| @cindex Airy functions |
| @cindex Ai(x) |
| @cindex Bi(x) |
| |
| The Airy functions @math{Ai(x)} and @math{Bi(x)} are defined by the |
| integral representations, |
| @tex |
| \beforedisplay |
| $$ |
| \eqalign{ |
| Ai(x) & = {1\over\pi} \int_0^\infty \cos(t^3/3 + xt ) \,dt, \cr |
| Bi(x) & = {1\over\pi} \int_0^\infty (e^{-t^3/3} + \sin(t^3/3 + xt)) \,dt. |
| } |
| $$ |
| \afterdisplay |
| @end tex |
| @ifinfo |
| |
| @example |
| Ai(x) = (1/\pi) \int_0^\infty \cos((1/3) t^3 + xt) dt |
| Bi(x) = (1/\pi) \int_0^\infty (e^(-(1/3) t^3) + \sin((1/3) t^3 + xt)) dt |
| @end example |
| |
| @end ifinfo |
| @noindent |
| For further information see Abramowitz & Stegun, Section 10.4. The Airy |
| functions are defined in the header file @file{gsl_sf_airy.h}. |
| |
| @menu |
| * Airy Functions:: |
| * Derivatives of Airy Functions:: |
| * Zeros of Airy Functions:: |
| * Zeros of Derivatives of Airy Functions:: |
| @end menu |
| |
| @node Airy Functions |
| @subsection Airy Functions |
| |
| @deftypefun double gsl_sf_airy_Ai (double @var{x}, gsl_mode_t @var{mode}) |
| @deftypefunx int gsl_sf_airy_Ai_e (double @var{x}, gsl_mode_t @var{mode}, gsl_sf_result * @var{result}) |
| These routines compute the Airy function @math{Ai(x)} with an accuracy |
| specified by @var{mode}. |
| @end deftypefun |
| |
| @deftypefun double gsl_sf_airy_Bi (double @var{x}, gsl_mode_t @var{mode}) |
| @deftypefunx int gsl_sf_airy_Bi_e (double @var{x}, gsl_mode_t @var{mode}, gsl_sf_result * @var{result}) |
| These routines compute the Airy function @math{Bi(x)} with an accuracy |
| specified by @var{mode}. |
| @end deftypefun |
| |
| @deftypefun double gsl_sf_airy_Ai_scaled (double @var{x}, gsl_mode_t @var{mode}) |
| @deftypefunx int gsl_sf_airy_Ai_scaled_e (double @var{x}, gsl_mode_t @var{mode}, gsl_sf_result * @var{result}) |
| These routines compute a scaled version of the Airy function |
| @math{S_A(x) Ai(x)}. For @math{x>0} the scaling factor @math{S_A(x)} is @c{$\exp(+(2/3) x^{3/2})$} |
| @math{\exp(+(2/3) x^(3/2))}, |
| and is 1 |
| for @math{x<0}. |
| @end deftypefun |
| |
| @deftypefun double gsl_sf_airy_Bi_scaled (double @var{x}, gsl_mode_t @var{mode}) |
| @deftypefunx int gsl_sf_airy_Bi_scaled_e (double @var{x}, gsl_mode_t @var{mode}, gsl_sf_result * @var{result}) |
| These routines compute a scaled version of the Airy function |
| @math{S_B(x) Bi(x)}. For @math{x>0} the scaling factor @math{S_B(x)} is @c{$\exp(-(2/3) x^{3/2})$} |
| @math{exp(-(2/3) x^(3/2))}, and is 1 for @math{x<0}. |
| @end deftypefun |
| |
| |
| @node Derivatives of Airy Functions |
| @subsection Derivatives of Airy Functions |
| |
| @deftypefun double gsl_sf_airy_Ai_deriv (double @var{x}, gsl_mode_t @var{mode}) |
| @deftypefunx int gsl_sf_airy_Ai_deriv_e (double @var{x}, gsl_mode_t @var{mode}, gsl_sf_result * @var{result}) |
| These routines compute the Airy function derivative @math{Ai'(x)} with |
| an accuracy specified by @var{mode}. |
| @end deftypefun |
| |
| @deftypefun double gsl_sf_airy_Bi_deriv (double @var{x}, gsl_mode_t @var{mode}) |
| @deftypefunx int gsl_sf_airy_Bi_deriv_e (double @var{x}, gsl_mode_t @var{mode}, gsl_sf_result * @var{result}) |
| These routines compute the Airy function derivative @math{Bi'(x)} with |
| an accuracy specified by @var{mode}. |
| @end deftypefun |
| |
| @deftypefun double gsl_sf_airy_Ai_deriv_scaled (double @var{x}, gsl_mode_t @var{mode}) |
| @deftypefunx int gsl_sf_airy_Ai_deriv_scaled_e (double @var{x}, gsl_mode_t @var{mode}, gsl_sf_result * @var{result}) |
| These routines compute the scaled Airy function derivative |
| @math{S_A(x) Ai'(x)}. |
| For @math{x>0} the scaling factor @math{S_A(x)} is @c{$\exp(+(2/3) x^{3/2})$} |
| @math{\exp(+(2/3) x^(3/2))}, and is 1 for @math{x<0}. |
| @end deftypefun |
| |
| @deftypefun double gsl_sf_airy_Bi_deriv_scaled (double @var{x}, gsl_mode_t @var{mode}) |
| @deftypefunx int gsl_sf_airy_Bi_deriv_scaled_e (double @var{x}, gsl_mode_t @var{mode}, gsl_sf_result * @var{result}) |
| These routines compute the scaled Airy function derivative |
| @math{S_B(x) Bi'(x)}. |
| For @math{x>0} the scaling factor @math{S_B(x)} is @c{$\exp(-(2/3) x^{3/2})$} |
| @math{exp(-(2/3) x^(3/2))}, and is 1 for @math{x<0}. |
| @end deftypefun |
| |
| |
| @node Zeros of Airy Functions |
| @subsection Zeros of Airy Functions |
| |
| @deftypefun double gsl_sf_airy_zero_Ai (unsigned int @var{s}) |
| @deftypefunx int gsl_sf_airy_zero_Ai_e (unsigned int @var{s}, gsl_sf_result * @var{result}) |
| These routines compute the location of the @var{s}-th zero of the Airy |
| function @math{Ai(x)}. |
| @end deftypefun |
| |
| @deftypefun double gsl_sf_airy_zero_Bi (unsigned int @var{s}) |
| @deftypefunx int gsl_sf_airy_zero_Bi_e (unsigned int @var{s}, gsl_sf_result * @var{result}) |
| These routines compute the location of the @var{s}-th zero of the Airy |
| function @math{Bi(x)}. |
| @end deftypefun |
| |
| |
| @node Zeros of Derivatives of Airy Functions |
| @subsection Zeros of Derivatives of Airy Functions |
| |
| @deftypefun double gsl_sf_airy_zero_Ai_deriv (unsigned int @var{s}) |
| @deftypefunx int gsl_sf_airy_zero_Ai_deriv_e (unsigned int @var{s}, gsl_sf_result * @var{result}) |
| These routines compute the location of the @var{s}-th zero of the Airy |
| function derivative @math{Ai'(x)}. |
| @end deftypefun |
| |
| @deftypefun double gsl_sf_airy_zero_Bi_deriv (unsigned int @var{s}) |
| @deftypefunx int gsl_sf_airy_zero_Bi_deriv_e (unsigned int @var{s}, gsl_sf_result * @var{result}) |
| These routines compute the location of the @var{s}-th zero of the Airy |
| function derivative @math{Bi'(x)}. |
| @end deftypefun |
| |