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.

Regular Cylindrical Bessel Functions

gsl_sf_bessel_J0(x)

This routine computes the regular cylindrical Bessel function of zeroth order, \(J_0(x)\).

gsl_sf_bessel_J1(x)

This routine computes the regular cylindrical Bessel function of first order, \(J_1(x)\).

gsl_sf_bessel_Jn(n, x)

This routine computes the regular cylindrical Bessel function of integer order \(n\), \(J_n(x)\).

Irregular Cylindrical Bessel Functions

gsl_sf_bessel_Y0(x)

This routine computes the irregular cylindrical Bessel function of zeroth order, \(Y_0(x)\), for \(x > 0\).

gsl_sf_bessel_Y1(x)

This routine computes the irregular cylindrical Bessel function of first order, \(Y_1(x)\), for \(x > 0\).

gsl_sf_bessel_Yn(n, x)

This routine computes the irregular cylindrical Bessel function of integer order \(n\), \(Y_n(x)\), for \(x > 0\).

Regular Modified Cylindrical Bessel Functions

gsl_sf_bessel_I0(x)

This routine computes the regular modified cylindrical Bessel function of zeroth order, \(I_0(x)\).

gsl_sf_bessel_I1(x)

This routine computes the regular modified cylindrical Bessel function of first order, \(I_1(x)\).

gsl_sf_bessel_In(n, x)

This routine computes the regular modified cylindrical Bessel function of integer order \(n\), \(I_n(x)\).

gsl_sf_bessel_I0_scaled(x)

This routine computes the scaled regular modified cylindrical Bessel function of zeroth order \(\exp(-|x|) I_0(x)\).

gsl_sf_bessel_I1_scaled(x)

This routine computes the scaled regular modified cylindrical Bessel function of first order \(\exp(-|x|) I_1(x)\).

gsl_sf_bessel_In_scaled(n, x)

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

Irregular Modified Cylindrical Bessel Functions

gsl_sf_bessel_K0(x)

This routine computes the irregular modified cylindrical Bessel function of zeroth order, \(K_0(x)\), for \(x > 0\).

gsl_sf_bessel_K1(x)

This routine computes the irregular modified cylindrical Bessel function of first order, \(K_1(x)\), for \(x > 0\).

gsl_sf_bessel_Kn(n, x)

This routine computes the irregular modified cylindrical Bessel function of integer order \(n\), \(K_n(x)\), for \(x > 0\).

gsl_sf_bessel_K0_scaled(x)

This routine 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)

This routine 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)

This routine computes the scaled irregular modified cylindrical Bessel function of integer order \(n\), \(\exp(x) K_n(x)\), for \(x > 0\).

Regular Spherical Bessel Functions

gsl_sf_bessel_j0(x)

This routine computes the regular spherical Bessel function of zeroth order, \(j_0(x) = \sin(x)/x\).

gsl_sf_bessel_j1(x)

This routine computes the regular spherical Bessel function of first order, \(j_1(x) = (\sin(x)/x - \cos(x))/x\).

gsl_sf_bessel_j2(x)

This routine computes the regular spherical Bessel function of second order, \(j_2(x) = ((3/x^2 - 1)\sin(x) - 3\cos(x)/x)/x\).

gsl_sf_bessel_jl(l, x)

This routine computes the regular spherical Bessel function of integer order \(l\), \(j_l(x)\), for \(l \geq 0\) and \(x \geq 0\).

Irregular Spherical Bessel Functions

gsl_sf_bessel_y0(x)

This routine computes the irregular spherical Bessel function of zeroth order, \(y_0(x) = -\cos(x)/x\).

gsl_sf_bessel_y1(x)

This routine computes the irregular spherical Bessel function of first order, \(y_1(x) = -(\cos(x)/x + \sin(x))/x\).

gsl_sf_bessel_y2(x)

This routine 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)

This routine computes the irregular spherical Bessel function of integer order \(l\), \(y_l(x)\), for \(l \geq 0\).

Regular Modified Spherical Bessel Functions

gsl_sf_bessel_i0_scaled(x)

This routine computes the regular modified spherical Bessel function of zeroth order, \(\exp(-|x|) i_0(x)\).

gsl_sf_bessel_i1_scaled(x)

This routine computes the regular modified spherical Bessel function of first order, \(\exp(-|x|) i_1(x)\).

gsl_sf_bessel_i2_scaled(x)

This routine computes the regular modified spherical Bessel function of second order, \(\exp(-|x|) i_2(x)\).

gsl_sf_bessel_il_scaled(l, x)

This routine computes the regular modified spherical Bessel function of integer order \(l\), \(\exp(-|x|) i_l(x)\).

Irregular Modified Spherical Bessel Functions

gsl_sf_bessel_k0_scaled(x)

This routine computes the scaled irregular modified spherical Bessel function of zeroth order, \(\exp(x) k_0(x)\), for \(x > 0\).

gsl_sf_bessel_k1_scaled(x)

This routine computes the scaled irregular modified spherical Bessel function of first order, \(\exp(x) k_1(x)\), for \(x > 0\).

gsl_sf_bessel_k2_scaled(x)

This routine computes the scaled irregular modified spherical Bessel function of second order, \(\exp(x) k_2(x)\), for \(x > 0\).

gsl_sf_bessel_kl_scaled(l, x)

This routine computes the scaled irregular modified spherical Bessel function of integer order \(l\), \(\exp(x) k_l(x)\), for \(x > 0\).

Regular Bessel Function - Fractional Order

gsl_sf_bessel_Jnu(nu, x)

This routine computes the regular cylindrical Bessel function of fractional order \(\nu\), \(J_\nu(x)\).

Irregular Bessel Function - Fractional Order

gsl_sf_bessel_Ynu(nu, x)

This routine computes the irregular cylindrical Bessel function of fractional order \(\nu\), \(Y_\nu(x)\).

Regular Modified Bessel Functions - Fractional Order

gsl_sf_bessel_Inu(nu, x)

This routine 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)

This routine computes 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

gsl_sf_bessel_Knu(nu, x)

This routine computes the irregular modified Bessel function of fractional order \(\nu\), \(K_\nu(x)\) for \(x > 0\), \(\nu > 0\).

gsl_sf_bessel_lnKnu(nu, x)

This routine 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)

This routine computes 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

gsl_sf_bessel_zero_J0(s)

This routine computes the location of the \(s\)-th positive zero of the Bessel function \(J_0(x)\).

gsl_sf_bessel_zero_J1(s)

This routine computes the location of the \(s\)-th positive zero of the Bessel function \(J_1(x)\).

gsl_sf_bessel_zero_Jnu(nu, s)

This routine 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.