path: root/gsl-1.9/doc/specfunc-bessel.texi

@cindex 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}.

@menu
* Regular Cylindrical Bessel Functions::  
* Irregular Cylindrical Bessel Functions::  
* Regular Modified Cylindrical Bessel Functions::  
* Irregular Modified Cylindrical Bessel Functions::  
* Regular Spherical Bessel Functions::  
* Irregular Spherical Bessel Functions::  
* Regular Modified Spherical Bessel Functions::  
* Irregular Modified Spherical Bessel Functions::  
* Regular Bessel Function - Fractional Order::  
* Irregular Bessel Functions - Fractional Order::  
* Regular Modified Bessel Functions - Fractional Order::  
* Irregular Modified Bessel Functions - Fractional Order::  
* Zeros of Regular Bessel Functions::  
@end menu

@node Regular Cylindrical Bessel Functions
@subsection Regular Cylindrical Bessel Functions
@cindex Cylindrical Bessel Functions
@cindex Regular Cylindrical Bessel Functions

@deftypefun double gsl_sf_bessel_J0 (double @var{x})
@deftypefunx int gsl_sf_bessel_J0_e (double @var{x}, gsl_sf_result * @var{result})
These routines compute the regular cylindrical Bessel function of zeroth
order, @math{J_0(x)}.
@comment Exceptional Return Values: none
@end deftypefun

@deftypefun double gsl_sf_bessel_J1 (double @var{x})
@deftypefunx int gsl_sf_bessel_J1_e (double @var{x}, gsl_sf_result * @var{result})
These routines compute the regular cylindrical Bessel function of first
order, @math{J_1(x)}.
@comment Exceptional Return Values: GSL_EUNDRFLW
@end deftypefun

@deftypefun double gsl_sf_bessel_Jn (int @var{n}, double @var{x})
@deftypefunx int gsl_sf_bessel_Jn_e (int @var{n}, double @var{x}, gsl_sf_result * @var{result})
These routines compute the regular cylindrical Bessel function of 
order @var{n}, @math{J_n(x)}.
@comment Exceptional Return Values: GSL_EUNDRFLW
@end deftypefun

@deftypefun int gsl_sf_bessel_Jn_array (int @var{nmin}, int @var{nmax}, double @var{x}, double @var{result_array}[])
This routine computes the values of the regular cylindrical Bessel
functions @math{J_n(x)} for @math{n} from @var{nmin} to @var{nmax}
inclusive, storing the results in the array @var{result_array}.  The
values are computed using recurrence relations for efficiency, and
therefore may differ slightly from the exact values.
@comment Exceptional Return Values: GSL_EDOM, GSL_EUNDRFLW
@end deftypefun


@node Irregular Cylindrical Bessel Functions
@subsection Irregular Cylindrical Bessel Functions
@cindex Irregular Cylindrical Bessel Functions

@deftypefun double gsl_sf_bessel_Y0 (double @var{x})
@deftypefunx int gsl_sf_bessel_Y0_e (double @var{x}, gsl_sf_result * @var{result})
These routines compute the irregular cylindrical Bessel function of zeroth
order, @math{Y_0(x)}, for @math{x>0}.
@comment Exceptional Return Values: GSL_EDOM, GSL_EUNDRFLW
@end deftypefun

@deftypefun double gsl_sf_bessel_Y1 (double @var{x})
@deftypefunx int gsl_sf_bessel_Y1_e (double @var{x}, gsl_sf_result * @var{result})
These routines compute the irregular cylindrical Bessel function of first
order, @math{Y_1(x)}, for @math{x>0}.
@comment Exceptional Return Values: GSL_EDOM, GSL_EOVRFLW, GSL_EUNDRFLW
@end deftypefun

@deftypefun double gsl_sf_bessel_Yn (int @var{n},double @var{x})
@deftypefunx int gsl_sf_bessel_Yn_e (int @var{n},double @var{x}, gsl_sf_result * @var{result})
These routines compute the irregular cylindrical Bessel function of 
order @var{n}, @math{Y_n(x)}, for @math{x>0}.
@comment Exceptional Return Values: GSL_EDOM, GSL_EOVRFLW, GSL_EUNDRFLW
@end deftypefun

@deftypefun int gsl_sf_bessel_Yn_array (int @var{nmin}, int @var{nmax}, double @var{x}, double @var{result_array}[])
This routine computes the values of the irregular cylindrical Bessel
functions @math{Y_n(x)} for @math{n} from @var{nmin} to @var{nmax}
inclusive, storing the results in the array @var{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.
@comment Exceptional Return Values: GSL_EDOM, GSL_EOVRFLW, GSL_EUNDRFLW
@end deftypefun


@node Regular Modified Cylindrical Bessel Functions
@subsection Regular Modified Cylindrical Bessel Functions
@cindex Modified Cylindrical Bessel Functions
@cindex Regular Modified Cylindrical Bessel Functions

@deftypefun double gsl_sf_bessel_I0 (double @var{x})
@deftypefunx int gsl_sf_bessel_I0_e (double @var{x}, gsl_sf_result * @var{result})
These routines compute the regular modified cylindrical Bessel function
of zeroth order, @math{I_0(x)}.
@comment Exceptional Return Values: GSL_EOVRFLW
@end deftypefun

@deftypefun double gsl_sf_bessel_I1 (double @var{x})
@deftypefunx int gsl_sf_bessel_I1_e (double @var{x}, gsl_sf_result * @var{result})
These routines compute the regular modified cylindrical Bessel function
of first order, @math{I_1(x)}.
@comment Exceptional Return Values: GSL_EOVRFLW, GSL_EUNDRFLW
@end deftypefun

@deftypefun double gsl_sf_bessel_In (int @var{n}, double @var{x})
@deftypefunx int gsl_sf_bessel_In_e (int @var{n}, double @var{x}, gsl_sf_result * @var{result})
These routines compute the regular modified cylindrical Bessel function
of order @var{n}, @math{I_n(x)}.
@comment Exceptional Return Values: GSL_EOVRFLW, GSL_EUNDRFLW
@end deftypefun

@deftypefun int gsl_sf_bessel_In_array (int @var{nmin}, int @var{nmax}, double @var{x}, double @var{result_array}[])
This routine computes the values of the regular modified cylindrical
Bessel functions @math{I_n(x)} for @math{n} from @var{nmin} to
@var{nmax} inclusive, storing the results in the array
@var{result_array}.  The start of the range @var{nmin} must be positive
or zero.  The values are computed using recurrence relations for
efficiency, and therefore may differ slightly from the exact values.
@comment Domain: nmin >=0, nmax >= nmin 
@comment Conditions: n=nmin,...,nmax, nmin >=0, nmax >= nmin 
@comment Exceptional Return Values: GSL_EDOM, GSL_EOVRFLW, GSL_EUNDRFLW
@end deftypefun

@deftypefun double gsl_sf_bessel_I0_scaled (double @var{x})
@deftypefunx int gsl_sf_bessel_I0_scaled_e (double @var{x}, gsl_sf_result * @var{result})
These routines compute the scaled regular modified cylindrical Bessel
function of zeroth order @math{\exp(-|x|) I_0(x)}.
@comment Exceptional Return Values: none
@end deftypefun

@deftypefun double gsl_sf_bessel_I1_scaled (double @var{x})
@deftypefunx int gsl_sf_bessel_I1_scaled_e (double @var{x}, gsl_sf_result * @var{result})
These routines compute the scaled regular modified cylindrical Bessel
function of first order @math{\exp(-|x|) I_1(x)}.
@comment Exceptional Return Values: GSL_EUNDRFLW
@end deftypefun

@deftypefun double gsl_sf_bessel_In_scaled (int @var{n}, double @var{x})
@deftypefunx int gsl_sf_bessel_In_scaled_e (int @var{n}, double @var{x}, gsl_sf_result * @var{result})
These routines compute the scaled regular modified cylindrical Bessel
function of order @var{n}, @math{\exp(-|x|) I_n(x)} 
@comment Exceptional Return Values: GSL_EUNDRFLW
@end deftypefun

@deftypefun int gsl_sf_bessel_In_scaled_array (int @var{nmin}, int @var{nmax}, double @var{x}, double @var{result_array}[])
This routine computes the values of the scaled regular cylindrical
Bessel functions @math{\exp(-|x|) I_n(x)} for @math{n} from
@var{nmin} to @var{nmax} inclusive, storing the results in the array
@var{result_array}. The start of the range @var{nmin} must be positive
or zero.  The values are computed using recurrence relations for
efficiency, and therefore may differ slightly from the exact values.
@comment Domain: nmin >=0, nmax >= nmin 
@comment Conditions:  n=nmin,...,nmax 
@comment Exceptional Return Values: GSL_EUNDRFLW
@end deftypefun


@node Irregular Modified Cylindrical Bessel Functions
@subsection Irregular Modified Cylindrical Bessel Functions
@cindex Irregular Modified Cylindrical Bessel Functions

@deftypefun double gsl_sf_bessel_K0 (double @var{x})
@deftypefunx int gsl_sf_bessel_K0_e (double @var{x}, gsl_sf_result * @var{result})
These routines compute the irregular modified cylindrical Bessel
function of zeroth order, @math{K_0(x)}, for @math{x > 0}.
@comment Domain: x > 0.0 
@comment Exceptional Return Values: GSL_EDOM, GSL_EUNDRFLW
@end deftypefun

@deftypefun double gsl_sf_bessel_K1 (double @var{x})
@deftypefunx int gsl_sf_bessel_K1_e (double @var{x}, gsl_sf_result * @var{result})
These routines compute the irregular modified cylindrical Bessel
function of first order, @math{K_1(x)}, for @math{x > 0}.
@comment Domain: x > 0.0 
@comment Exceptional Return Values: GSL_EDOM, GSL_EOVRFLW, GSL_EUNDRFLW
@end deftypefun

@deftypefun double gsl_sf_bessel_Kn (int @var{n}, double @var{x})
@deftypefunx int gsl_sf_bessel_Kn_e (int @var{n}, double @var{x}, gsl_sf_result * @var{result})
These routines compute the irregular modified cylindrical Bessel
function of order @var{n}, @math{K_n(x)}, for @math{x > 0}.
@comment Domain: x > 0.0 
@comment Exceptional Return Values: GSL_EDOM, GSL_EOVRFLW, GSL_EUNDRFLW
@end deftypefun

@deftypefun int gsl_sf_bessel_Kn_array (int @var{nmin}, int @var{nmax}, double @var{x}, double @var{result_array}[])
This routine computes the values of the irregular modified cylindrical
Bessel functions @math{K_n(x)} for @math{n} from @var{nmin} to
@var{nmax} inclusive, storing the results in the array
@var{result_array}. The start of the range @var{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.
@comment Conditions: n=nmin,...,nmax 
@comment Domain: x > 0.0, nmin>=0, nmax >= nmin
@comment Exceptional Return Values: GSL_EDOM, GSL_EOVRFLW, GSL_EUNDRFLW
@end deftypefun

@deftypefun double gsl_sf_bessel_K0_scaled (double @var{x})
@deftypefunx int gsl_sf_bessel_K0_scaled_e (double @var{x}, gsl_sf_result * @var{result})
These routines compute the scaled irregular modified cylindrical Bessel
function of zeroth order @math{\exp(x) K_0(x)} for @math{x>0}.
@comment Domain: x > 0.0 
@comment Exceptional Return Values: GSL_EDOM
@end deftypefun

@deftypefun double gsl_sf_bessel_K1_scaled (double @var{x}) 
@deftypefunx int gsl_sf_bessel_K1_scaled_e (double @var{x}, gsl_sf_result * @var{result})
These routines compute the scaled irregular modified cylindrical Bessel
function of first order @math{\exp(x) K_1(x)} for @math{x>0}.
@comment Domain: x > 0.0 
@comment Exceptional Return Values: GSL_EDOM, GSL_EUNDRFLW
@end deftypefun

@deftypefun double gsl_sf_bessel_Kn_scaled (int @var{n}, double @var{x})
@deftypefunx int gsl_sf_bessel_Kn_scaled_e (int @var{n}, double @var{x}, gsl_sf_result * @var{result})
These routines compute the scaled irregular modified cylindrical Bessel
function of order @var{n}, @math{\exp(x) K_n(x)}, for @math{x>0}.
@comment Domain: x > 0.0 
@comment Exceptional Return Values: GSL_EDOM, GSL_EUNDRFLW
@end deftypefun

@deftypefun int gsl_sf_bessel_Kn_scaled_array (int @var{nmin}, int @var{nmax}, double @var{x}, double @var{result_array}[])
This routine computes the values of the scaled irregular cylindrical
Bessel functions @math{\exp(x) K_n(x)} for @math{n} from @var{nmin} to
@var{nmax} inclusive, storing the results in the array
@var{result_array}. The start of the range @var{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.
@comment Domain: x > 0.0, nmin >=0, nmax >= nmin 
@comment Conditions: n=nmin,...,nmax 
@comment Exceptional Return Values: GSL_EDOM, GSL_EUNDRFLW
@end deftypefun


@node Regular Spherical Bessel Functions
@subsection Regular Spherical Bessel Functions
@cindex Spherical Bessel Functions
@cindex Regular Spherical Bessel Functions

@deftypefun double gsl_sf_bessel_j0 (double @var{x})
@deftypefunx int gsl_sf_bessel_j0_e (double @var{x}, gsl_sf_result * @var{result})
These routines compute the regular spherical Bessel function of zeroth
order, @math{j_0(x) = \sin(x)/x}.
@comment Exceptional Return Values: none
@end deftypefun

@deftypefun double gsl_sf_bessel_j1 (double @var{x})
@deftypefunx int gsl_sf_bessel_j1_e (double @var{x}, gsl_sf_result * @var{result})
These routines compute the regular spherical Bessel function of first
order, @math{j_1(x) = (\sin(x)/x - \cos(x))/x}.
@comment Exceptional Return Values: GSL_EUNDRFLW
@end deftypefun

@deftypefun double gsl_sf_bessel_j2 (double @var{x})
@deftypefunx int gsl_sf_bessel_j2_e (double @var{x}, gsl_sf_result * @var{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}.
@comment Exceptional Return Values: GSL_EUNDRFLW
@end deftypefun

@deftypefun double gsl_sf_bessel_jl (int @var{l}, double @var{x})
@deftypefunx int gsl_sf_bessel_jl_e (int @var{l}, double @var{x}, gsl_sf_result * @var{result})
These routines compute the regular spherical Bessel function of 
order @var{l}, @math{j_l(x)}, for @c{$l \geq 0$}
@math{l >= 0} and @c{$x \geq 0$}
@math{x >= 0}.
@comment Domain: l >= 0, x >= 0.0 
@comment Exceptional Return Values: GSL_EDOM, GSL_EUNDRFLW
@end deftypefun

@deftypefun int gsl_sf_bessel_jl_array (int @var{lmax}, double @var{x}, double @var{result_array}[])
This routine computes the values of the regular spherical Bessel
functions @math{j_l(x)} for @math{l} from 0 to @var{lmax}
inclusive  for @c{$lmax \geq 0$}
@math{lmax >= 0} and @c{$x \geq 0$}
@math{x >= 0}, storing the results in the array @var{result_array}.
The values are computed using recurrence relations for
efficiency, and therefore may differ slightly from the exact values.
@comment Domain: lmax >= 0 
@comment Conditions: l=0,1,...,lmax 
@comment Exceptional Return Values: GSL_EDOM, GSL_EUNDRFLW
@end deftypefun

@deftypefun int gsl_sf_bessel_jl_steed_array (int @var{lmax}, double @var{x}, double * @var{jl_x_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
@var{lmax} inclusive for @c{$lmax \geq 0$}
@math{lmax >= 0} and @c{$x \geq 0$}
@math{x >= 0}, storing the results in the array
@var{result_array}.
The Steed/Barnett algorithm is described in @cite{Comp. Phys. Comm.} 21,
297 (1981).  Steed's method is more stable than the
recurrence used in the other functions but is also slower.
@comment Domain: lmax >= 0 
@comment Conditions: l=0,1,...,lmax 
@comment Exceptional Return Values: GSL_EDOM, GSL_EUNDRFLW
@end deftypefun


@node Irregular Spherical Bessel Functions
@subsection Irregular Spherical Bessel Functions
@cindex Irregular Spherical Bessel Functions

@deftypefun double gsl_sf_bessel_y0 (double @var{x})
@deftypefunx int gsl_sf_bessel_y0_e (double @var{x}, gsl_sf_result * @var{result})
These routines compute the irregular spherical Bessel function of zeroth
order, @math{y_0(x) = -\cos(x)/x}.
@comment Exceptional Return Values: none
@end deftypefun

@deftypefun double gsl_sf_bessel_y1 (double @var{x})
@deftypefunx int gsl_sf_bessel_y1_e (double @var{x}, gsl_sf_result * @var{result})
These routines compute the irregular spherical Bessel function of first
order, @math{y_1(x) = -(\cos(x)/x + \sin(x))/x}.
@comment Exceptional Return Values: GSL_EUNDRFLW
@end deftypefun

@deftypefun double gsl_sf_bessel_y2 (double @var{x})
@deftypefunx int gsl_sf_bessel_y2_e (double @var{x}, gsl_sf_result * @var{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)}.
@comment Exceptional Return Values: GSL_EUNDRFLW
@end deftypefun

@deftypefun double gsl_sf_bessel_yl (int @var{l}, double @var{x})
@deftypefunx int gsl_sf_bessel_yl_e (int @var{l}, double @var{x}, gsl_sf_result * @var{result})
These routines compute the irregular spherical Bessel function of 
order @var{l}, @math{y_l(x)}, for @c{$l \geq 0$}
@math{l >= 0}.
@comment Exceptional Return Values: GSL_EUNDRFLW
@end deftypefun

@deftypefun int gsl_sf_bessel_yl_array (int @var{lmax}, double @var{x}, double @var{result_array}[])
This routine computes the values of the irregular spherical Bessel
functions @math{y_l(x)} for @math{l} from 0 to @var{lmax}
inclusive  for @c{$lmax \geq 0$}
@math{lmax >= 0}, storing the results in the array @var{result_array}.
The values are computed using recurrence relations for
efficiency, and therefore may differ slightly from the exact values.
@comment Domain: lmax >= 0 
@comment Conditions: l=0,1,...,lmax 
@comment Exceptional Return Values: GSL_EUNDRFLW
@end deftypefun


@node Regular Modified Spherical Bessel Functions
@subsection Regular Modified Spherical Bessel Functions
@cindex Modified Spherical Bessel Functions
@cindex Regular Modified Spherical Bessel Functions

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

@deftypefun double gsl_sf_bessel_i0_scaled (double @var{x})
@deftypefunx int gsl_sf_bessel_i0_scaled_e (double @var{x}, gsl_sf_result * @var{result})
These routines compute the scaled regular modified spherical Bessel
function of zeroth order, @math{\exp(-|x|) i_0(x)}.
@comment Exceptional Return Values: none
@end deftypefun

@deftypefun double gsl_sf_bessel_i1_scaled (double @var{x})
@deftypefunx int gsl_sf_bessel_i1_scaled_e (double @var{x}, gsl_sf_result * @var{result})
These routines compute the scaled regular modified spherical Bessel
function of first order, @math{\exp(-|x|) i_1(x)}.
@comment Exceptional Return Values: GSL_EUNDRFLW
@end deftypefun

@deftypefun double gsl_sf_bessel_i2_scaled (double @var{x})
@deftypefunx int gsl_sf_bessel_i2_scaled_e (double @var{x}, gsl_sf_result * @var{result})
These routines compute the scaled regular modified spherical Bessel
function of second order, @math{ \exp(-|x|) i_2(x) } 
@comment Exceptional Return Values: GSL_EUNDRFLW
@end deftypefun

@deftypefun double gsl_sf_bessel_il_scaled (int @var{l}, double @var{x})
@deftypefunx int gsl_sf_bessel_il_scaled_e (int @var{l}, double @var{x}, gsl_sf_result * @var{result})
These routines compute the scaled regular modified spherical Bessel
function of order @var{l}, @math{ \exp(-|x|) i_l(x) }
@comment Domain: l >= 0 
@comment Exceptional Return Values: GSL_EDOM, GSL_EUNDRFLW
@end deftypefun

@deftypefun int gsl_sf_bessel_il_scaled_array (int @var{lmax}, double @var{x}, double @var{result_array}[])
This routine computes the values of the scaled regular modified
cylindrical Bessel functions @math{\exp(-|x|) i_l(x)} for @math{l} from
0 to @var{lmax} inclusive for @c{$lmax \geq 0$}
@math{lmax >= 0}, storing the results in
the array @var{result_array}. 
The values are computed using recurrence relations for
efficiency, and therefore may differ slightly from the exact values.
@comment Domain: lmax >= 0 
@comment Conditions: l=0,1,...,lmax 
@comment Exceptional Return Values: GSL_EUNDRFLW
@end deftypefun


@node Irregular Modified Spherical Bessel Functions
@subsection Irregular Modified Spherical Bessel Functions
@cindex Irregular Modified Spherical Bessel Functions

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

@deftypefun double gsl_sf_bessel_k0_scaled (double @var{x})
@deftypefunx int gsl_sf_bessel_k0_scaled_e (double @var{x}, gsl_sf_result * @var{result})
These routines compute the scaled irregular modified spherical Bessel
function of zeroth order, @math{\exp(x) k_0(x)}, for @math{x>0}.
@comment Domain: x > 0.0 
@comment Exceptional Return Values: GSL_EDOM, GSL_EUNDRFLW
@end deftypefun

@deftypefun double gsl_sf_bessel_k1_scaled (double @var{x})
@deftypefunx int gsl_sf_bessel_k1_scaled_e (double @var{x}, gsl_sf_result * @var{result})
These routines compute the scaled irregular modified spherical Bessel
function of first order, @math{\exp(x) k_1(x)}, for @math{x>0}.
@comment Domain: x > 0.0 
@comment Exceptional Return Values: GSL_EDOM, GSL_EUNDRFLW, GSL_EOVRFLW
@end deftypefun

@deftypefun double gsl_sf_bessel_k2_scaled (double @var{x})
@deftypefunx int gsl_sf_bessel_k2_scaled_e (double @var{x}, gsl_sf_result * @var{result})
These routines compute the scaled irregular modified spherical Bessel
function of second order, @math{\exp(x) k_2(x)}, for @math{x>0}.
@comment Domain: x > 0.0 
@comment Exceptional Return Values: GSL_EDOM, GSL_EUNDRFLW, GSL_EOVRFLW
@end deftypefun

@deftypefun double gsl_sf_bessel_kl_scaled (int @var{l}, double @var{x})
@deftypefunx int gsl_sf_bessel_kl_scaled_e (int @var{l}, double @var{x}, gsl_sf_result * @var{result})
These routines compute the scaled irregular modified spherical Bessel
function of order @var{l}, @math{\exp(x) k_l(x)}, for @math{x>0}.
@comment Domain: x > 0.0 
@comment Exceptional Return Values: GSL_EDOM, GSL_EUNDRFLW
@end deftypefun

@deftypefun int gsl_sf_bessel_kl_scaled_array (int @var{lmax}, double @var{x}, double @var{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 @var{lmax} inclusive for @c{$lmax \geq 0$}
@math{lmax >= 0} and @math{x>0}, storing the results in
the array @var{result_array}. 
The values are computed using recurrence relations for
efficiency, and therefore may differ slightly from the exact values.
@comment Domain: lmax >= 0 
@comment Conditions: l=0,1,...,lmax 
@comment Exceptional Return Values: GSL_EDOM, GSL_EUNDRFLW
@end deftypefun


@node Regular Bessel Function - Fractional Order
@subsection Regular Bessel Function---Fractional Order
@cindex Fractional Order Bessel Functions
@cindex Bessel Functions, Fractional Order
@cindex Regular Bessel Functions, Fractional Order

@deftypefun double gsl_sf_bessel_Jnu (double @var{nu}, double @var{x})
@deftypefunx int gsl_sf_bessel_Jnu_e (double @var{nu}, double @var{x}, gsl_sf_result * @var{result})
These routines compute the regular cylindrical Bessel function of
fractional order @math{\nu}, @math{J_\nu(x)}.
@comment Exceptional Return Values: GSL_EDOM, GSL_EUNDRFLW
@end deftypefun

@deftypefun int gsl_sf_bessel_sequence_Jnu_e (double @var{nu}, gsl_mode_t @var{mode}, size_t @var{size}, double @var{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 @var{v} of length @var{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)}.
@comment Exceptional Return Values: GSL_EDOM, GSL_EINVAL
@end deftypefun


@node Irregular Bessel Functions - Fractional Order
@subsection Irregular Bessel Functions---Fractional Order

@deftypefun double gsl_sf_bessel_Ynu (double @var{nu}, double @var{x})
@deftypefunx int gsl_sf_bessel_Ynu_e (double @var{nu}, double @var{x}, gsl_sf_result * @var{result})
These routines compute the irregular cylindrical Bessel function of
fractional order @math{\nu}, @math{Y_\nu(x)}.
@comment Exceptional Return Values: 
@end deftypefun


@node Regular Modified Bessel Functions - Fractional Order
@subsection Regular Modified Bessel Functions---Fractional Order
@cindex Modified Bessel Functions, Fractional Order
@cindex Regular Modified Bessel Functions, Fractional Order

@deftypefun double gsl_sf_bessel_Inu (double @var{nu}, double @var{x})
@deftypefunx int gsl_sf_bessel_Inu_e (double @var{nu}, double @var{x}, gsl_sf_result * @var{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}.
@comment Domain: x >= 0, nu >= 0 
@comment Exceptional Return Values: GSL_EDOM, GSL_EOVRFLW
@end deftypefun

@deftypefun double gsl_sf_bessel_Inu_scaled (double @var{nu}, double @var{x})
@deftypefunx int gsl_sf_bessel_Inu_scaled_e (double @var{nu}, double @var{x}, gsl_sf_result * @var{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}.
@comment @math{ \exp(-|x|) I_@{\nu@}(x) } 
@comment Domain: x >= 0, nu >= 0 
@comment Exceptional Return Values: GSL_EDOM
@end deftypefun


@node Irregular Modified Bessel Functions - Fractional Order
@subsection Irregular Modified Bessel Functions---Fractional Order
@cindex Irregular Modified Bessel Functions, Fractional Order

@deftypefun double gsl_sf_bessel_Knu (double @var{nu}, double @var{x})
@deftypefunx int gsl_sf_bessel_Knu_e (double @var{nu}, double @var{x}, gsl_sf_result * @var{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}.
@comment Domain: x > 0, nu >= 0 
@comment Exceptional Return Values: GSL_EDOM, GSL_EUNDRFLW
@end deftypefun

@deftypefun double gsl_sf_bessel_lnKnu (double @var{nu}, double @var{x})
@deftypefunx int gsl_sf_bessel_lnKnu_e (double @var{nu}, double @var{x}, gsl_sf_result * @var{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}. 
@comment Domain: x > 0, nu >= 0 
@comment Exceptional Return Values: GSL_EDOM
@end deftypefun

@deftypefun double gsl_sf_bessel_Knu_scaled (double @var{nu}, double @var{x})
@deftypefunx int gsl_sf_bessel_Knu_scaled_e (double @var{nu}, double @var{x}, gsl_sf_result * @var{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}.
@comment Domain: x > 0, nu >= 0 
@comment Exceptional Return Values: GSL_EDOM
@end deftypefun

@node Zeros of Regular Bessel Functions
@subsection Zeros of Regular Bessel Functions
@cindex Zeros of Regular Bessel Functions
@cindex Regular Bessel Functions, Zeros of 

@deftypefun double gsl_sf_bessel_zero_J0 (unsigned int @var{s})
@deftypefunx int gsl_sf_bessel_zero_J0_e (unsigned int @var{s}, gsl_sf_result * @var{result})
These routines compute the location of the @var{s}-th positive zero of
the Bessel function @math{J_0(x)}.
@comment Exceptional Return Values: 
@end deftypefun

@deftypefun double gsl_sf_bessel_zero_J1 (unsigned int @var{s})
@deftypefunx int gsl_sf_bessel_zero_J1_e (unsigned int @var{s}, gsl_sf_result * @var{result})
These routines compute the location of the @var{s}-th positive zero of
the Bessel function @math{J_1(x)}.
@comment Exceptional Return Values: 
@end deftypefun

@deftypefun double gsl_sf_bessel_zero_Jnu (double @var{nu}, unsigned int @var{s})
@deftypefunx int gsl_sf_bessel_zero_Jnu_e (double @var{nu}, unsigned int @var{s}, gsl_sf_result * @var{result})
These routines compute the location of the @var{s}-th positive zero of
the Bessel function @math{J_\nu(x)}.  The current implementation does not
support negative values of @var{nu}. 
@comment Exceptional Return Values: 
@end deftypefun