specfunc-airy.texi

The GNU Scientific Library (GSL) is a numerical library for C and C++ programmers.

@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
$$
\eqalign{
Ai(x) & = {1\over\pi} \int_0^\infty \cos( {1\over 3} t^3 + xt ) dt, \cr
Bi(x) & = {1\over\pi} \int_0^\infty (e^{-t^3/3} + \sin({1\over 3} t^3 + xt)) dt.
}
$$
\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 derivative of the scaled Airy
function @math{S_A(x) Ai(x)}.
@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 derivative of the scaled Airy function
@math{S_B(x) Bi(x)}.
@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