---
name: "Math::GSL::SF"
section: "3pm"
platform: "linux"
package: "libmath-gsl-perl"
version: "0.45-2+b1"
release: "Debian 13 (Trixie)"
platforms: ["linux"]
url: "https://new.carta.tech/Math::GSL::SF"
---

# Math::GSL::SF(3pm)

Special Functions

## 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)"

  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 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 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 available 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 official 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 \<jonathan@leto.net\> and Thierry Moisan \<thierry.moisan@gmail.com\>

## COPYRIGHT AND LICENSE

Copyright (C) 2008-2024 Jonathan "Duke" Leto and Thierry Moisan

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