@cindex Bessel functionsThe routines described in this section compute the Cylindrical Besselfunctions @math{J_n(x)}, @math{Y_n(x)}, Modified cylindrical Besselfunctions @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 zerothorder, @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 firstorder, @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 Besselfunctions @math{J_n(x)} for @math{n} from @var{nmin} to @var{nmax}inclusive, storing the results in the array @var{result_array}.  Thevalues are computed using recurrence relations, for efficiency, andtherefore 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 zerothorder, @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 firstorder, @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 Besselfunctions @math{Y_n(x)} for @math{n} from @var{nmin} to @var{nmax}inclusive, storing the results in the array @var{result_array}.  Thedomain of the function is @math{x>0}.  The values are computed usingrecurrence relations, for efficiency, and therefore may differ slightlyfrom 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 functionof 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 functionof 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 functionof 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 cylindricalBessel 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 positiveor zero.  The values are computed using recurrence relations, forefficiency, 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 Besselfunction 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 Besselfunction 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 Besselfunction 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 cylindricalBessel 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 positiveor zero.  The values are computed using recurrence relations, forefficiency, 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 Besselfunction 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 Besselfunction 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 Besselfunction 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 cylindricalBessel 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 positiveor zero. The domain of the function is @math{x>0}. The values arecomputed using recurrence relations, for efficiency, and thereforemay 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 Besselfunction 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 Besselfunction 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 Besselfunction 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 cylindricalBessel 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 positiveor zero.  The domain of the function is @math{x>0}. The values arecomputed using recurrence relations, for efficiency, and thereforemay 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 zerothorder, @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 firstorder, @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 secondorder, @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 Besselfunctions @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, forefficiency, and therefore may differ slightly from the exact values.