# 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.