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diff --git a/gsl-1.9/doc/specfunc-airy.texi b/gsl-1.9/doc/specfunc-airy.texi new file mode 100644 index 0000000..726f516 --- /dev/null +++ b/gsl-1.9/doc/specfunc-airy.texi @@ -0,0 +1,130 @@ +@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 + |