| @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 |
| |
