.. index:: Bessel functions

The routines described in this section compute the Cylindrical Bessel

functions :math:`J_n(x)`, :math:`Y_n(x)`, Modified cylindrical Bessel

functions :math:`I_n(x)`, :math:`K_n(x)`, Spherical Bessel functions

:math:`j_l(x)`, :math:`y_l(x)`, and Modified Spherical Bessel functions

:math:`i_l(x)`, :math:`k_l(x)`.  For more information see Abramowitz & Stegun,

Chapters 9 and 10.  The Bessel functions are defined in the header file

:file:`gsl_sf_bessel.h`.

Regular Cylindrical Bessel Functions

------------------------------------

.. index:: Cylindrical Bessel Functions

.. index:: Regular Cylindrical Bessel Functions

.. index::

   single: J(x), Bessel Functions

.. function:: double gsl_sf_bessel_J0 (double x)

              int gsl_sf_bessel_J0_e (double x, gsl_sf_result * result)

   These routines compute the regular cylindrical Bessel function of zeroth

   order, :math:`J_0(x)`.

.. function:: double gsl_sf_bessel_J1 (double x)

              int gsl_sf_bessel_J1_e (double x, gsl_sf_result * result)

   These routines compute the regular cylindrical Bessel function of first

   order, :math:`J_1(x)`.

.. function:: double gsl_sf_bessel_Jn (int n, double x)

              int gsl_sf_bessel_Jn_e (int n, double x, gsl_sf_result * result)

   These routines compute the regular cylindrical Bessel function of 

   order :data:`n`, :math:`J_n(x)`.

.. Exceptional Return Values: GSL_EUNDRFLW

.. 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 :math:`J_n(x)` for :math:`n` from :data:`nmin` to :data:`nmax`

   inclusive, storing the results in the array :data:`result_array`.  The

   values are computed using recurrence relations for efficiency, and

   therefore may differ slightly from the exact values.

.. Exceptional Return Values: GSL_EDOM, GSL_EUNDRFLW

Irregular Cylindrical Bessel Functions

--------------------------------------

.. index:: Irregular Cylindrical Bessel Functions

.. index::

   single: Y(x), Bessel Functions

.. function:: double gsl_sf_bessel_Y0 (double x)

              int gsl_sf_bessel_Y0_e (double x, gsl_sf_result * result)

   These routines compute the irregular cylindrical Bessel function of zeroth

   order, :math:`Y_0(x)`, for :math:`x>0`.

.. Exceptional Return Values: GSL_EDOM, GSL_EUNDRFLW

.. function:: double gsl_sf_bessel_Y1 (double x)

              int gsl_sf_bessel_Y1_e (double x, gsl_sf_result * result)

   These routines compute the irregular cylindrical Bessel function of first

   order, :math:`Y_1(x)`, for :math:`x>0`.

.. Exceptional Return Values: GSL_EDOM, GSL_EOVRFLW, GSL_EUNDRFLW

.. function:: double gsl_sf_bessel_Yn (int n, double x)

              int gsl_sf_bessel_Yn_e (int n, double x, gsl_sf_result * result)

   These routines compute the irregular cylindrical Bessel function of 

   order :data:`n`, :math:`Y_n(x)`, for :math:`x>0`.

.. Exceptional Return Values: GSL_EDOM, GSL_EOVRFLW, GSL_EUNDRFLW

.. 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 :math:`Y_n(x)` for :math:`n` from :data:`nmin` to :data:`nmax`

   inclusive, storing the results in the array :data:`result_array`.  The

   domain of the function is :math:`x>0`.  The values are computed using

   recurrence relations for efficiency, and therefore may differ slightly

   from the exact values.

.. Exceptional Return Values: GSL_EDOM, GSL_EOVRFLW, GSL_EUNDRFLW

Regular Modified Cylindrical Bessel Functions

---------------------------------------------

.. index:: Modified Cylindrical Bessel Functions

.. index:: Regular Modified Cylindrical Bessel Functions

.. index::

   single: I(x), Bessel Functions

.. function:: double gsl_sf_bessel_I0 (double x)

              int gsl_sf_bessel_I0_e (double x, gsl_sf_result * result)

   These routines compute the regular modified cylindrical Bessel function

   of zeroth order, :math:`I_0(x)`.

.. Exceptional Return Values: GSL_EOVRFLW

.. function:: double gsl_sf_bessel_I1 (double x)

              int gsl_sf_bessel_I1_e (double x, gsl_sf_result * result)

   These routines compute the regular modified cylindrical Bessel function

   of first order, :math:`I_1(x)`.

.. Exceptional Return Values: GSL_EOVRFLW, GSL_EUNDRFLW

.. function:: double gsl_sf_bessel_In (int n, double x)

              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 :data:`n`, :math:`I_n(x)`.

.. Exceptional Return Values: GSL_EOVRFLW, GSL_EUNDRFLW

.. 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 :math:`I_n(x)` for :math:`n` from :data:`nmin` to

   :data:`nmax` inclusive, storing the results in the array

   :data:`result_array`.  The start of the range :data:`nmin` must be positive

   or zero.  The values are computed using recurrence relations for

   efficiency, and therefore may differ slightly from the exact values.

.. Domain: nmin >=0, nmax >= nmin 

.. Conditions: n=nmin,...,nmax, nmin >=0, nmax >= nmin 

.. Exceptional Return Values: GSL_EDOM, GSL_EOVRFLW, GSL_EUNDRFLW

.. function:: double gsl_sf_bessel_I0_scaled (double x)

              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 :math:`\exp(-|x|) I_0(x)`.

.. Exceptional Return Values: none

.. function:: double gsl_sf_bessel_I1_scaled (double x)

              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 :math:`\exp(-|x|) I_1(x)`.

.. Exceptional Return Values: GSL_EUNDRFLW

.. function:: double gsl_sf_bessel_In_scaled (int n, double x)

              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 :data:`n`, :math:`\exp(-|x|) I_n(x)` 

.. Exceptional Return Values: GSL_EUNDRFLW

.. 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 :math:`\exp(-|x|) I_n(x)` for :math:`n` from

   :data:`nmin` to :data:`nmax` inclusive, storing the results in the array

   :data:`result_array`. The start of the range :data:`nmin` must be positive

   or zero.  The values are computed using recurrence relations for

   efficiency, and therefore may differ slightly from the exact values.

.. Domain: nmin >=0, nmax >= nmin 

.. Conditions:  n=nmin,...,nmax 

.. Exceptional Return Values: GSL_EUNDRFLW

Irregular Modified Cylindrical Bessel Functions

-----------------------------------------------

.. index:: Irregular Modified Cylindrical Bessel Functions

.. index::

   single: K(x), Bessel Functions

.. function:: double gsl_sf_bessel_K0 (double x)

              int gsl_sf_bessel_K0_e (double x, gsl_sf_result * result)

   These routines compute the irregular modified cylindrical Bessel

   function of zeroth order, :math:`K_0(x)`, for :math:`x > 0`.

.. Domain: x > 0.0 

.. Exceptional Return Values: GSL_EDOM, GSL_EUNDRFLW

.. function:: double gsl_sf_bessel_K1 (double x)

              int gsl_sf_bessel_K1_e (double x, gsl_sf_result * result)

   These routines compute the irregular modified cylindrical Bessel

   function of first order, :math:`K_1(x)`, for :math:`x > 0`.

.. Domain: x > 0.0 

.. Exceptional Return Values: GSL_EDOM, GSL_EOVRFLW, GSL_EUNDRFLW

.. function:: double gsl_sf_bessel_Kn (int n, double x)

              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 :data:`n`, :math:`K_n(x)`, for :math:`x > 0`.

.. Domain: x > 0.0 

.. Exceptional Return Values: GSL_EDOM, GSL_EOVRFLW, GSL_EUNDRFLW

.. 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 :math:`K_n(x)` for :math:`n` from :data:`nmin` to

   :data:`nmax` inclusive, storing the results in the array

   :data:`result_array`. The start of the range :data:`nmin` must be positive

   or zero. The domain of the function is :math:`x>0`. The values are

   computed using recurrence relations for efficiency, and therefore

   may differ slightly from the exact values.

.. Conditions: n=nmin,...,nmax 

.. Domain: x > 0.0, nmin>=0, nmax >= nmin

.. Exceptional Return Values: GSL_EDOM, GSL_EOVRFLW, GSL_EUNDRFLW

.. function:: double gsl_sf_bessel_K0_scaled (double x)

              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 :math:`\exp(x) K_0(x)` for :math:`x>0`.

.. Domain: x > 0.0 

.. Exceptional Return Values: GSL_EDOM

.. function:: double gsl_sf_bessel_K1_scaled (double x) 

              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 :math:`\exp(x) K_1(x)` for :math:`x>0`.

.. Domain: x > 0.0 

.. Exceptional Return Values: GSL_EDOM, GSL_EUNDRFLW

.. function:: double gsl_sf_bessel_Kn_scaled (int n, double x)

              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 :data:`n`, :math:`\exp(x) K_n(x)`, for :math:`x>0`.

.. Domain: x > 0.0 

.. Exceptional Return Values: GSL_EDOM, GSL_EUNDRFLW

.. 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 :math:`\exp(x) K_n(x)` for :math:`n` from :data:`nmin` to

   :data:`nmax` inclusive, storing the results in the array

   :data:`result_array`. The start of the range :data:`nmin` must be positive

   or zero.  The domain of the function is :math:`x>0`. The values are

   computed using recurrence relations for efficiency, and therefore

   may differ slightly from the exact values.

.. Domain: x > 0.0, nmin >=0, nmax >= nmin 

.. Conditions: n=nmin,...,nmax 

.. Exceptional Return Values: GSL_EDOM, GSL_EUNDRFLW

Regular Spherical Bessel Functions

----------------------------------

.. index:: Spherical Bessel Functions

.. index:: Regular Spherical Bessel Functions

.. index::

   single: j(x), Bessel Functions

.. function:: double gsl_sf_bessel_j0 (double x)

              int gsl_sf_bessel_j0_e (double x, gsl_sf_result * result)

   These routines compute the regular spherical Bessel function of zeroth

   order, :math:`j_0(x) = \sin(x)/x`.

.. Exceptional Return Values: none

.. function:: double gsl_sf_bessel_j1 (double x)

              int gsl_sf_bessel_j1_e (double x, gsl_sf_result * result)

   These routines compute the regular spherical Bessel function of first

   order, :math:`j_1(x) = (\sin(x)/x - \cos(x))/x`.

.. Exceptional Return Values: GSL_EUNDRFLW

.. function:: double gsl_sf_bessel_j2 (double x)

              int gsl_sf_bessel_j2_e (double x, gsl_sf_result * result)

   These routines compute the regular spherical Bessel function of second

   order, :math:`j_2(x) = ((3/x^2 - 1)\sin(x) - 3\cos(x)/x)/x`.

.. Exceptional Return Values: GSL_EUNDRFLW

.. function:: double gsl_sf_bessel_jl (int l, double x)

              int gsl_sf_bessel_jl_e (int l, double x, gsl_sf_result * result)

   These routines compute the regular spherical Bessel function of 

   order :data:`l`, :math:`j_l(x)`, for

   :math:`l \geq 0` and :math:`x \geq 0`.

.. Domain: l >= 0, x >= 0.0 

.. Exceptional Return Values: GSL_EDOM, GSL_EUNDRFLW

.. 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 :math:`j_l(x)` for :math:`l` from 0 to :data:`lmax`

   inclusive  for

   :math:`lmax \geq 0` and

   :math:`x \geq 0`, storing the results in the array :data:`result_array`.

   The values are computed using recurrence relations for

   efficiency, and therefore may differ slightly from the exact values.

.. Domain: lmax >= 0 

.. Conditions: l=0,1,...,lmax 

.. Exceptional Return Values: GSL_EDOM, GSL_EUNDRFLW

.. function:: int gsl_sf_bessel_jl_steed_array (int lmax, double x, double * result_array)

   This routine uses Steed's method to compute the values of the regular

   spherical Bessel functions :math:`j_l(x)` for :math:`l` from 0 to

   :data:`lmax` inclusive for

   :math:`lmax \geq 0` and

   :math:`x \geq 0`, storing the results in the array

   :data:`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.

.. Domain: lmax >= 0 

.. Conditions: l=0,1,...,lmax 

.. Exceptional Return Values: GSL_EDOM, GSL_EUNDRFLW

Irregular Spherical Bessel Functions

------------------------------------

.. index:: Irregular Spherical Bessel Functions

.. index::

   single: y(x), Bessel Functions

.. function:: double gsl_sf_bessel_y0 (double x)

              int gsl_sf_bessel_y0_e (double x, gsl_sf_result * result)

   These routines compute the irregular spherical Bessel function of zeroth

   order, :math:`y_0(x) = -\cos(x)/x`.

.. Exceptional Return Values: none

.. function:: double gsl_sf_bessel_y1 (double x)

              int gsl_sf_bessel_y1_e (double x, gsl_sf_result * result)

   These routines compute the irregular spherical Bessel function of first

   order, :math:`y_1(x) = -(\cos(x)/x + \sin(x))/x`.

.. Exceptional Return Values: GSL_EUNDRFLW

.. function:: double gsl_sf_bessel_y2 (double x)

              int gsl_sf_bessel_y2_e (double x, gsl_sf_result * result)

   These routines compute the irregular spherical Bessel function of second

   order, :math:`y_2(x) = (-3/x^3 + 1/x)\cos(x) - (3/x^2)\sin(x)`.

.. Exceptional Return Values: GSL_EUNDRFLW

.. function:: double gsl_sf_bessel_yl (int l, double x)

              int gsl_sf_bessel_yl_e (int l, double x, gsl_sf_result * result)

   These routines compute the irregular spherical Bessel function of 

   order :data:`l`, :math:`y_l(x)`, for

   :math:`l \geq 0`.

.. Exceptional Return Values: GSL_EUNDRFLW

.. 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 :math:`y_l(x)` for :math:`l` from 0 to :data:`lmax`

   inclusive for

   :math:`lmax \geq 0`, storing the results in the array :data:`result_array`.

   The values are computed using recurrence relations for

   efficiency, and therefore may differ slightly from the exact values.

.. Domain: lmax >= 0 

.. Conditions: l=0,1,...,lmax 

.. Exceptional Return Values: GSL_EUNDRFLW

Regular Modified Spherical Bessel Functions

-------------------------------------------

.. index:: Modified Spherical Bessel Functions

.. index:: Regular Modified Spherical Bessel Functions

.. index::

   single: i(x), Bessel Functions

The regular modified spherical Bessel functions :math:`i_l(x)` 

are related to the modified Bessel functions of fractional order,

:math:`i_l(x) = \sqrt{\pi/(2x)} I_{l+1/2}(x)`

.. function:: double gsl_sf_bessel_i0_scaled (double x)

              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, :math:`\exp(-|x|) i_0(x)`.

.. Exceptional Return Values: none

.. function:: double gsl_sf_bessel_i1_scaled (double x)

              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, :math:`\exp(-|x|) i_1(x)`.

.. Exceptional Return Values: GSL_EUNDRFLW

.. function:: double gsl_sf_bessel_i2_scaled (double x)

              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, :math:`\exp(-|x|) i_2(x)` 

.. Exceptional Return Values: GSL_EUNDRFLW

.. function:: double gsl_sf_bessel_il_scaled (int l, double x)

              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 :data:`l`, :math:`\exp(-|x|) i_l(x)`

.. Domain: l >= 0 

.. Exceptional Return Values: GSL_EDOM, GSL_EUNDRFLW

.. 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

   spherical Bessel functions :math:`\exp(-|x|) i_l(x)` for :math:`l` from

   0 to :data:`lmax` inclusive for

   :math:`lmax \geq 0`, storing the results in

   the array :data:`result_array`. 

   The values are computed using recurrence relations for

   efficiency, and therefore may differ slightly from the exact values.

.. Domain: lmax >= 0 

.. Conditions: l=0,1,...,lmax 

.. Exceptional Return Values: GSL_EUNDRFLW

Irregular Modified Spherical Bessel Functions

---------------------------------------------

.. index:: Irregular Modified Spherical Bessel Functions

.. index::

   single: k(x), Bessel Functions

The irregular modified spherical Bessel functions :math:`k_l(x)`

are related to the irregular modified Bessel functions of fractional order,

:math:`k_l(x) = \sqrt{\pi/(2x)} K_{l+1/2}(x)`.

.. function:: double gsl_sf_bessel_k0_scaled (double x)

              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, :math:`\exp(x) k_0(x)`, for :math:`x>0`.

.. Domain: x > 0.0 

.. Exceptional Return Values: GSL_EDOM, GSL_EUNDRFLW

.. function:: double gsl_sf_bessel_k1_scaled (double x)

              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, :math:`\exp(x) k_1(x)`, for :math:`x>0`.

.. Domain: x > 0.0 

.. Exceptional Return Values: GSL_EDOM, GSL_EUNDRFLW, GSL_EOVRFLW

.. function:: double gsl_sf_bessel_k2_scaled (double x)

              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, :math:`\exp(x) k_2(x)`, for :math:`x>0`.

.. Domain: x > 0.0 

.. Exceptional Return Values: GSL_EDOM, GSL_EUNDRFLW, GSL_EOVRFLW

.. function:: double gsl_sf_bessel_kl_scaled (int l, double x)

              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 :data:`l`, :math:`\exp(x) k_l(x)`, for :math:`x>0`.

.. Domain: x > 0.0 

.. Exceptional Return Values: GSL_EDOM, GSL_EUNDRFLW

.. 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 :math:`\exp(x) k_l(x)` for :math:`l` from

   0 to :data:`lmax` inclusive for

   :math:`lmax \geq 0` and :math:`x>0`, storing the results in

   the array :data:`result_array`. 

   The values are computed using recurrence relations for

   efficiency, and therefore may differ slightly from the exact values.

.. Domain: lmax >= 0 

.. Conditions: l=0,1,...,lmax 

.. Exceptional Return Values: GSL_EDOM, GSL_EUNDRFLW

Regular Bessel Function---Fractional Order

------------------------------------------

.. index::

   single: Fractional Order Bessel Functions

   single: Bessel Functions, Fractional Order

   single: Regular Bessel Functions, Fractional Order

.. function:: double gsl_sf_bessel_Jnu (double nu, double x)

              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 :math:`\nu`, :math:`J_\nu(x)`.

.. Exceptional Return Values: GSL_EDOM, GSL_EUNDRFLW

.. 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 :math:`\nu`, :math:`J_\nu(x)`, evaluated at a series of

   :math:`x` values.  The array :data:`v` of length :data:`size` contains the

   :math:`x` values.  They are assumed to be strictly ordered and positive.

   The array is over-written with the values of :math:`J_\nu(x_i)`.

.. Exceptional Return Values: GSL_EDOM, GSL_EINVAL

Irregular Bessel Functions---Fractional Order

---------------------------------------------

.. function:: double gsl_sf_bessel_Ynu (double nu, double x)

              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 :math:`\nu`, :math:`Y_\nu(x)`.

.. Exceptional Return Values: 

Regular Modified Bessel Functions---Fractional Order

----------------------------------------------------

.. index::

   single: Modified Bessel Functions, Fractional Order

   single: Regular Modified Bessel Functions, Fractional Order

.. function:: double gsl_sf_bessel_Inu (double nu, double x)

              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 :math:`\nu`, :math:`I_\nu(x)` for :math:`x>0`,

   :math:`\nu>0`.

.. Domain: x >= 0, nu >= 0 

.. Exceptional Return Values: GSL_EDOM, GSL_EOVRFLW

.. function:: double gsl_sf_bessel_Inu_scaled (double nu, double x)

              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 :math:`\nu`, :math:`\exp(-|x|)I_\nu(x)` for :math:`x>0`,

   :math:`\nu>0`.

.. Domain: x >= 0, nu >= 0 

.. Exceptional Return Values: GSL_EDOM

Irregular Modified Bessel Functions---Fractional Order

------------------------------------------------------

.. index::

   single: Irregular Modified Bessel Functions, Fractional Order

.. function:: double gsl_sf_bessel_Knu (double nu, double x)

              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 :math:`\nu`, :math:`K_\nu(x)` for :math:`x>0`,

   :math:`\nu>0`.

.. Domain: x > 0, nu >= 0 

.. Exceptional Return Values: GSL_EDOM, GSL_EUNDRFLW

.. function:: double gsl_sf_bessel_lnKnu (double nu, double x)

              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 :math:`\nu`, :math:`\ln(K_\nu(x))` for

   :math:`x>0`, :math:`\nu>0`. 

.. Domain: x > 0, nu >= 0 

.. Exceptional Return Values: GSL_EDOM

.. function:: double gsl_sf_bessel_Knu_scaled (double nu, double x)

              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 :math:`\nu`, :math:`\exp(+|x|) K_\nu(x)` for :math:`x>0`,

   :math:`\nu>0`.

.. Domain: x > 0, nu >= 0 

.. Exceptional Return Values: GSL_EDOM

Zeros of Regular Bessel Functions

---------------------------------

.. index::

   single: Zeros of Regular Bessel Functions

   single: Regular Bessel Functions, Zeros of 

.. function:: double gsl_sf_bessel_zero_J0 (unsigned int s)

              int gsl_sf_bessel_zero_J0_e (unsigned int s, gsl_sf_result * result)

   These routines compute the location of the :data:`s`-th positive zero of

   the Bessel function :math:`J_0(x)`.

.. Exceptional Return Values: 

.. function:: double gsl_sf_bessel_zero_J1 (unsigned int s)

              int gsl_sf_bessel_zero_J1_e (unsigned int s, gsl_sf_result * result)

   These routines compute the location of the :data:`s`-th positive zero of

   the Bessel function :math:`J_1(x)`.

.. Exceptional Return Values: 

.. function:: double gsl_sf_bessel_zero_Jnu (double nu, unsigned int s)

              int gsl_sf_bessel_zero_Jnu_e (double nu, unsigned int s, gsl_sf_result * result)

   These routines compute the location of the :data:`s`-th positive zero of

   the Bessel function :math:`J_\nu(x)`.  The current implementation does not

   support negative values of :data:`nu`. 

.. Exceptional Return Values: