1 Bessel Functions

1.1 Regular Cylindrical Bessel Functions

GSL::Sf::bessel_J0(x)
Computes the regular cylindrical Bessel function of zeroth order, J_0(x).
GSL::Sf::bessel_J1(x)
Computes the regular cylindrical Bessel function of first order, J_1(x).
GSL::Sf::bessel_Jn(n, x)
Computes the regular cylindrical Bessel function of order n, J_n(x).
GSL::Sf::bessel_Jn_array(nmin, nmax, x)
Computes the values of the regular cylindrical Bessel functions J_n(x) for n from nmin to nmax inclusive, and returns the results as a GSL::Vector object. The values are computed using recurrence relations, for efficiency, and therefore may differ slightly from the exact values.

1.2 Irregular Cylindrical Bessel Functions

GSL::Sf::bessel_Y0(x)
Computes the irregular cylindrical Bessel function of zeroth order, Y_0(x).
GSL::Sf::bessel_Y1(x)
Computes the irregular cylindrical Bessel function of first order, Y_1(x).
GSL::Sf::bessel_Yn(n, x)
Computes the irregular cylindrical Bessel function of order n, Y_n(x).
GSL::Sf::bessel_Yn_array(nmin, nmax, x)
Computes the values of the irregular cylindrical Bessel functions Y_n(x) for n from nmin to nmax inclusive, and returns the results as a GSL::Vector object. The values are computed using recurrence relations, for efficiency, and therefore may differ slightly from the exact values.

1.3 Regular Modified Cylindrical Bessel Functions

GSL::Sf::bessel_I0(x)
Computes the regular modified cylindrical Bessel function of zeroth order, I_0(x).
GSL::Sf::bessel_I1(x)
Computes the regular modified cylindrical Bessel function of first order, I_1(x).
GSL::Sf::bessel_In(n, x)
Computes the regular modified cylindrical Bessel function of order n, I_n(x).
GSL::Sf::bessel_In_array(nmin, nmax, x)
Computes the values of the regular modified cylindrical Bessel functions I_n(x) for n from nmin to nmax inclusive, and returns the results as a GSL::Vector object. 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(x)
Computes the scaled regular modified cylindrical Bessel function of zeroth order, exp(-|x|) I_0(x).
GSL::Sf::bessel_I1_scaled(x)
Computes the scaled regular modified cylindrical Bessel function of first order, exp(-|x|)I_1(x).
GSL::Sf::bessel_In_scaled(n, x)
Computes the scaled regular modified cylindrical Bessel function of order n, exp(-|x|) I_n(x).
GSL::Sf::bessel_In_scaled_array(nmin, nmax, x)
Computes the values of the scaled regular modified cylindrical Bessel functions exp(-|x|) I_n(x) for n from nmin to nmax inclusive, and returns the results as a GSL::Vector object. 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.

1.4 Irregular Modified Cylindrical Bessel Functions

GSL::Sf::bessel_K0(x)
Computes the irregular modified cylindrical Bessel function of zeroth order, K_0(x), for x > 0.
GSL::Sf::bessel_K1(x)
Computes the irregular modified cylindrical Bessel function of first order, K_1(x), for x > 0.
GSL::Sf::bessel_Kn(n, x)
Computes the irregular modified cylindrical Bessel function of order n, K_n(x), for x > 0.
GSL::Sf::bessel_Kn_array(nmin, nmax, x)
Computes the values of the irregular modified cylindrical Bessel functions K_n(x) for n from nmin to nmax inclusive, and returns the results as a GSL::Vector object. 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_K0_scaled(x)
Computes the scaled irregular modified cylindrical Bessel function of zeroth order exp(x) K_0(x) for x>0.
GSL::Sf::bessel_K1_scaled(x)
Computes the scaled irregular modified cylindrical Bessel function of first order exp(x) K_1(x) for x>0
GSL::Sf::bessel_Kn_scaled(n, x)
Computes the scaled irregular modified cylindrical Bessel function of order n, exp(x) K_n(x), for x>0.
GSL::Sf::bessel_Kn_scaled_array(nmin, nmax, x)
Computes the values of the scaled irregular cylindrical Bessel functions exp(x) K_n(x) for n from nmin to nmax inclusive, and returns the results as a GSL::Vector object. 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.

1.5 Regular Spherical Bessel Functions

GSL::Sf::bessel_j0(x)
Computes the regular spherical Bessel function of zeroth order, j0(x) = sin(x)/x.
GSL::Sf::bessel_j1(x)
Computes the regular spherical Bessel function of first order, j1(x) = (sin(x)/x - cos(x))/x.
GSL::Sf::bessel_j2(x)
Computes the regular spherical Bessel function of second order, j2(x) = ((3/x^2 - 1)sin(x) - 3cos(x)/x)/x.
GSL::Sf::bessel_jl(l, x)
Computes the regular spherical Bessel function of order l, j_l(x), for l >= 0 and x >= 0.
GSL::Sf::bessel_jl_array(lmax, x)
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, and returns the results as a GSL::Vector object. 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 method 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, and returns the results as a GSL::Vector object. 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.

1.6 Irregular Spherical Bessel Functions

GSL::Sf::bessel_y0(x)
Computes the irregular spherical Bessel function of zeroth order, y_0(x) = -cos(x)/x.
GSL::Sf::bessel_y1(x)
Computes the irregular spherical Bessel function of first order, y_1(x) = -(cos(x)/x + sin(x))/x.
GSL::Sf::bessel_y2(x)
Computes the irregular spherical Bessel function of second order, y_2(x) = (-3/x^3 + 1/x)cos(x) - (3/x^2)sin(x).
GSL::Sf::bessel_yl(l, x)
Computes the irregular spherical Bessel function of order l, y_l(x), for l >= 0.
GSL::Sf::bessel_yl_array(lmax, x)
This method computes the values of the irregular spherical Bessel functions y_l(x) for l from 0 to lmax inclusive for lmax >= 0), and returns the results as a GSL::Vector object. The values are computed using recurrence relations, for efficiency, and therefore may differ slightly from the exact values.

1.7 Regular Modified Spherical Bessel Functions

GSL::Sf::bessel_i0_scaled(x)
Computes the scaled regular modified spherical Bessel function of zeroth order, exp(-|x|) i_0(x).
GSL::Sf::bessel_i1_scaled(x)
Computes the scaled regular modified spherical Bessel function of first order, exp(-|x|) i_1(x).
GSL::Sf::bessel_i2_scaled(x)
Computes the scaled regular modified spherical Bessel function of second order, exp(-|x|) i_2(x).
GSL::Sf::bessel_il_scaled(l, x)
Computes the scaled regular modified spherical Bessel function of order l, exp(-|x|) i_l(x).
GSL::Sf::bessel_il_scaled_array(lmax, x)
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, and returns the results as a GSL::Vector object. The values are computed using recurrence relations, for efficiency, and therefore may differ slightly from the exact values.

1.8 Irregular Modified Spherical Bessel Functions

GSL::Sf::bessel_k0_scaled(x)
Computes the scaled irregular modified spherical Bessel function of zeroth order, exp(-|x|) k_0(x).
GSL::Sf::bessel_k1_scaled(x)
Computes the scaled irregular modified spherical Bessel function of first order, exp(-|x|) k_1(x).
GSL::Sf::bessel_k2_scaled(x)
Computes the scaled irregular modified spherical Bessel function of second order, exp(-|x|) k_2(x).
GSL::Sf::bessel_kl_scaled(l, x)
Computes the scaled irregular modified spherical Bessel function of order l, exp(-|x|) k_l(x).
GSL::Sf::bessel_kl_scaled_array(lmax, x)
This routine computes the values of the scaled irregular modified cylindrical Bessel functions exp(-|x|) k_l(x) for l from 0 to lmax inclusive for lmax >= 0, and returns the results as a GSL::Vector object. The values are computed using recurrence relations, for efficiency, and therefore may differ slightly from the exact values.

1.9 Regular Bessel Function - Fractional Order

GSL::Sf::bessel_Jnu(nu, x)
Computes the regular cylindrical Bessel function of fractional order nu, J_nu(x).
GSL::Sf::bessel_sequence_Jnu_e(nu, v)
GSL::Sf::bessel_sequence_Jnu_e(nu, mode, v)
These compute the regular cylindrical Bessel function of fractional order nu, J_nu(x), evaluated at a series of x values. The GSL::Vector object v contains the x values. They are assumed to be strictly ordered and positive. The vector is over-written with the values of J_nu(x_i).

1.10 Irregular Bessel Functions - Fractional Order

GSL::Sf::bessel_Ynu(nu, x)
Computes the irregular cylindrical Bessel function of fractional order nu, Y_nu(x).

1.11 Regular Modified Bessel Functions - Fractional Order

GSL::Sf::bessel_Inu(nu, x)
Computes the regular modified Bessel function of fractional order nu, I_nu(x) for x>0, nu>0.
GSL::Sf::bessel_Inu_scaled(nu, x)
Computes the scaled regular modified Bessel function of fractional order nu, exp(-|x|) I_nu(x) for x>0, nu>0.

1.12 Irregular Modified Bessel Functions - Fractional Order

GSL::Sf::bessel_Knu(nu, x)
Computes the irregular modified Bessel function of fractional order nu, K_nu(x) for x>0, nu>0.
GSL::Sf::bessel_lnKnu(nu, x)
Computes the logarithm of the irregular modified Bessel function of fractional order nu, ln(K_nu(x)) for x>0, nu>0
GSL::Sf::bessel_Knu_scaled(nu, x)
Computes the scaled irregular modified Bessel function of fractional order nu, exp(+|x|) K_nu(x) for x>0, nu>0.

1.13 Zeros of Regular Bessel Functions

GSL::Sf::bessel_zero_J0(s)
Computes the location of the s-th positive zero of the Bessel function J_0(x).
GSL::Sf::bessel_zero_J1(s)
Computes the location of the s-th positive zero of the Bessel function J_1(x).
GSL::Sf::bessel_zero_Jnu(nu, s)
Computes the location of the s-th positive zero of the Bessel function J_nu(x). The current implementation does not support negative values of nu.