expint_E1_e($x, $result), gsl_sf_expint_E1($x) — computes 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) — computes 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, $x) — computes the exponential integral E_n(x) of order n, E_n(x) := \Re \int_1^\infty dt \exp(-xt)/t^n.
gsl_sf_expint_E1_scaled_e, gsl_sf_expint_E1_scaled — listed, no blurb.
gsl_sf_expint_E2_scaled_e, gsl_sf_expint_E2_scaled — listed, no blurb.
gsl_sf_expint_En_scaled_e, gsl_sf_expint_En_scaled — listed, no blurb.
gsl_sf_expint_Ei_e($x, $result), gsl_sf_expint_Ei($x) — computes the exponential integral Ei(x), Ei(x) := - PV(\int_{-x}^\infty dt \exp(-t)/t) where PV denotes the principal value of the integral. Principal value, main character energy. 👑
gsl_sf_expint_Ei_scaled_e, gsl_sf_expint_Ei_scaled — listed, no blurb.
gsl_sf_Shi_e($x, $result), gsl_sf_Shi($x) — computes the integral Shi(x) = \int_0^x dt \sinh(t)/t.
gsl_sf_Chi_e($x, $result), gsl_sf_Chi($x) — computes 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 $M_EULER from the Math::GSL::Const module). Cute little cameo from another module, love that for us. 🫶
gsl_sf_expint_3_e($x, $result), gsl_sf_expint_3($x) — computes 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) — computes the Sine integral Si(x) = \int_0^x dt \sin(t)/t.
gsl_sf_Ci_e($x, $result), gsl_sf_Ci($x) — computes the Cosine integral Ci(x) = -\int_x^\infty dt \cos(t)/t for x > 0.

🌈 Fermi-Dirac gang 🌈

gsl_sf_fermi_dirac_m1_e($x, $result), gsl_sf_fermi_dirac_m1($x) — computes the complete Fermi-Dirac integral with an index of -1, given by F_{-1}(x) = e^x / (1 + e^x).
gsl_sf_fermi_dirac_0_e($x, $result), gsl_sf_fermi_dirac_0($x) — computes the complete Fermi-Dirac integral with an index of 0, given by F_0(x) = \ln(1 + e^x).
gsl_sf_fermi_dirac_1_e($x, $result), gsl_sf_fermi_dirac_1($x) — computes 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) — computes 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, $x) — computes the complete Fermi-Dirac integral with an integer index of j, F_j(x) = (1/\Gamma(j+1)) \int_0^\infty dt (t^j /(\exp(t-x)+1)).
gsl_sf_fermi_dirac_mhalf_e($x, $result), gsl_sf_fermi_dirac_mhalf($x) — computes the complete Fermi-Dirac integral F_{-1/2}(x).
gsl_sf_fermi_dirac_half_e($x, $result), gsl_sf_fermi_dirac_half($x) — computes the complete Fermi-Dirac integral F_{1/2}(x).
gsl_sf_fermi_dirac_3half_e($x, $result), gsl_sf_fermi_dirac_3half($x) — computes 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) — computes the incomplete Fermi-Dirac integral with an index of zero, F_0(x,b) = \ln(1 + e^{b-x}) - (b-x).

🧚 Legendre polynomials 🧚

gsl_sf_legendre_Pl_e($l, $x, $result), gsl_sf_legendre_Pl($l, $x) — evaluates the Legendre polynomial P_l(x) for a specific value of l, x, subject to l >= 0, |x| <= 1.
gsl_sf_legendre_Pl_array, gsl_sf_legendre_Pl_deriv_array — listed, no blurb.
gsl_sf_legendre_P1_e($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 evaluate the Legendre polynomials P_l(x) using explicit representations for l=1, 2, 3. Hardcoded and proud of it.
gsl_sf_legendre_Q0_e($x, $result), gsl_sf_legendre_Q0($x) — computes the Legendre function Q_0(x) for x > -1, x != 1.
gsl_sf_legendre_Q1_e($x, $result), gsl_sf_legendre_Q1($x) — computes the Legendre function Q_1(x) for x > -1, x != 1.
gsl_sf_legendre_Ql_e($l, $x, $result), gsl_sf_legendre_Ql($l, $x) — computes the Legendre function Q_l(x) for x > -1, x != 1 and l >= 0.
gsl_sf_legendre_Plm_e($l, $m, $x, $result), gsl_sf_legendre_Plm($l, $m, $x) — computes the associated Legendre polynomial P_l^m(x) for m >= 0, l >= m, |x| <= 1.
gsl_sf_legendre_Plm_array, gsl_sf_legendre_Plm_deriv_array — listed, no blurb.
gsl_sf_legendre_sphPlm_e($l, $m, $x, $result), gsl_sf_legendre_sphPlm($l, $m, $x) — computes 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. These routines avoid the overflows that occur for the standard normalization of P_l^m(x). Anti-overflow protection: iconic behavior. 💖
gsl_sf_legendre_sphPlm_array, gsl_sf_legendre_sphPlm_deriv_array — listed, no blurb.
gsl_sf_legendre_array_size — listed, no blurb.

🎀 Gamma, factorials and friends 🎀

gsl_sf_lngamma_e($x, $result), gsl_sf_lngamma($x) — computes 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)|). Computed using the real Lanczos method. So like, negative integers and zero are a hard no. 🚫
gsl_sf_lngamma_sgn_e($x, $result_lg) — 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. Computed using the real Lanczos method. You can reconstruct the value of the gamma function with the relation \Gamma(x) = sgn * \exp(result_lg).
gsl_sf_gamma_e, gsl_sf_gamma — listed, no blurb.
gsl_sf_gammastar_e, gsl_sf_gammastar — listed, no blurb.
gsl_sf_gammainv_e, gsl_sf_gammainv — listed, no blurb.
gsl_sf_lngamma_complex_e — listed, no blurb.
gsl_sf_gamma_inc_Q_e, gsl_sf_gamma_inc_Q — listed, no blurb.
gsl_sf_gamma_inc_P_e, gsl_sf_gamma_inc_P — listed, no blurb.
gsl_sf_gamma_inc_e, gsl_sf_gamma_inc — listed, no blurb.
gsl_sf_taylorcoeff_e, gsl_sf_taylorcoeff — listed, no blurb.
gsl_sf_fact_e, gsl_sf_fact — listed, no blurb.
gsl_sf_doublefact_e, gsl_sf_doublefact — listed, no blurb.
gsl_sf_lnfact_e, gsl_sf_lnfact — listed, no blurb.
gsl_sf_lndoublefact_e, gsl_sf_lndoublefact — listed, no blurb.
gsl_sf_lnchoose_e, gsl_sf_lnchoose — listed, no blurb.
gsl_sf_choose_e, gsl_sf_choose — listed, no blurb.
gsl_sf_lnpoch_e, gsl_sf_lnpoch — listed, no blurb.
gsl_sf_lnpoch_sgn_e — listed, no blurb.
gsl_sf_poch_e, gsl_sf_poch — listed, no blurb.
gsl_sf_pochrel_e, gsl_sf_pochrel — listed, no blurb.
gsl_sf_lnbeta_e, gsl_sf_lnbeta — listed, no blurb.
gsl_sf_lnbeta_sgn_e — listed, no blurb.
gsl_sf_beta_e, gsl_sf_beta — listed, no blurb.
gsl_sf_beta_inc_e, gsl_sf_beta_inc — listed, no blurb.

Heads up bestie: those pairs above really do show up in the page with zero description attached — not me being lazy, that's just what's there. 🙈

🌸 Orthogonal-adjacent and hypergeometric era 🌸

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 — listed, no blurb.
gsl_sf_gegenpoly_n_e, gsl_sf_gegenpoly_n — listed, no blurb.
gsl_sf_gegenpoly_array — listed, no blurb.
gsl_sf_hyperg_0F1_e, gsl_sf_hyperg_0F1 — listed, no blurb.
gsl_sf_hyperg_1F1_int_e, gsl_sf_hyperg_1F1_int — listed, no blurb.
gsl_sf_hyperg_1F1_e, gsl_sf_hyperg_1F1 — listed, no blurb.
gsl_sf_hyperg_U_int_e, gsl_sf_hyperg_U_int — listed, no blurb.
gsl_sf_hyperg_U_int_e10_e — listed, no blurb.
gsl_sf_hyperg_U_e, gsl_sf_hyperg_U — listed, no blurb.
gsl_sf_hyperg_U_e10_e — listed, no blurb.
gsl_sf_hyperg_2F1_e, gsl_sf_hyperg_2F1 — listed, no blurb.
gsl_sf_hyperg_2F1_conj_e, gsl_sf_hyperg_2F1_conj — listed, no blurb.
gsl_sf_hyperg_2F1_renorm_e, gsl_sf_hyperg_2F1_renorm — listed, no blurb.
gsl_sf_hyperg_2F1_conj_renorm_e, gsl_sf_hyperg_2F1_conj_renorm — listed, no blurb.
gsl_sf_hyperg_2F0_e, gsl_sf_hyperg_2F0 — listed, no blurb.
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 — listed, no blurb.
gsl_sf_laguerre_n_e, gsl_sf_laguerre_n — listed, no blurb.
gsl_sf_lambert_W0_e, gsl_sf_lambert_W0 — listed, no blurb.
gsl_sf_lambert_Wm1_e, gsl_sf_lambert_Wm1 — listed, no blurb.
gsl_sf_conicalP_half_e, gsl_sf_conicalP_half — listed, no blurb.
gsl_sf_conicalP_mhalf_e, gsl_sf_conicalP_mhalf — listed, no blurb.
gsl_sf_conicalP_0_e, gsl_sf_conicalP_0 — listed, no blurb.
gsl_sf_conicalP_1_e, gsl_sf_conicalP_1 — listed, no blurb.
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 — okay bestie these ones 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, and they get used as the mode in a bunch of those functions up there 👆 :

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. ✨

So how do you actually get at these functions, you ask?? 🫶 You import the ones you want by handing Math::GSL::SF a space separated list when you use the package. And yes, you can absolutely just pull in everything:

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

to use all available functions of the module. 💅 Heads up though — the tag names begin with a colon. There are other tags available too, and here is the complete list of all the tags for this module, no gatekeeping:

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

For more info on the functions themselves, we refer you to the GSL official documentation: http://www.gnu.org/software/gsl/manual/html_node/ 📚

✨ EXAMPLES ✨

Okay this example computes the dilogarithm of 1/10, watch this 🍓:

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

There's also an example using Math::GSL::SF and gnuplot chillin' in the examples/sf folder of the source code. 👑

👑 AUTHORS 👑

Jonathan "Duke" Leto jonathan@leto.net and Thierry Moisan thierry.moisan@gmail.com — legends, fr fr. 🫶

💖 COPYRIGHT AND LICENSE 💖

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

And that's the whole page bestie, we literally made it to the end 🌈 — credit where it's due: this is perl v5.42.3, dated 2026-08-20, and the page you just read is Math::GSL::SF(3pm). Go forth and compute, main character 💅💖✨

📦 where this page comes from

ubuntu stonking: libmath-gsl-perl 0.49-1build1 · Artistic-1.0-Perl OR GPL-1.0-or-later
💖 girlified by deepseek-ai/DeepSeek-V4.1-Flash on 2026-11-03: a language model's rewrite of the original, under its terms