GNU Scientific Library -- Reference Manual - Special Functions

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# Special Functions

This chapter describes the GSL special function library. The library includes routines for calculating the values of Airy functions, Bessel functions, Clausen functions, Coulomb wave functions, Coupling coefficients, the Dawson function, Debye functions, Dilogarithms, Elliptic integrals, Jacobi elliptic functions, Error functions, Exponential integrals, Fermi-Dirac functions, Gamma functions, Gegenbauer functions, Hypergeometric functions, Laguerre functions, Legendre functions and Spherical Harmonics, the Psi (Digamma) Function, Synchrotron functions, Transport functions, Trigonometric functions and Zeta functions. Each routine also computes an estimate of the numerical error in the calculated value of the function.

The functions are declared in individual header files, such as ``gsl_sf_airy.h'`, ``gsl_sf_bessel.h'`, etc. The complete set of header files can be included using the file ``gsl_sf.h'`.

## Usage

The special functions are available in two calling conventions, a natural form which returns the numerical value of the function and an error-handling form which returns an error code. The two types of function provide alternative ways of accessing the same underlying code.

The natural form returns only the value of the function and can be used directly in mathematical expressions.. For example, the following function call will compute the value of the Bessel function J_0(x),

    double y = gsl_sf_bessel_J0 (x);

There is no way to access an error code or to estimate the error using this method. To allow access to this information the alternative error-handling form stores the value and error in a modifiable argument,

    gsl_sf_result result;
    int status = gsl_sf_bessel_J0_e (x, &result);

The error-handling functions have the suffix `_e`. The returned status value indicates error conditions such as overflow, underflow or loss of precision. If there are no errors the error-handling functions return `GSL_SUCCESS`.

## The gsl_sf_result struct

The error handling form of the special functions always calculate an error estimate along with the value of the result. Therefore, structures are provided for amalgamating a value and error estimate. These structures are declared in the header file ``gsl_sf_result.h'`.

The `gsl_sf_result` struct contains value and error fields.

    typedef struct
    {
      double val;
      double err;
    } gsl_sf_result;

The field val contains the value and the field err contains an estimate of the absolute error in the value.

In some cases, an overflow or underflow can be detected and handled by a function. In this case, it may be possible to return a scaling exponent as well as an error/value pair in order to save the result from exceeding the dynamic range of the built-in types. The `gsl_sf_result_e10` struct contains value and error fields as well as an exponent field such that the actual result is obtained as `result * 10^(e10)`.

    typedef struct
    {
      double val;
      double err;
      int    e10;
    } gsl_sf_result_e10;

## Modes

The goal of the library is to achieve double precision accuracy wherever possible. However the cost of evaluating some special functions to double precision can be significant, particularly where very high order terms are required. In these cases a `mode` argument allows the accuracy of the function to be reduced in order to improve performance. The following precision levels are available for the mode argument,

`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.

The approximate mode provides the fastest evaluation at the lowest accuracy.

## Airy Functions and Derivatives

The Airy functions Ai(x) and Bi(x) are defined by the integral representations,

    Ai(x) = (1/\pi) \int_0^\infty \cos((1/3) t^3 + xt) dt
    Bi(x) = (1/\pi) \int_0^\infty (e^(-(1/3) t^3) + \sin((1/3) t^3 + xt)) dt

For further information see Abramowitz & Stegun, Section 10.4. The Airy functions are defined in the header file ``gsl_sf_airy.h'`.

### Airy Functions

Function: double gsl_sf_airy_Ai (double x, gsl_mode_t mode)
Function: int gsl_sf_airy_Ai_e (double x, gsl_mode_t mode, gsl_sf_result * result)
  These routines compute the Airy function Ai(x) with an accuracy specified by mode.

Function: double gsl_sf_airy_Bi (double x, gsl_mode_t mode)
Function: int gsl_sf_airy_Bi_e (double x, gsl_mode_t mode, gsl_sf_result * result)
  These routines compute the Airy function Bi(x) with an accuracy specified by mode.

Function: double gsl_sf_airy_Ai_scaled (double x, gsl_mode_t mode)
Function: int gsl_sf_airy_Ai_scaled_e (double x, gsl_mode_t mode, gsl_sf_result * result)
  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.

Function: double gsl_sf_airy_Bi_scaled (double x, gsl_mode_t mode)
Function: int gsl_sf_airy_Bi_scaled_e (double x, gsl_mode_t mode, gsl_sf_result * result)
  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.

### Derivatives of Airy Functions

Function: double gsl_sf_airy_Ai_deriv (double x, gsl_mode_t mode)
Function: int gsl_sf_airy_Ai_deriv_e (double x, gsl_mode_t mode, gsl_sf_result * result)
  These routines compute the Airy function derivative Ai'(x) with an accuracy specified by mode.

Function: double gsl_sf_airy_Bi_deriv (double x, gsl_mode_t mode)
Function: int gsl_sf_airy_Bi_deriv_e (double x, gsl_mode_t mode, gsl_sf_result * result)
  These routines compute the Airy function derivative Bi'(x) with an accuracy specified by mode.

Function: double gsl_sf_airy_Ai_deriv_scaled (double x, gsl_mode_t mode)
Function: int gsl_sf_airy_Ai_deriv_scaled_e (double x, gsl_mode_t mode, gsl_sf_result * result)
  These routines compute the derivative of the scaled Airy function S_A(x) Ai(x).

Function: double gsl_sf_airy_Bi_deriv_scaled (double x, gsl_mode_t mode)
Function: int gsl_sf_airy_Bi_deriv_scaled_e (double x, gsl_mode_t mode, gsl_sf_result * result)
  These routines compute the derivative of the scaled Airy function S_B(x) Bi(x).

### Zeros of Airy Functions

Function: double gsl_sf_airy_zero_Ai (unsigned int s)
Function: int gsl_sf_airy_zero_Ai_e (unsigned int s, gsl_sf_result * result)
  These routines compute the location of the s-th zero of the Airy function Ai(x).

Function: double gsl_sf_airy_zero_Bi (unsigned int s)
Function: int gsl_sf_airy_zero_Bi_e (unsigned int s, gsl_sf_result * result)
  These routines compute the location of the s-th zero of the Airy function Bi(x).

### Zeros of Derivatives of Airy Functions

Function: double gsl_sf_airy_zero_Ai_deriv (unsigned int s)
Function: int gsl_sf_airy_zero_Ai_deriv_e (unsigned int s, gsl_sf_result * result)
  These routines compute the location of the s-th zero of the Airy function derivative Ai'(x).

Function: double gsl_sf_airy_zero_Bi_deriv (unsigned int s)
Function: int gsl_sf_airy_zero_Bi_deriv_e (unsigned int s, gsl_sf_result * result)
  These routines compute the location of the s-th zero of the Airy function derivative Bi'(x).

## Bessel Functions

The routines described in this section compute the Cylindrical Bessel functions J_n(x), Y_n(x), Modified cylindrical Bessel functions I_n(x), K_n(x), Spherical Bessel functions j_l(x), y_l(x), and Modified Spherical Bessel functions i_l(x), k_l(x). For more information see Abramowitz & Stegun, Chapters 9 and 10. The Bessel functions are defined in the header file ``gsl_sf_bessel.h'`.

### Regular Cylindrical Bessel Functions

Function: double gsl_sf_bessel_J0 (double x)
Function: int gsl_sf_bessel_J0_e (double x, gsl_sf_result * result)
  These routines compute the regular cylindrical Bessel function of zeroth order, J_0(x).

Function: double gsl_sf_bessel_J1 (double x)
Function: int gsl_sf_bessel_J1_e (double x, gsl_sf_result * result)
  These routines compute the regular cylindrical Bessel function of first order, J_1(x).

Function: double gsl_sf_bessel_Jn (int n, double x)
Function: int gsl_sf_bessel_Jn_e (int n, double x, gsl_sf_result * result)
  These routines compute the regular cylindrical Bessel function of order n, J_n(x).

Function: int gsl_sf_bessel_Jn_array (int nmin, int nmax, double x, double result_array[])
  This routine computes the values of the regular cylindrical Bessel functions J_n(x) for n from nmin to nmax inclusive, storing the results in the array result_array. The values are computed using recurrence relations, for efficiency, and therefore may differ slightly from the exact values.

### Irregular Cylindrical Bessel Functions

Function: double gsl_sf_bessel_Y0 (double x)
Function: int gsl_sf_bessel_Y0_e (double x, gsl_sf_result * result)
  These routines compute the irregular cylindrical Bessel function of zeroth order, Y_0(x), for x>0.

Function: double gsl_sf_bessel_Y1 (double x)
Function: int gsl_sf_bessel_Y1_e (double x, gsl_sf_result * result)
  These routines compute the irregular cylindrical Bessel function of first order, Y_1(x), for x>0.

Function: double gsl_sf_bessel_Yn (int n,double x)
Function: int gsl_sf_bessel_Yn_e (int n,double x, gsl_sf_result * result)
  These routines compute the irregular cylindrical Bessel function of order n, Y_n(x), for x>0.

Function: int gsl_sf_bessel_Yn_array (int nmin, int nmax, double x, double result_array[])
  This routine computes the values of the irregular cylindrical Bessel functions Y_n(x) for n from nmin to nmax inclusive, storing the results in the array result_array. 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.

### Regular Modified Cylindrical Bessel Functions

Function: double gsl_sf_bessel_I0 (double x)
Function: int gsl_sf_bessel_I0_e (double x, gsl_sf_result * result)
  These routines compute the regular modified cylindrical Bessel function of zeroth order, I_0(x).

Function: double gsl_sf_bessel_I1 (double x)
Function: int gsl_sf_bessel_I1_e (double x, gsl_sf_result * result)
  These routines compute the regular modified cylindrical Bessel function of first order, I_1(x).

Function: double gsl_sf_bessel_In (int n, double x)
Function: int gsl_sf_bessel_In_e (int n, double x, gsl_sf_result * result)
  These routines compute the regular modified cylindrical Bessel function of order n, I_n(x).

Function: int gsl_sf_bessel_In_array (int nmin, int nmax, double x, double result_array[])
  This routine computes the values of the regular modified cylindrical Bessel functions I_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 values are computed using recurrence relations, for efficiency, and therefore may differ slightly from the exact values.

Function: double gsl_sf_bessel_I0_scaled (double x)
Function: int gsl_sf_bessel_I0_scaled_e (double x, gsl_sf_result * result)
  These routines compute the scaled regular modified cylindrical Bessel function of zeroth order \exp(-|x|) I_0(x).

Function: double gsl_sf_bessel_I1_scaled (double x)
Function: int gsl_sf_bessel_I1_scaled_e (double x, gsl_sf_result * result)
  These routines compute the scaled regular modified cylindrical Bessel function of first order \exp(-|x|) I_1(x).

Function: double gsl_sf_bessel_In_scaled (int n, double x)
Function: int gsl_sf_bessel_In_scaled_e (int n, double x, gsl_sf_result * result)
  These routines compute the scaled regular modified cylindrical Bessel function of order n, \exp(-|x|) I_n(x)

Function: int gsl_sf_bessel_In_scaled_array (int nmin, int nmax, double x, double result_array[])
  This routine computes the values of the scaled regular cylindrical Bessel functions \exp(-|x|) I_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 values are computed using recurrence relations, for efficiency, and therefore may differ slightly from the exact values.

### Irregular Modified Cylindrical Bessel Functions

Function: double gsl_sf_bessel_K0 (double x)
Function: int gsl_sf_bessel_K0_e (double x, gsl_sf_result * result)
  These routines compute the irregular modified cylindrical Bessel function of zeroth order, K_0(x), for x > 0.

Function: double gsl_sf_bessel_K1 (double x)
Function: int gsl_sf_bessel_K1_e (double x, gsl_sf_result * result)
  These routines compute the irregular modified cylindrical Bessel function of first order, K_1(x), for x > 0.

Function: double gsl_sf_bessel_Kn (int n, double x)
Function: int gsl_sf_bessel_Kn_e (int n, double x, gsl_sf_result * result)
  These routines compute the irregular modified cylindrical Bessel function of order n, K_n(x), for x > 0.

Function: int gsl_sf_bessel_Kn_array (int nmin, int nmax, double x, double result_array[])
  This routine computes the values of the irregular modified cylindrical Bessel functions 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.

Function: double gsl_sf_bessel_K0_scaled (double x)
Function: int gsl_sf_bessel_K0_scaled_e (double x, gsl_sf_result * result)
  These routines compute the scaled irregular modified cylindrical Bessel function of zeroth order \exp(x) K_0(x) for x>0.

Function: double gsl_sf_bessel_K1_scaled (double x)
Function: int gsl_sf_bessel_K1_scaled_e (double x, gsl_sf_result * result)
  These routines compute the scaled irregular modified cylindrical Bessel function of first order \exp(x) K_1(x) for x>0.

Function: double gsl_sf_bessel_Kn_scaled (int n, double x)
Function: int gsl_sf_bessel_Kn_scaled_e (int n, double x, gsl_sf_result * result)
  These routines compute the scaled irregular modified cylindrical Bessel function of order n, \exp(x) K_n(x), for x>0.

Function: int gsl_sf_bessel_Kn_scaled_array (int nmin, int nmax, double x, double result_array[])
  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.

### Regular Spherical Bessel Functions

Function: double gsl_sf_bessel_j0 (double x)
Function: int gsl_sf_bessel_j0_e (double x, gsl_sf_result * result)
  These routines compute the regular spherical Bessel function of zeroth order, j_0(x) = \sin(x)/x.

Function: double gsl_sf_bessel_j1 (double x)
Function: int gsl_sf_bessel_j1_e (double x, gsl_sf_result * result)
  These routines compute the regular spherical Bessel function of first order, j_1(x) = (\sin(x)/x - \cos(x))/x.

Function: double gsl_sf_bessel_j2 (double x)
Function: int gsl_sf_bessel_j2_e (double x, gsl_sf_result * result)
  These routines compute the regular spherical Bessel function of second order, j_2(x) = ((3/x^2 - 1)\sin(x) - 3\cos(x)/x)/x.

Function: double gsl_sf_bessel_jl (int l, double x)
Function: int gsl_sf_bessel_jl_e (int l, double x, gsl_sf_result * result)
  These routines compute the regular spherical Bessel function of order l, j_l(x), for l >= 0 and x >= 0.

Function: int gsl_sf_bessel_jl_array (int lmax, double x, double result_array[])
  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, storing the results in the array result_array. The values are computed using recurrence relations, for efficiency, and therefore may differ slightly from the exact values.

Function: int gsl_sf_bessel_jl_steed_array (int lmax, double x, double * jl_x_array)
  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.

### Irregular Spherical Bessel Functions

Function: double gsl_sf_bessel_y0 (double x)
Function: int gsl_sf_bessel_y0_e (double x, gsl_sf_result * result)
  These routines compute the irregular spherical Bessel function of zeroth order, y_0(x) = -\cos(x)/x.

Function: double gsl_sf_bessel_y1 (double x)
Function: int gsl_sf_bessel_y1_e (double x, gsl_sf_result * result)
  These routines compute the irregular spherical Bessel function of first order, y_1(x) = -(\cos(x)/x + \sin(x))/x.

Function: double gsl_sf_bessel_y2 (double x)
Function: int gsl_sf_bessel_y2_e (double x, gsl_sf_result * result)
  These routines compute the irregular spherical Bessel function of second order, y_2(x) = (-3/x^2 + 1/x)\cos(x) - (3/x^2)\sin(x).

Function: double gsl_sf_bessel_yl (int l, double x)
Function: int gsl_sf_bessel_yl_e (int l, double x, gsl_sf_result * result)
  These routines compute the irregular spherical Bessel function of order l, y_l(x), for l >= 0.

Function: int gsl_sf_bessel_yl_array (int lmax, double x, double result_array[])
  This routine computes the values of the irregular spherical Bessel functions y_l(x) for l from 0 to lmax inclusive for lmax >= 0, storing the results in the array result_array. The values are computed using recurrence relations, for efficiency, and therefore may differ slightly from the exact values.

### Regular Modified Spherical Bessel Functions

The regular modified spherical Bessel functions i_l(x) are related to the modified Bessel functions of fractional order, i_l(x) = \sqrt{\pi/(2x)} I_{l+1/2}(x)

Function: double gsl_sf_bessel_i0_scaled (double x)
Function: int gsl_sf_bessel_i0_scaled_e (double x, gsl_sf_result * result)
  These routines compute the scaled regular modified spherical Bessel function of zeroth order, \exp(-|x|) i_0(x).

Function: double gsl_sf_bessel_i1_scaled (double x)
Function: int gsl_sf_bessel_i1_scaled_e (double x, gsl_sf_result * result)
  These routines compute the scaled regular modified spherical Bessel function of first order, \exp(-|x|) i_1(x).

Function: double gsl_sf_bessel_i2_scaled (double x)
Function: int gsl_sf_bessel_i2_scaled_e (double x, gsl_sf_result * result)
  These routines compute the scaled regular modified spherical Bessel function of second order, \exp(-|x|) i_2(x)

Function: double gsl_sf_bessel_il_scaled (int l, double x)
Function: int gsl_sf_bessel_il_scaled_e (int l, double x, gsl_sf_result * result)
  These routines compute the scaled regular modified spherical Bessel function of order l, \exp(-|x|) i_l(x)

Function: int gsl_sf_bessel_il_scaled_array (int lmax, double x, double result_array[])
  This routine computes the values of the scaled regular modified cylindrical Bessel functions \exp(-|x|) i_l(x) for l from 0 to lmax inclusive for lmax >= 0, storing the results in the array result_array. The values are computed using recurrence relations, for efficiency, and therefore may differ slightly from the exact values.

### Irregular Modified Spherical Bessel Functions

The irregular modified spherical Bessel functions k_l(x) are related to the irregular modified Bessel functions of fractional order, k_l(x) = \sqrt{\pi/(2x)} K_{l+1/2}(x).

Function: double gsl_sf_bessel_k0_scaled (double x)
Function: int gsl_sf_bessel_k0_scaled_e (double x, gsl_sf_result * result)
  These routines compute the scaled irregular modified spherical Bessel function of zeroth order, \exp(x) k_0(x), for x>0.

Function: double gsl_sf_bessel_k1_scaled (double x)
Function: int gsl_sf_bessel_k1_scaled_e (double x, gsl_sf_result * result)
  These routines compute the scaled irregular modified spherical Bessel function of first order, \exp(x) k_1(x), for x>0.

Function: double gsl_sf_bessel_k2_scaled (double x)
Function: int gsl_sf_bessel_k2_scaled_e (double x, gsl_sf_result * result)
  These routines compute the scaled irregular modified spherical Bessel function of second order, \exp(x) k_2(x), for x>0.

Function: double gsl_sf_bessel_kl_scaled (int l, double x)
Function: int gsl_sf_bessel_kl_scaled_e (int l, double x, gsl_sf_result * result)
  These routines compute the scaled irregular modified spherical Bessel function of order l, \exp(x) k_l(x), for x>0.

Function: int gsl_sf_bessel_kl_scaled_array (int lmax, double x, double result_array[])
  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, storing the results in the array result_array. The values are computed using recurrence relations, for efficiency, and therefore may differ slightly from the exact values.

### Regular Bessel Function - Fractional Order

Function: double gsl_sf_bessel_Jnu (double nu, double x)
Function: int gsl_sf_bessel_Jnu_e (double nu, double x, gsl_sf_result * result)
  These routines compute the regular cylindrical Bessel function of fractional order nu, J_\nu(x).

Function: int gsl_sf_bessel_sequence_Jnu_e (double nu, gsl_mode_t mode, size_t size, double v[])
  This function computes the regular cylindrical Bessel function of fractional order \nu, J_\nu(x), evaluated at a series of x values. The array v of length size contains the x values. They are assumed to be strictly ordered and positive. The array is over-written with the values of J_\nu(x_i).

### Irregular Bessel Functions - Fractional Order

Function: double gsl_sf_bessel_Ynu (double nu, double x)
Function: int gsl_sf_bessel_Ynu_e (double nu, double x, gsl_sf_result * result)
  These routines compute the irregular cylindrical Bessel function of fractional order nu, Y_\nu(x).

### Regular Modified Bessel Functions - Fractional Order

Function: double gsl_sf_bessel_Inu (double nu, double x)
Function: int gsl_sf_bessel_Inu_e (double nu, double x, gsl_sf_result * result)
  These routines compute the regular modified Bessel function of fractional order nu, I_\nu(x) for x>0, \nu>0.

Function: double gsl_sf_bessel_Inu_scaled (double nu, double x)
Function: int gsl_sf_bessel_Inu_scaled_e (double nu, double x, gsl_sf_result * result)
  These routines compute the scaled regular modified Bessel function of fractional order nu, \exp(-|x|)I_\nu(x) for x>0, \nu>0.

### Irregular Modified Bessel Functions - Fractional Order

Function: double gsl_sf_bessel_Knu (double nu, double x)
Function: int gsl_sf_bessel_Knu_e (double nu, double x, gsl_sf_result * result)
  These routines compute the irregular modified Bessel function of fractional order nu, K_\nu(x) for x>0, \nu>0.

Function: double gsl_sf_bessel_lnKnu (double nu, double x)
Function: int gsl_sf_bessel_lnKnu_e (double nu, double x, gsl_sf_result * result)
  These routines compute the logarithm of the irregular modified Bessel function of fractional order nu, \ln(K_\nu(x)) for x>0, \nu>0.

Function: double gsl_sf_bessel_Knu_scaled (double nu, double x)
Function: int gsl_sf_bessel_Knu_scaled_e (double nu, double x, gsl_sf_result * result)
  These routines compute the scaled irregular modified Bessel function of fractional order nu, \exp(+|x|) K_\nu(x) for x>0, \nu>0.

### Zeros of Regular Bessel Functions

Function: double gsl_sf_bessel_zero_J0 (unsigned int s)
Function: int gsl_sf_bessel_zero_J0_e (unsigned int s, gsl_sf_result * result)
  These routines compute the location of the s-th positive zero of the Bessel function J_0(x).

Function: double gsl_sf_bessel_zero_J1 (unsigned int s)
Function: int gsl_sf_bessel_zero_J1_e (unsigned int s, gsl_sf_result * result)
  These routines compute the location of the s-th positive zero of the Bessel function J_1(x).

Function: double gsl_sf_bessel_zero_Jnu (double nu, unsigned int s)
Function: int gsl_sf_bessel_zero_Jnu_e (double nu, unsigned int s, gsl_sf_result * result)
  These routines compute the location of the s-th positive zero of the Bessel function J_\nu(x).

## Clausen Functions

The Clausen function is defined by the following integral,

    Cl_2(x) = - \int_0^x dt \log(2 \sin(t/2))

It is related to the dilogarithm by Cl_2(\theta) = \Im Li_2(\exp(i \theta)). The Clausen functions are declared in the header file ``gsl_sf_clausen.h'`.

Function: double gsl_sf_clausen (double x)
Function: int gsl_sf_clausen_e (double x, gsl_sf_result * result)
  These routines compute the Clausen integral Cl_2(x).

## Coulomb Functions

The Coulomb functions are declared in the header file ``gsl_sf_coulomb.h'`. Both bound state and scattering solutions are available.

### Normalized Hydrogenic Bound States

Function: double gsl_sf_hydrogenicR_1 (double Z, double r)
Function: int gsl_sf_hydrogenicR_1_e (double Z, double r, gsl_sf_result * result)
  These routines compute the lowest-order normalized hydrogenic bound state radial wavefunction R_1 := 2Z \sqrt{Z} \exp(-Z r).

Function: double gsl_sf_hydrogenicR (int n, int l, double Z, double r)
Function: int gsl_sf_hydrogenicR_e (int n, int l, double Z, double r, gsl_sf_result * result)
  These routines compute the n-th normalized hydrogenic bound state radial wavefunction,

      R_n := 2 (Z^{3/2}/n^2) \sqrt{(n-l-1)!/(n+l)!} \exp(-Z r/n) (2Z/n)^l
                L^{2l+1}_{n-l-1}(2Z/n r).

  The normalization is chosen such that the wavefunction \psi is given by \psi(n,l,r) = R_n Y_{lm}.

### Coulomb Wave Functions

The Coulomb wave functions F_L(\eta,x), G_L(\eta,x) are described in Abramowitz & Stegun, Chapter 14. Because there can be a large dynamic range of values for these functions, overflows are handled gracefully. If an overflow occurs, `GSL_EOVRFLW` is signalled and exponent(s) are returned through the modifiable parameters exp_F, exp_G. The full solution can be reconstructed from the following relations,

    F_L(eta,x)  =  fc[k_L] * exp(exp_F)
    G_L(eta,x)  =  gc[k_L] * exp(exp_G)

    F_L'(eta,x) = fcp[k_L] * exp(exp_F)
    G_L'(eta,x) = gcp[k_L] * exp(exp_G)

Function: int gsl_sf_coulomb_wave_FG_e (double eta, double x, double L_F, int k, gsl_sf_result * F, gsl_sf_result * Fp, gsl_sf_result * G, gsl_sf_result * Gp, double * exp_F, double * exp_G)
  This function computes the coulomb wave functions F_L(\eta,x), G_{L-k}(\eta,x) and their derivatives with respect to x, F'_L(\eta,x) G'_{L-k}(\eta,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. If an overflow occurs, `GSL_EOVRFLW` is returned and scaling exponents are stored in the modifiable parameters exp_F, exp_G.

Function: int gsl_sf_coulomb_wave_F_array (double L_min, int kmax, double eta, double x, double fc_array[], double * F_exponent)
  This function computes the function F_L(eta,x) for L = Lmin \dots Lmin + kmax storing the results in fc_array. In the case of overflow the exponent is stored in F_exponent.

Function: int gsl_sf_coulomb_wave_FG_array (double L_min, int kmax, double eta, double x, double fc_array[], double gc_array[], double * F_exponent, double * G_exponent)
  This function computes the functions F_L(\eta,x), G_L(\eta,x) for L = Lmin \dots Lmin + kmax storing the results in fc_array and gc_array. In the case of overflow the exponents are stored in F_exponent and G_exponent.

Function: int gsl_sf_coulomb_wave_FGp_array (double L_min, int kmax, double eta, double x, double fc_array[], double fcp_array[], double gc_array[], double gcp_array[], double * F_exponent, double * G_exponent)
  This function computes the functions F_L(\eta,x), G_L(\eta,x) and their derivatives F'_L(\eta,x), G'_L(\eta,x) for L = Lmin \dots Lmin + kmax storing the results in fc_array, gc_array, fcp_array and gcp_array. In the case of overflow the exponents are stored in F_exponent and G_exponent.

Function: int gsl_sf_coulomb_wave_sphF_array (double L_min, int kmax, double eta, double x, double fc_array[], double F_exponent[])
  This function computes the Coulomb wave function divided by the argument F_L(\eta, x)/x for L = Lmin \dots Lmin + kmax, storing the results in fc_array. In the case of overflow the exponent is stored in F_exponent. This function reduces to spherical Bessel functions in the limit \eta \to 0.

### Coulomb Wave Function Normalization Constant

The Coulomb wave function normalization constant is defined in Abramowitz 14.1.7.

Function: int gsl_sf_coulomb_CL_e (double L, double eta, gsl_sf_result * result)
  This function computes the Coulomb wave function normalization constant C_L(\eta) for L > -1.

Function: int gsl_sf_coulomb_CL_array (double Lmin, int kmax, double eta, double cl[])
  This function computes the coulomb wave function normalization constant C_L(\eta) for L = Lmin \dots Lmin + kmax, Lmin > -1.

## Coupling Coefficients

The Wigner 3-j, 6-j and 9-j symbols give the coupling coefficients for combined angular momentum vectors. Since the arguments of the standard coupling coefficient functions are integer or half-integer, the arguments of the following functions are, by convention, integers equal to twice the actual spin value. For information on the 3-j coefficients see Abramowitz & Stegun, Section 27.9. The functions described in this section are declared in the header file ``gsl_sf_coupling.h'`.

### 3-j Symbols

Function: double gsl_sf_coupling_3j (int two_ja, int two_jb, int two_jc, int two_ma, int two_mb, int two_mc)
Function: int gsl_sf_coupling_3j_e (int two_ja, int two_jb, int two_jc, int two_ma, int two_mb, int two_mc, gsl_sf_result * result)
  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.

### 6-j Symbols

Function: double gsl_sf_coupling_6j (int two_ja, int two_jb, int two_jc, int two_jd, int two_je, int two_jf)
Function: int gsl_sf_coupling_6j_e (int two_ja, int two_jb, int two_jc, int two_jd, int two_je, int two_jf, gsl_sf_result * result)
  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.

### 9-j Symbols

Function: double gsl_sf_coupling_9j (int two_ja, int two_jb, int two_jc, int two_jd, int two_je, int two_jf, int two_jg, int two_jh, int two_ji)
Function: int gsl_sf_coupling_9j_e (int two_ja, int two_jb, int two_jc, int two_jd, int two_je, int two_jf, int two_jg, int two_jh, int two_ji, gsl_sf_result * result)
  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.

## Dawson Function

The Dawson integral is defined by \exp(-x^2) \int_0^x dt \exp(t^2). A table of Dawson's integral can be found in Abramowitz & Stegun, Table 7.5. The Dawson functions are declared in the header file ``gsl_sf_dawson.h'`.

Function: double gsl_sf_dawson (double x)
Function: int gsl_sf_dawson_e (double x, gsl_sf_result * result)
  These routines compute the value of Dawson's integral for x.

## Debye Functions

The Debye functions are defined by the integral D_n(x) = n/x^n \int_0^x dt (t^n/(e^t - 1)). For further information see Abramowitz & Stegun, Section 27.1. The Debye functions are declared in the header file ``gsl_sf_debye.h'`.

Function: double gsl_sf_debye_1 (double x)
Function: int gsl_sf_debye_1_e (double x, gsl_sf_result * result)
  These routines compute the first-order Debye function D_1(x) = (1/x) \int_0^x dt (t/(e^t - 1)).

Function: double gsl_sf_debye_2 (double x)
Function: int gsl_sf_debye_2_e (double x, gsl_sf_result * result)
  These routines compute the second-order Debye function D_2(x) = (2/x^2) \int_0^x dt (t^2/(e^t - 1)).

Function: double gsl_sf_debye_3 (double x)
Function: int gsl_sf_debye_3_e (double x, gsl_sf_result * result)
  These routines compute the third-order Debye function D_3(x) = (3/x^3) \int_0^x dt (t^3/(e^t - 1)).

Function: double gsl_sf_debye_4 (double x)
Function: int gsl_sf_debye_4_e (double x, gsl_sf_result * result)
  These routines compute the fourth-order Debye function D_4(x) = (4/x^4) \int_0^x dt (t^4/(e^t - 1)).

## Dilogarithm

The functions described in this section are declared in the header file ``gsl_sf_dilog.h'`.

### Real Argument

Function: double gsl_sf_dilog (double x)
Function: int gsl_sf_dilog_e (double x, gsl_sf_result * result)
  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.

### Complex Argument

Function: int gsl_sf_complex_dilog_e (double r, double theta, gsl_sf_result * result_re, gsl_sf_result * 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 result_re, result_im.

## Elementary Operations

The following functions allow for the propagation of errors when combining quantities by multiplication. The functions are declared in the header file ``gsl_sf_elementary.h'`.

Function: int gsl_sf_multiply_e (double x, double y, gsl_sf_result * result)
  This function multiplies x and y storing the product and its associated error in result.

Function: int gsl_sf_multiply_err_e (double x, double dx, double y, double dy, gsl_sf_result * 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.

## Elliptic Integrals

The functions described in this section are declared in the header file ``gsl_sf_ellint.h'`.

### Definition of Legendre Forms

The Legendre forms of elliptic integrals F(\phi,k), E(\phi,k) and P(\phi,k,n) are defined by,

      F(\phi,k) = \int_0^\phi dt 1/\sqrt((1 - k^2 \sin^2(t)))

      E(\phi,k) = \int_0^\phi dt   \sqrt((1 - k^2 \sin^2(t)))

    P(\phi,k,n) = \int_0^\phi dt 1/((1 + n \sin^2(t))\sqrt(1 - k^2 \sin^2(t)))

The complete Legendre forms are denoted by K(k) = F(\pi/2, k) and E(k) = E(\pi/2, k). Further information on the Legendre forms of elliptic integrals can be found in Abramowitz & Stegun, Chapter 17. The notation used here is based on Carlson, Numerische Mathematik 33 (1979) 1 and differs slightly from that used by Abramowitz & Stegun.

### Definition of Carlson Forms

The Carlson symmetric forms of elliptical integrals RC(x,y), RD(x,y,z), RF(x,y,z) and RJ(x,y,z,p) are defined by,

        RC(x,y) = 1/2 \int_0^\infty dt (t+x)^(-1/2) (t+y)^(-1)

      RD(x,y,z) = 3/2 \int_0^\infty dt (t+x)^(-1/2) (t+y)^(-1/2) (t+z)^(-3/2)

      RF(x,y,z) = 1/2 \int_0^\infty dt (t+x)^(-1/2) (t+y)^(-1/2) (t+z)^(-1/2)

    RJ(x,y,z,p) = 3/2 \int_0^\infty dt
                     (t+x)^(-1/2) (t+y)^(-1/2) (t+z)^(-1/2) (t+p)^(-1)

### Legendre Form of Complete Elliptic Integrals

Function: double gsl_sf_ellint_Kcomp (double k, gsl_mode_t mode)
Function: int gsl_sf_ellint_Kcomp_e (double k, gsl_mode_t mode, gsl_sf_result * result)
  These routines compute the complete elliptic integral K(k) to the accuracy specified by the mode variable mode.

Function: double gsl_sf_ellint_Ecomp (double k, gsl_mode_t mode)
Function: int gsl_sf_ellint_Ecomp_e (double k, gsl_mode_t mode, gsl_sf_result * result)
  These routines compute the complete elliptic integral E(k) to the accuracy specified by the mode variable mode.

### Legendre Form of Incomplete Elliptic Integrals

Function: double gsl_sf_ellint_F (double phi, double k, gsl_mode_t mode)
Function: int gsl_sf_ellint_F_e (double phi, double k, gsl_mode_t mode, gsl_sf_result * result)
  These routines compute the incomplete elliptic integral F(\phi,k) to the accuracy specified by the mode variable mode.

Function: double gsl_sf_ellint_E (double phi, double k, gsl_mode_t mode)
Function: int gsl_sf_ellint_E_e (double phi, double k, gsl_mode_t mode, gsl_sf_result * result)
  These routines compute the incomplete elliptic integral E(\phi,k) to the accuracy specified by the mode variable mode.

Function: double gsl_sf_ellint_P (double phi, double k, double n, gsl_mode_t mode)
Function: int gsl_sf_ellint_P_e (double phi, double k, double n, gsl_mode_t mode, gsl_sf_result * result)
  These routines compute the incomplete elliptic integral P(\phi,k,n) to the accuracy specified by the mode variable mode.

Function: double gsl_sf_ellint_D (double phi, double k, double n, gsl_mode_t mode)
Function: int gsl_sf_ellint_D_e (double phi, double k, double n, gsl_mode_t mode, gsl_sf_result * result)
  These functions compute the incomplete elliptic integral D(\phi,k,n) which is defined through the Carlson form RD(x,y,z) by the following relation,

      D(\phi,k,n) = RD (1-\sin^2(\phi), 1-k^2 \sin^2(\phi), 1).

### Carlson Forms

Function: double gsl_sf_ellint_RC (double x, double y, gsl_mode_t mode)
Function: int gsl_sf_ellint_RC_e (double x, double y, gsl_mode_t mode, gsl_sf_result * result)
  These routines compute the incomplete elliptic integral RC(x,y) to the accuracy specified by the mode variable mode.

Function: double gsl_sf_ellint_RD (double x, double y, double z, gsl_mode_t mode)
Function: int gsl_sf_ellint_RD_e (double x, double y, double z, gsl_mode_t mode, gsl_sf_result * result)
  These routines compute the incomplete elliptic integral RD(x,y,z) to the accuracy specified by the mode variable mode.

Function: double gsl_sf_ellint_RF (double x, double y, double z, gsl_mode_t mode)
Function: int gsl_sf_ellint_RF_e (double x, double y, double z, gsl_mode_t mode, gsl_sf_result * result)
  These routines compute the incomplete elliptic integral RF(x,y,z) to the accuracy specified by the mode variable mode.

Function: double gsl_sf_ellint_RJ (double x, double y, double z, double p, gsl_mode_t mode)
Function: int gsl_sf_ellint_RJ_e (double x, double y, double z, double p, gsl_mode_t mode, gsl_sf_result * result)
  These routines compute the incomplete elliptic integral RJ(x,y,z,p) to the accuracy specified by the mode variable mode.

## Elliptic Functions (Jacobi)

The Jacobian Elliptic functions are defined in Abramowitz & Stegun, Chapter 16. The functions are declared in the header file ``gsl_sf_elljac.h'`.

Function: int gsl_sf_elljac_e (double u, double m, double * sn, double * cn, double * dn)
  This function computes the Jacobian elliptic functions sn(u|m), cn(u|m), dn(u|m) by descending Landen transformations.

## Error Functions

The error function is described in Abramowitz & Stegun, Chapter 7. The functions in this section are declared in the header file ``gsl_sf_erf.h'`.

### Error Function

Function: double gsl_sf_erf (double x)
Function: int gsl_sf_erf_e (double x, gsl_sf_result * result)
  These routines compute the error function erf(x) = (2/\sqrt(\pi)) \int_0^x dt \exp(-t^2).

### Complementary Error Function

Function: double gsl_sf_erfc (double x)
Function: int gsl_sf_erfc_e (double x, gsl_sf_result * result)
  These routines compute the complementary error function erfc(x) = 1 - erf(x) = (2/\sqrt(\pi)) \int_x^\infty \exp(-t^2).

### Log Complementary Error Function

Function: double gsl_sf_log_erfc (double x)
Function: int gsl_sf_log_erfc_e (double x, gsl_sf_result * result)
  These routines compute the logarithm of the complementary error function \log(\erfc(x)).

### Probability functions

The probability functions for the Normal or Gaussian distribution are described in Abramowitz & Stegun, Section 26.2.

Function: double gsl_sf_erf_Z (double x)
Function: int gsl_sf_erf_Z_e (double x, gsl_sf_result * result)
  These routines compute the Gaussian probability function Z(x) = (1/(2\pi)) \exp(-x^2/2).

Function: double gsl_sf_erf_Q (double x)
Function: int gsl_sf_erf_Q_e (double x, gsl_sf_result * result)
  These routines compute the upper tail of the Gaussian probability function Q(x) = (1/(2\pi)) \int_x^\infty dt \exp(-t^2/2).

## Exponential Functions

The functions described in this section are declared in the header file ``gsl_sf_exp.h'`.

### Exponential Function

Function: double gsl_sf_exp (double x)
Function: int gsl_sf_exp_e (double x, gsl_sf_result * result)
  These routines provide an exponential function \exp(x) using GSL semantics and error checking.

Function: int gsl_sf_exp_e10_e (double x, gsl_sf_result_e10 * result)
  This function computes the exponential \exp(x) using the `gsl_sf_result_e10` type to return a result with extended range. This function may be useful if the value of \exp(x) would overflow the numeric range of `double`.

Function: double gsl_sf_exp_mult (double x, double y)
Function: int gsl_sf_exp_mult_e (double x, double y, gsl_sf_result * result)
  These routines exponentiate x and multiply by the factor y to return the product y \exp(x).

Function: int gsl_sf_exp_mult_e10_e (const double x, const double y, gsl_sf_result_e10 * result)
  This function computes the product y \exp(x) using the `gsl_sf_result_e10` type to return a result with extended numeric range.

### Relative Exponential Functions

Function: double gsl_sf_expm1 (double x)
Function: int gsl_sf_expm1_e (double x, gsl_sf_result * result)
  These routines compute the quantity \exp(x)-1 using an algorithm that is accurate for small x.

Function: double gsl_sf_exprel (double x)
Function: int gsl_sf_exprel_e (double x, gsl_sf_result * result)
  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.

Function: double gsl_sf_exprel_2 (double x)
Function: int gsl_sf_exprel_2_e (double x, gsl_sf_result * result)
  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.

Function: double gsl_sf_exprel_n (int n, double x)
Function: int gsl_sf_exprel_n_e (int n, double x, gsl_sf_result * result)
  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)

### Exponentiation With Error Estimate

Function: int gsl_sf_exp_err_e (double x, double dx, gsl_sf_result * result)
  This function exponentiates x with an associated absolute error dx.

Function: int gsl_sf_exp_err_e10_e (double x, double dx, gsl_sf_result_e10 * result)
  This functions exponentiate a quantity x with an associated absolute error dx using the `gsl_sf_result_e10` type to return a result with extended range.

Function: int gsl_sf_exp_mult_err_e (double x, double dx, double y, double dy, gsl_sf_result * result)
  This routine computes the product y \exp(x) for the quantities x, y with associated absolute errors dx, dy.

Function: int gsl_sf_exp_mult_err_e10_e (double x, double dx, double y, double dy, gsl_sf_result_e10 * result)
  This routine computes the product y \exp(x) for the quantities x, y with associated absolute errors dx, dy using the `gsl_sf_result_e10` type to return a result with extended range.

## Exponential Integrals

Information on the exponential integrals can be found in Abramowitz & Stegun, Chapter 5. These functions are declared in the header file ``gsl_sf_expint.h'`.

### Exponential Integral

Function: double gsl_sf_expint_E1 (double x)
Function: int gsl_sf_expint_E1_e (double x, gsl_sf_result * result)
  These routines compute the exponential integral E_1(x),

      E_1(x) := Re \int_1^\infty dt \exp(-xt)/t.

Function: double gsl_sf_expint_E2 (double x)
Function: int gsl_sf_expint_E2_e (double x, gsl_sf_result * result)
  These routines compute the second-order exponential integral E_2(x),

      E_2(x) := \Re \int_1^\infty dt \exp(-xt)/t^2.

### Ei(x)

Function: double gsl_sf_expint_Ei (double x)
Function: int gsl_sf_expint_Ei_e (double x, gsl_sf_result * result)
  These routines compute the exponential integral E_i(x),

      Ei(x) := PV(\int_{-x}^\infty dt \exp(-t)/t)

  where PV denotes the principal value of the integral.

### Hyperbolic Integrals

Function: double gsl_sf_Shi (double x)
Function: int gsl_sf_Shi_e (double x, gsl_sf_result * result)
  These routines compute the integral Shi(x) = \int_0^x dt \sinh(t)/t.

Function: double gsl_sf_Chi (double x)
Function: int gsl_sf_Chi_e (double x, gsl_sf_result * result)
  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 the macro `M_EULER`).

### Ei_3(x)

Function: double gsl_sf_expint_3 (double x)
Function: int gsl_sf_expint_3_e (double x, gsl_sf_result * result)
  These routines compute the exponential integral Ei_3(x) = \int_0^x dt \exp(-t^3) for x >= 0.

### Trigonometric Integrals

Function: double gsl_sf_Si (const double x)
Function: int gsl_sf_Si_e (double x, gsl_sf_result * result)
  These routines compute the Sine integral Si(x) = \int_0^x dt \sin(t)/t.

Function: double gsl_sf_Ci (const double x)
Function: int gsl_sf_Ci_e (double x, gsl_sf_result * result)
  These routines compute the Cosine integral Ci(x) = -\int_x^\infty dt \cos(t)/t for x > 0.

### Arctangent Integral

Function: double gsl_sf_atanint (double x)
Function: int gsl_sf_atanint_e (double x, gsl_sf_result * result)
  These routines compute the Arctangent integral AtanInt(x) = \int_0^x dt \arctan(t)/t.

## Fermi-Dirac Function

The functions described in this section are declared in the header file ``gsl_sf_fermi_dirac.h'`.

### Complete Fermi-Dirac Integrals

The complete Fermi-Dirac integral F_j(x) is given by,

    F_j(x)   := (1/r\Gamma(j+1)) \int_0^\infty (t^j / (\exp(t-x) + 1))

Function: double gsl_sf_fermi_dirac_m1 (double x)
Function: int gsl_sf_fermi_dirac_m1_e (double x, gsl_sf_result * result)
  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).

Function: double gsl_sf_fermi_dirac_0 (double x)
Function: int gsl_sf_fermi_dirac_0_e (double x, gsl_sf_result * result)
  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).

Function: double gsl_sf_fermi_dirac_1 (double x)
Function: int gsl_sf_fermi_dirac_1_e (double x, gsl_sf_result * result)
  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)).

Function: double gsl_sf_fermi_dirac_2 (double x)
Function: int gsl_sf_fermi_dirac_2_e (double x, gsl_sf_result * result)
  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)).

Function: double gsl_sf_fermi_dirac_int (int j, double x)
Function: int gsl_sf_fermi_dirac_int_e (int j, double x, gsl_sf_result * result)
  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)).

Function: double gsl_sf_fermi_dirac_mhalf (double x)
Function: int gsl_sf_fermi_dirac_mhalf_e (double x, gsl_sf_result * result)
  These routines compute the complete Fermi-Dirac integral F_{-1/2}(x).

Function: double gsl_sf_fermi_dirac_half (double x)
Function: int gsl_sf_fermi_dirac_half_e (double x, gsl_sf_result * result)
  These routines compute the complete Fermi-Dirac integral F_{1/2}(x).

Function: double gsl_sf_fermi_dirac_3half (double x)
Function: int gsl_sf_fermi_dirac_3half_e (double x, gsl_sf_result * result)
  These routines compute the complete Fermi-Dirac integral F_{3/2}(x).

### Incomplete Fermi-Dirac Integrals

The incomplete Fermi-Dirac integral F_j(x,b) is given by,

    F_j(x,b)   := (1/\Gamma(j+1)) \int_b^\infty (t^j / (\Exp(t-x) + 1))

Function: double gsl_sf_fermi_dirac_inc_0 (double x, double b)
Function: int gsl_sf_fermi_dirac_inc_0_e (double x, double b, gsl_sf_result * 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).

## Gamma Function

The Gamma function is defined by the following integral,

    \Gamma(x) = \int_0^t dt  t^{x-1} \exp(-t)

Further information on the Gamma function can be found in Abramowitz & Stegun, Chapter 6. The functions described in this section are declared in the header file ``gsl_sf_gamma.h'`.

Function: double gsl_sf_gamma (double x)
Function: int gsl_sf_gamma_e (double x, gsl_sf_result * result)
  These routines compute the Gamma function \Gamma(x), subject to x not being a negative integer. The function is computed using the real Lanczos method. The maximum value of x such that \Gamma(x) is not considered an overflow is given by the macro `GSL_SF_GAMMA_XMAX` and is 171.0.

Function: double gsl_sf_lngamma (double x)
Function: int gsl_sf_lngamma_e (double x, gsl_sf_result * result)
  These routines compute the logarithm of the Gamma function, \log(\Gamma(x)), subject to x not a being negative integer. 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.

Function: int gsl_sf_lngamma_sgn_e (double x, gsl_sf_result * result_lg, double * sgn)
  This routine computes the sign of the gamma function and the logarithm its magnitude, subject to x not being a negative integer. 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).

Function: double gsl_sf_gammastar (double x)
Function: int gsl_sf_gammastar_e (double x, gsl_sf_result * result)
  These routines compute the regulated Gamma Function \Gamma^*(x) for x > 0. The regulated gamma function is given by,

      \Gamma^*(x) = \Gamma(x)/(\sqrt{2\pi} x^{(x-1/2)} \exp(-x))
                  = (1 + (1/12x) + ...)  for x \to \infty

  and is a useful suggestion of Temme.

Function: double gsl_sf_gammainv (double x)
Function: int gsl_sf_gammainv_e (double x, gsl_sf_result * result)
  These routines compute the reciprocal of the gamma function, 1/\Gamma(x) using the real Lanczos method.

Function: int gsl_sf_lngamma_complex_e (double zr, double zi, gsl_sf_result * lnr, gsl_sf_result * arg)
  This routine computes \log(\Gamma(z)) for complex z=z_r+i z_i and z not a negative integer, using the complex Lanczos method. The returned parameters are lnr = \log|\Gamma(z)| and arg = \arg(\Gamma(z)) in (-\pi,\pi]. Note that the phase part (arg) is not well-determined when |z| is very large, due to inevitable roundoff in restricting to (-\pi,\pi]. This will result in a `GSL_ELOSS` error when it occurs. The absolute value part (lnr), however, never suffers from loss of precision.

Function: double gsl_sf_taylorcoeff (int n, double x)
Function: int gsl_sf_taylorcoeff_e (int n, double x, gsl_sf_result * result)
  These routines compute the Taylor coefficient x^n / n! for x >= 0, n >= 0.

Function: double gsl_sf_fact (unsigned int n)
Function: int gsl_sf_fact_e (unsigned int n, gsl_sf_result * result)
  These routines compute the factorial n!. The factorial is related to the Gamma function by n! = \Gamma(n+1).

Function: double gsl_sf_doublefact (unsigned int n)
Function: int gsl_sf_doublefact_e (unsigned int n, gsl_sf_result * result)
  These routines compute the double factorial n!! = n(n-2)(n-4) \dots.

Function: double gsl_sf_lnfact (unsigned int n)
Function: int gsl_sf_lnfact_e (unsigned int n, gsl_sf_result * result)
  These routines compute the logarithm of the factorial of n, \log(n!). The algorithm is faster than computing \ln(\Gamma(n+1)) via `gsl_sf_lngamma` for n < 170, but defers for larger n.

Function: double gsl_sf_lndoublefact (unsigned int n)
Function: int gsl_sf_lndoublefact_e (unsigned int n, gsl_sf_result * result)
  These routines compute the logarithm of the double factorial of n, \log(n!!).

Function: double gsl_sf_choose (unsigned int n, unsigned int m)
Function: int gsl_sf_choose_e (unsigned int n, unsigned int m, gsl_sf_result * result)
  These routines compute the combinatorial factor `n choose m` = n!/(m!(n-m)!)

Function: double gsl_sf_lnchoose (unsigned int n, unsigned int m)
Function: int gsl_sf_lnchoose_e (unsigned int n, unsigned int m, gsl_sf_result * result)
  These routines compute the logarithm of `n choose m`. This is equivalent to the sum \log(n!) - \log(m!) - \log((n-m)!).

Function: double gsl_sf_poch (double a, double x)
Function: int gsl_sf_poch_e (double a, double x, gsl_sf_result * result)
  These routines compute the Pochhammer symbol (a)_x := \Gamma(a + x)/\Gamma(x), subject to a and a+x not being negative integers. The Pochhammer symbol is also known as the Apell symbol.

Function: double gsl_sf_lnpoch (double a, double x)
Function: int gsl_sf_lnpoch_e (double a, double x, gsl_sf_result * result)
  These routines compute the logarithm of the Pochhammer symbol, \log((a)_x) = \log(\Gamma(a + x)/\Gamma(a)) for a > 0, a+x > 0.

Function: int gsl_sf_lnpoch_sgn_e (double a, double x, gsl_sf_result * result, double * sgn)
  These routines compute the sign of the Pochhammer symbol and the logarithm of its magnitude. The computed parameters are result = \log(|(a)_x|) and sgn = sgn((a)_x) where (a)_x := \Gamma(a + x)/\Gamma(a), subject to a, a+x not being negative integers.

Function: double gsl_sf_pochrel (double a, double x)
Function: int gsl_sf_pochrel_e (double a, double x, gsl_sf_result * result)
  These routines compute the relative Pochhammer symbol ((a,x) - 1)/x where (a,x) = (a)_x := \Gamma(a + x)/\Gamma(a).

Function: double gsl_sf_gamma_inc_Q (double a, double x)
Function: int gsl_sf_gamma_inc_Q_e (double a, double x, gsl_sf_result * result)
  These routines compute the normalized incomplete Gamma Function P(a,x) = 1/\Gamma(a) \int_x\infty dt t^{a-1} \exp(-t) for a > 0, x >= 0.

Function: double gsl_sf_gamma_inc_P (double a, double x)
Function: int gsl_sf_gamma_inc_P_e (double a, double x, gsl_sf_result * result)
  These routines compute the complementary normalized incomplete Gamma Function P(a,x) = 1/\Gamma(a) \int_0^x dt t^{a-1} \exp(-t) for a > 0, x >= 0.

Function: double gsl_sf_beta (double a, double b)
Function: int gsl_sf_beta_e (double a, double b, gsl_sf_result * result)
  These routines compute the Beta Function, B(a,b) = \Gamma(a)\Gamma(b)/\Gamma(a+b) for a > 0, b > 0.

Function: double gsl_sf_lnbeta (double a, double b)
Function: int gsl_sf_lnbeta_e (double a, double b, gsl_sf_result * result)
  These routines compute the logarithm of the Beta Function, \log(B(a,b)) for a > 0, b > 0.

Function: double gsl_sf_beta_inc (double a, double b, double x)
Function: int gsl_sf_beta_inc_e (double a, double b, double x, gsl_sf_result * result)
  These routines compute the normalize incomplete Beta function B_x(a,b)/B(a,b) for a > 0, b > 0, and 0 <= x <= 1.

## Gegenbauer Functions

The Gegenbauer polynomials are defined in Abramowitz & Stegun, Chapter 22, where they are known as Ultraspherical polynomials. The functions described in this section are declared in the header file ``gsl_sf_gegenbauer.h'`.

Function: double gsl_sf_gegenpoly_1 (double lambda, double x)
Function: double gsl_sf_gegenpoly_2 (double lambda, double x)
Function: double gsl_sf_gegenpoly_3 (double lambda, double x)
Function: int gsl_sf_gegenpoly_1_e (double lambda, double x, gsl_sf_result * result)
Function: int gsl_sf_gegenpoly_2_e (double lambda, double x, gsl_sf_result * result)
Function: int gsl_sf_gegenpoly_3_e (double lambda, double x, gsl_sf_result * result)
  These functions evaluate the Gegenbauer polynomials C^{(\lambda)}_n(x) using explicit representations for n =1, 2, 3.

Function: double gsl_sf_gegenpoly_n (int n, double lambda, double x)
Function: int gsl_sf_gegenpoly_n_e (int n, double lambda, double x, gsl_sf_result * result)
  These functions evaluate the Gegenbauer polynomial C^{(\lambda)}_n(x) for a specific value of n, lambda, x subject to \lambda > -1/2, n >= 0.

Function: int gsl_sf_gegenpoly_array (int nmax, double lambda, double x, double result_array[])
  This function computes an array of Gegenbauer polynomials C^{(\lambda)}_n(x) for n = 0, 1, 2, \dots, nmax, subject to \lambda > -1/2, nmax >= 0.

## Hypergeometric Functions

Hypergeometric functions are described in Abramowitz & Stegun, Chapters 13 and 15. These functions are declared in the header file ``gsl_sf_hyperg.h'`.

Function: double gsl_sf_hyperg_0F1 (double c, double x)
Function: int gsl_sf_hyperg_0F1_e (double c, double x, gsl_sf_result * result)
  These routines compute the hypergeometric function 0F1(c,x).

Function: double gsl_sf_hyperg_1F1_int (int m, int n, double x)
Function: int gsl_sf_hyperg_1F1_int_e (int m, int n, double x, gsl_sf_result * result)
  These routines compute the confluent hypergeometric function 1F1(m,n,x) = M(m,n,x) for integer parameters m, n.

Function: double gsl_sf_hyperg_1F1 (double a, double b, double x)
Function: int gsl_sf_hyperg_1F1_e (double a, double b, double x, gsl_sf_result * result)
  These routines compute the confluent hypergeometric function 1F1(a,b,x) = M(a,b,x) for general parameters a, b.

Function: double gsl_sf_hyperg_U_int (int m, int n, double x)
Function: int gsl_sf_hyperg_U_int_e (int m, int n, double x, gsl_sf_result * result)
  These routines compute the confluent hypergeometric function U(m,n,x) for integer parameters m, n.

Function: int gsl_sf_hyperg_U_int_e10_e (int m, int n, double x, gsl_sf_result_e10 * result)
  This routine computes the confluent hypergeometric function U(m,n,x) for integer parameters m, n using the `gsl_sf_result_e10` type to return a result with extended range.

Function: double gsl_sf_hyperg_U (double a, double b, double x)
Function: int gsl_sf_hyperg_U_e (double a, double b, double x)
  These routines compute the confluent hypergeometric function U(a,b,x).

Function: int gsl_sf_hyperg_U_e10_e (double a, double b, double x, gsl_sf_result_e10 * result)
  This routine computes the confluent hypergeometric function U(a,b,x) using the `gsl_sf_result_e10` type to return a result with extended range.

Function: double gsl_sf_hyperg_2F1 (double a, double b, double c, double x)
Function: int gsl_sf_hyperg_2F1_e (double a, double b, double c, double x, gsl_sf_result * result)
  These routines compute the Gauss hypergeometric function 2F1(a,b,c,x) for |x| < 1.

  If the arguments (a,b,c,x) are too close to a singularity then the function can return the error code `GSL_EMAXITER` when the series approximation converges too slowly. This occurs in the region of x=1, c - a - b = m for integer m.

Function: double gsl_sf_hyperg_2F1_conj (double aR, double aI, double c, double x)
Function: int gsl_sf_hyperg_2F1_conj_e (double aR, double aI, double c, double x, gsl_sf_result * result)
  These routines compute the Gauss hypergeometric function 2F1(a_R + i a_I, a_R - i a_I, c, x) with complex parameters for |x| < 1. exceptions:

Function: double gsl_sf_hyperg_2F1_renorm (double a, double b, double c, double x)
Function: int gsl_sf_hyperg_2F1_renorm_e (double a, double b, double c, double x, gsl_sf_result * result)
  These routines compute the renormalized Gauss hypergeometric function 2F1(a,b,c,x) / \Gamma(c) for |x| < 1.

Function: double gsl_sf_hyperg_2F1_conj_renorm (double aR, double aI, double c, double x)
Function: int gsl_sf_hyperg_2F1_conj_renorm_e (double aR, double aI, double c, double x, gsl_sf_result * result)
  These routines compute the renormalized Gauss hypergeometric function 2F1(a_R + i a_I, a_R - i a_I, c, x) / \Gamma(c) for |x| < 1.

Function: double gsl_sf_hyperg_2F0 (double a, double b, double x)
Function: int gsl_sf_hyperg_2F0_e (double a, double b, double x, gsl_sf_result * result)
  These routines compute the hypergeometric function 2F0(a,b,x). The series representation is a divergent hypergeometric series. However, for x < 0 we have 2F0(a,b,x) = (-1/x)^a U(a,1+a-b,-1/x)

## Laguerre Functions

The Laguerre polynomials are defined in terms of confluent hypergeometric functions as L^a_n(x) = ((a+1)_n / n!) 1F1(-n,a+1,x). These functions are declared in the header file ``gsl_sf_laguerre.h'`.

Function: double gsl_sf_laguerre_1 (double a, double x)
Function: double gsl_sf_laguerre_2 (double a, double x)
Function: double gsl_sf_laguerre_3 (double a, double x)
Function: int gsl_sf_laguerre_1_e (double a, double x, gsl_sf_result * result)
Function: int gsl_sf_laguerre_2_e (double a, double x, gsl_sf_result * result)
Function: int gsl_sf_laguerre_3_e (double a, double x, gsl_sf_result * result)
  These routines evaluate the generalized Laguerre polynomials L^a_1(x), L^a_2(x), L^a_3(x) using explicit representations.

Function: double gsl_sf_laguerre_n (const int n, const double a, const double x)
Function: int gsl_sf_laguerre_n_e (int n, double a, double x, gsl_sf_result * result)
  Thse routines evaluate the generalized Laguerre polynomials L^a_n(x) for a > -1, n >= 0.

## Lambert W Functions

Lambert's W functions, W(x), are defined to be solutions of the equation W(x) \exp(W(x)) = x. This function has multiple branches for x < 0; however, it has only two real-valued branches. We define W_0(x) to be the principal branch, where W > -1 for x < 0, and W_{-1}(x) to be the other real branch, where W < -1 for x < 0. The Lambert functions are declared in the header file ``gsl_sf_lambert.h'`.

Function: double gsl_sf_lambert_W0 (double x)
Function: int gsl_sf_lambert_W0_e (double x, gsl_sf_result * result)
  These compute the principal branch of the Lambert W function, W_0(x).

Function: double gsl_sf_lambert_Wm1 (double x)
Function: int gsl_sf_lambert_Wm1_e (double x, gsl_sf_result * result)
  These compute the secondary real-valued branch of the Lambert W function, W_{-1}(x).

## Legendre Functions and Spherical Harmonics

The Legendre Functions and Legendre Polynomials are described in Abramowitz & Stegun, Chapter 8. These functions are declared in the header file ``gsl_sf_legendre.h'`.

### Legendre Polynomials

Function: double gsl_sf_legendre_P1 (double x)
Function: double gsl_sf_legendre_P2 (double x)
Function: double gsl_sf_legendre_P3 (double x)
Function: int gsl_sf_legendre_P1_e (double x, gsl_sf_result * result)
Function: int gsl_sf_legendre_P2_e (double x, gsl_sf_result * result)
Function: int gsl_sf_legendre_P3_e (double x, gsl_sf_result * result)
  These functions evaluate the Legendre polynomials P_l(x) using explicit representations for l=1, 2, 3.

Function: double gsl_sf_legendre_Pl (int l, double x)
Function: int gsl_sf_legendre_Pl_e (int l, double x, gsl_sf_result * result)
  These functions evaluate the Legendre polynomial P_l(x) for a specific value of l, x subject to l >= 0, |x| <= 1

Function: int gsl_sf_legendre_Pl_array (int lmax, double x, double result_array[])
  This function computes an array of Legendre polynomials P_l(x) for l = 0, \dots, lmax, |x| <= 1

Function: double gsl_sf_legendre_Q0 (double x)
Function: int gsl_sf_legendre_Q0_e (double x, gsl_sf_result * result)
  These routines compute the Legendre function Q_0(x) for x > -1, x != 1.

Function: double gsl_sf_legendre_Q1 (double x)
Function: int gsl_sf_legendre_Q1_e (double x, gsl_sf_result * result)
  These routines compute the Legendre function Q_1(x) for x > -1, x != 1.

Function: double gsl_sf_legendre_Ql (int l, double x)
Function: int gsl_sf_legendre_Ql_e (int l, double x, gsl_sf_result * result)
  These routines compute the Legendre function Q_l(x) for x > -1, x != 1 and l >= 0.

### Associated Legendre Polynomials and Spherical Harmonics

The following functions compute the associated Legendre Polynomials P_l^m(x). Note that this function grows combinatorially with l and can overflow for l larger than about 150. There is no trouble for small m, but overflow occurs when m and l are both large. Rather than allow overflows, these functions refuse to calculate P_l^m(x) and return `GSL_EOVRFLW` when they can sense that l and m are too big.

If you want to calculate a spherical harmonic, then do not use these functions. Instead use `gsl_sf_legendre_sphPlm()` below, which uses a similar recursion, but with the normalized functions.

Function: double gsl_sf_legendre_Plm (int l, int m, double x)
Function: int gsl_sf_legendre_Plm_e (int l, int m, double x, gsl_sf_result * result)
  These routines compute the associated Legendre polynomial P_l^m(x) for m >= 0, l >= m, |x| <= 1.

Function: int gsl_sf_legendre_Plm_array (int lmax, int m, double x, double result_array[])
  This function computes an array of Legendre polynomials P_l^m(x) for m >= 0, l = |m|, ..., lmax, |x| <= 1.

Function: double gsl_sf_legendre_sphPlm (int l, int m, double x)
Function: int gsl_sf_legendre_sphPlm_e (int l, int m, double x, gsl_sf_result * result)
  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).

Function: int gsl_sf_legendre_sphPlm_array (int lmax, int m, double x, double result_array[])
  This function computes an array of normalized associated Legendre functions $\sqrt{(2l+1)/(4\pi)} \sqrt{(l-m)!/(l+m)!} P_l^m(x)$ for m >= 0, l = |m|, ..., lmax, |x| <= 1.0

Function: int gsl_sf_legendre_array_size (const int lmax, const int m)
  This functions returns the size of result_array[] needed for the array versions of P_l^m(x), lmax - m + 1.

### Conical Functions

The Conical Functions P^\mu_{-(1/2)+i\lambda}(x), Q^\mu_{-(1/2)+i\lambda} are described in Abramowitz & Stegun, Section 8.12.

Function: double gsl_sf_conicalP_half (double lambda, double x)
Function: int gsl_sf_conicalP_half_e (double lambda, double x, gsl_sf_result * result)
  These routines compute the irregular Spherical Conical Function P^{1/2}_{-1/2 + i \lambda}(x) for x > -1.

Function: double gsl_sf_conicalP_mhalf (double lambda, double x)
Function: int gsl_sf_conicalP_mhalf_e (double lambda, double x, gsl_sf_result * result)
  These routines compute the regular Spherical Conical Function P^{-1/2}_{-1/2 + i \lambda}(x) for x > -1.

Function: double gsl_sf_conicalP_0 (double lambda, double x)
Function: int gsl_sf_conicalP_0_e (double lambda, double x, gsl_sf_result * result)
  These routines compute the conical function P^0_{-1/2 + i \lambda}(x) for x > -1.

Function: double gsl_sf_conicalP_1 (double lambda, double x)
Function: int gsl_sf_conicalP_1_e (double lambda, double x, gsl_sf_result * result)
  These routines compute the conical function P^1_{-1/2 + i \lambda}(x) for x > -1.

Function: double gsl_sf_conicalP_sph_reg (int l, double lambda, double x)
Function: int gsl_sf_conicalP_sph_reg_e (int l, double lambda, double x, gsl_sf_result * result)
  These routines compute the Regular Spherical Conical Function P^{-1/2-l}_{-1/2 + i \lambda}(x) for x > -1, l >= -1.

Function: double gsl_sf_conicalP_cyl_reg (int m, double lambda, double x)
Function: int gsl_sf_conicalP_cyl_reg_e (int m, double lambda, double x, gsl_sf_result * result)
  These routines compute the Regular Cylindrical Conical Function P^{-m}_{-1/2 + i \lambda}(x) for x > -1, m >= -1.

### Radial Functions for Hyperbolic Space

The following spherical functions are specializations of Legendre functions which give the regular eigenfunctions of the Laplacian on a 3-dimensional hyperbolic space H3d. Of particular interest is the flat limit, \lambda \to \infty, \eta \to 0, \lambda\eta fixed.

Function: double gsl_sf_legendre_H3d_0 (double lambda, double eta)
Function: int gsl_sf_legendre_H3d_0_e (double lambda, double eta, gsl_sf_result * result)
  These routines compute the zeroth radial eigenfunction of the Laplacian on the 3-dimensional hyperbolic space, L^{H3d}_0(\lambda,\eta) := \sin(\lambda\eta)/(\lambda\sinh(\eta)) for \eta >= 0. In the flat limit this takes the form L^{H3d}_0(\lambda,\eta) = j_0(\lambda\eta)

Function: double gsl_sf_legendre_H3d_1 (double lambda, double eta)
Function: int gsl_sf_legendre_H3d_1_e (double lambda, double eta, gsl_sf_result * result)
  These routines compute the first radial eigenfunction of the Laplacian on the 3-dimensional hyperbolic space, L^{H3d}_1(\lambda,\eta) := 1/\sqrt{\lambda^2 + 1} \sin(\lambda \eta)/(\lambda \sinh(\eta)) (\coth(\eta) - \lambda \cot(\lambda\eta)) for \eta >= 0. In the flat limit this takes the form L^{H3d}_1(\lambda,\eta) = j_1(\lambda\eta).

Function: double gsl_sf_legendre_H3d (int l, double lambda, double eta)
Function: int gsl_sf_legendre_H3d_e (int l, double lambda, double eta, gsl_sf_result * result)
  These routines compute the l'th radial eigenfunction of the Laplacian on the 3-dimensional hyperbolic space \eta >= 0, l >= 0. In the flat limit this takes the form L^{H3d}_l(\lambda,\eta) = j_l(\lambda\eta).

Function: int gsl_sf_legendre_H3d_array (int lmax, double lambda, double eta, double result_array[])
  This function computes an array of radial eigenfunctions L^{H3d}_l(\lambda, \eta) for 0 <= l <= lmax.

## Logarithm and Related Functions

Information on the properties of the Logarithm function can be found in Abramowitz & Stegun, Chapter 4. The functions described in this section are declared in the header file ``gsl_sf_log.h'`.

Function: double gsl_sf_log (double x)
Function: int gsl_sf_log_e (double x, gsl_sf_result * result)
  These routines compute the logarithm of x, \log(x), for x > 0.

Function: double gsl_sf_log_abs (double x)
Function: int gsl_sf_log_abs_e (double x, gsl_sf_result * result)
  These routines compute the logarithm of the magnitude of x, \log(|x|), for x \ne 0.

Function: int gsl_sf_complex_log_e (double zr, double zi, gsl_sf_result * lnr, gsl_sf_result * theta)
  This routine computes the complex logarithm of z = z_r + i z_i. The results are returned as lnr, theta such that \exp(lnr + i \theta) = z_r + i z_i, where \theta lies in the range [-\pi,\pi].

Function: double gsl_sf_log_1plusx (double x)
Function: int gsl_sf_log_1plusx_e (double x, gsl_sf_result * result)
  These routines compute \log(1 + x) for x > -1 using an algorithm that is accurate for small x.

Function: double gsl_sf_log_1plusx_mx (double x)
Function: int gsl_sf_log_1plusx_mx_e (double x, gsl_sf_result * result)
  These routines compute \log(1 + x) - x for x > -1 using an algorithm that is accurate for small x.

## Power Function

The following functions are equivalent to the function `gsl_pow_int` (see section Small integer powers) with an error estimate. These functions are declared in the header file ``gsl_sf_pow_int.h'`.

Function: double gsl_sf_pow_int (double x, int n)
Function: int gsl_sf_pow_int_e (double x, int n, gsl_sf_result * result)
  These routines compute the power x^n for integer n. The power is computed using the minimum number of multiplications. For example, x^8 is computed as ((x^2)^2)^2, requiring only 3 multiplications. For reasons of efficiency, these functions do not check for overflow or underflow conditions.

    #include <gsl/gsl_sf_pow_int.h>
    /* compute 3.0**12 */
    double y = gsl_sf_pow_int(3.0, 12);

## Psi (Digamma) Function

The polygamma functions of order m defined by \psi^{(m)}(x) = (d/dx)^m \psi(x) = (d/dx)^{m+1} \log(\Gamma(x)), where \psi(x) = \Gamma'(x)/\Gamma(x) is known as the digamma function. These functions are declared in the header file ``gsl_sf_psi.h'`.

### Digamma Function

Function: double gsl_sf_psi_int (int n)
Function: int gsl_sf_psi_int_e (int n, gsl_sf_result * result)
  These routines compute the digamma function \psi(n) for positive integer n. The digamma function is also called the Psi function.

Function: double gsl_sf_psi (double x)
Function: int gsl_sf_psi_e (double x, gsl_sf_result * result)
  These routines compute the digamma function \psi(x) for general x, x \ne 0.

Function: double gsl_sf_psi_1piy (double y)
Function: int gsl_sf_psi_1piy_e (double y, gsl_sf_result * result)
  These routines compute the real part of the digamma function on the line 1+i y, Re[\psi(1 + i y)].

### Trigamma Function

Function: double gsl_sf_psi_1_int (int n)
Function: int gsl_sf_psi_1_int_e (int n, gsl_sf_result * result)
  These routines compute the Trigamma function \psi'(n) for positive integer n.

### Polygamma Function

Function: double gsl_sf_psi_n (int m, double x)
Function: int gsl_sf_psi_n_e (int m, double x, gsl_sf_result * result)
  These routines compute the polygamma function \psi^{(m)}(x) for m >= 0, x > 0.

## Synchrotron Functions

The functions described in this section are declared in the header file ``gsl_sf_synchrotron.h'`.

Function: double gsl_sf_synchrotron_1 (double x)
Function: int gsl_sf_synchrotron_1_e (double x, gsl_sf_result * result)
  These routines compute the first synchrotron function x \int_x^\infty dt K_{5/3}(t) for x >= 0.

Function: double gsl_sf_synchrotron_2 (double x)
Function: int gsl_sf_synchrotron_2_e (double x, gsl_sf_result * result)
  These routines compute the second synchrotron function x K_{2/3}(x) for x >= 0.

## Transport Functions

The transport functions J(n,x) are defined by the integral representations J(n,x) := \int_0^x dt t^n e^t /(e^t - 1)^2. They are declared in the header file ``gsl_sf_transport.h'`.

Function: double gsl_sf_transport_2 (double x)
Function: int gsl_sf_transport_2_e (double x, gsl_sf_result * result)
  These routines compute the transport function J(2,x).

Function: double gsl_sf_transport_3 (double x)
Function: int gsl_sf_transport_3_e (double x, gsl_sf_result * result)
  These routines compute the transport function J(3,x).

Function: double gsl_sf_transport_4 (double x)
Function: int gsl_sf_transport_4_e (double x, gsl_sf_result * result)
  These routines compute the transport function J(4,x).

Function: double gsl_sf_transport_5 (double x)
Function: int gsl_sf_transport_5_e (double x, gsl_sf_result * result)
  These routines compute the transport function J(5,x).

## Trigonometric Functions

The library includes its own trigonometric functions in order to provide consistency across platforms and reliable error estimates. These functions are declared in the header file ``gsl_sf_trig.h'`.

### Circular Trigonometric Functions

Function: double gsl_sf_sin (double x)
Function: int gsl_sf_sin_e (double x, gsl_sf_result * result)
  These routines compute the sine function \sin(x).

Function: double gsl_sf_cos (double x)
Function: int gsl_sf_cos_e (double x, gsl_sf_result * result)
  These routines compute the cosine function \cos(x).

Function: double gsl_sf_hypot (double x, double y)
Function: int gsl_sf_hypot_e (double x, double y, gsl_sf_result * result)
  These routines compute the hypotenuse function \sqrt{x^2 + y^2} avoiding overflow and underflow.

Function: double gsl_sf_sinc (double x)
Function: int gsl_sf_sinc_e (double x, gsl_sf_result * result)
  These routines compute \sinc(x) = \sin(\pi x) / (\pi x) for any value of x.

### Trigonometric Functions for Complex Arguments

Function: int gsl_sf_complex_sin_e (double zr, double zi, gsl_sf_result * szr, gsl_sf_result * szi)
  This function computes the complex sine, \sin(z_r + i z_i) storing the real and imaginary parts in szr, szi.

Function: int gsl_sf_complex_cos_e (double zr, double zi, gsl_sf_result * czr, gsl_sf_result * czi)
  This function computes the complex cosine, \cos(z_r + i z_i) storing the real and imaginary parts in szr, szi.

Function: int gsl_sf_complex_logsin_e (double zr, double zi, gsl_sf_result * lszr, gsl_sf_result * lszi)
  This function computes the logarithm of the complex sine, \log(\sin(z_r + i z_i)) storing the real and imaginary parts in szr, szi.

### Hyperbolic Trigonometric Functions

Function: double gsl_sf_lnsinh (double x)
Function: int gsl_sf_lnsinh_e (double x, gsl_sf_result * result)
  These routines compute \log(\sinh(x)) for x > 0.

Function: double gsl_sf_lncosh (double x)
Function: int gsl_sf_lncosh_e (double x, gsl_sf_result * result)
  These routines compute \log(\cosh(x)) for any x.

### Conversion Functions

Function: int gsl_sf_polar_to_rect (double r, double theta, gsl_sf_result * x, gsl_sf_result * y);
  This function converts the polar coordinates (r,theta) to rectilinear coordinates (x,y), x = r\cos(\theta), y = r\sin(\theta).

Function: int gsl_sf_rect_to_polar (double x, double y, gsl_sf_result * r, gsl_sf_result * theta)
  This function converts the rectilinear coordinates (x,y) to polar coordinates (r,theta), such that x = r\cos(\theta), y = r\sin(\theta). The argument theta lies in the range [-\pi, \pi].

### Restriction Functions

Function: double gsl_sf_angle_restrict_symm (double theta)
Function: int gsl_sf_angle_restrict_symm_e (double * theta)
  These routines force the angle theta to lie in the range (-\pi,\pi].

Function: double gsl_sf_angle_restrict_pos (double theta)
Function: int gsl_sf_angle_restrict_pos_e (double * theta)
  These routines force the angle theta to lie in the range [0, 2\pi).

### Trigonometric Functions With Error Estimates

Function: double gsl_sf_sin_err (double x, double dx)
Function: int gsl_sf_sin_err_e (double x, double dx, gsl_sf_result * result)
  These routines compute the sine of an angle x with an associated absolute error dx, \sin(x \pm dx).

Function: double gsl_sf_cos_err (double x, double dx)
Function: int gsl_sf_cos_err_e (double x, double dx, gsl_sf_result * result)
  These routines compute the cosine of an angle x with an associated absolute error dx, \cos(x \pm dx).

## Zeta Functions

The Riemann zeta function is defined in Abramowitz & Stegun, Section 23.2. The functions described in this section are declared in the header file ``gsl_sf_zeta.h'`.

### Riemann Zeta Function

The Riemann zeta function is defined by the infinite sum \zeta(s) = \sum_{k=1}^\infty k^{-s}.

Function: double gsl_sf_zeta_int (int n)
Function: int gsl_sf_zeta_int_e (int n, gsl_sf_result * result)
  These routines compute the Riemann zeta function \zeta(n) for integer n, n \ne 1.

Function: double gsl_sf_zeta (double s)
Function: int gsl_sf_zeta_e (double s, gsl_sf_result * result)
  These routines compute the Riemann zeta function \zeta(s) for arbitrary s, s \ne 1.

### Hurwitz Zeta Function

The Hurwitz zeta function is defined by \zeta(s,q) = \sum_0^\infty (k+q)^{-s}.

Function: double gsl_sf_hzeta (double s, double q)
Function: int gsl_sf_hzeta_e (double s, double q, gsl_sf_result * result)
  These routines compute the Hurwitz zeta function \zeta(s,q) for s > 1, q > 0.

### Eta Function

The eta function is defined by \eta(s) = (1-2^{1-s}) \zeta(s).

Function: double gsl_sf_eta_int (int n)
Function: int gsl_sf_eta_int_e (int n, gsl_sf_result * result)
  These routines compute the eta function \eta(n) for integer n.

Function: double gsl_sf_eta (double s)
Function: int gsl_sf_eta_e (double s, gsl_sf_result * result)
  These routines compute the eta function \eta(s) for arbitrary s.

## Examples

The following example demonstrates the use of the error handling form of the special functions, in this case to compute the Bessel function J_0(5.0),

    #include <stdio.h>
    #include <gsl/gsl_sf_bessel.h>

    int
    main (void)
    {
      double x = 5.0;
      gsl_sf_result result;

      double expected = -0.17759677131433830434739701;

      int status = gsl_sf_bessel_J0_e (x, &result);

      printf("status  = %s\n", gsl_strerror(status));
      printf("J0(5.0) = %.18f\n"
             "      +/- % .18f\n",
             result.val, result.err);
      printf("exact   = %.18f\n", expected);
      return status;
    }

Here are the results of running the program,

    $ ./a.out
    status  = success
    J0(5.0) = -0.177596771314338292
          +/-  0.000000000000000193
    exact   = -0.177596771314338292

The next program computes the same quantity using the natural form of the function. In this case the error term result.err and return status are not accessible.

    #include <stdio.h>
    #include <gsl/gsl_sf_bessel.h>

    int
    main (void)
    {
      double x = 5.0;
      double expected = -0.17759677131433830434739701;

      double y = gsl_sf_bessel_J0 (x);

      printf("J0(5.0) = %.18f\n", y);
      printf("exact   = %.18f\n", expected);
      return 0;
    }

The results of the function are the same,

    $ ./a.out
    J0(5.0) = -0.177596771314338292
    exact   = -0.177596771314338292

## References and Further Reading

The library follows the conventions of Abramowitz & Stegun where possible,

- Abramowitz & Stegun (eds.), Handbook of Mathematical Functions

The following papers contain information on the algorithms used to compute the special functions,

- MISCFUN: A software package to compute uncommon special functions. ACM Trans. Math. Soft., vol. 22, 1996, 288-301
- G.N. Watson, A Treatise on the Theory of Bessel Functions, 2nd Edition (Cambridge University Press, 1944).
- G. Nemeth, Mathematical Approximations of Special Functions, Nova Science Publishers, ISBN 1-56072-052-2
- B.C. Carlson, Special Functions of Applied Mathematics (1977)
- W.J. Thompson, Atlas for Computing Mathematical Functions, John Wiley & Sons, New York (1997).
- Y.Y. Luke, Algorithms for the Computation of Mathematical Functions, Academic Press, New York (1977).

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