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diff --git a/gsl-1.9/doc/specfunc-bessel.texi b/gsl-1.9/doc/specfunc-bessel.texi new file mode 100644 index 0000000..e3b09c9 --- /dev/null +++ b/gsl-1.9/doc/specfunc-bessel.texi @@ -0,0 +1,585 @@ +@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 + |