The Airy functions Ai(x) and Bi(x) are defined by the integral representations,

     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

For further information see Abramowitz & Stegun, Section 10.4. The Airy functions are defined in the header file gsl_sf_airy.h.

Airy Functions
--------------

 -- Function: double gsl_sf_airy_Ai (double X, gsl_mode_t MODE)
 -- Function: int gsl_sf_airy_Ai_e (double X, gsl_mode_t MODE, gsl_sf_result * RESULT)
     These routines compute the Airy function Ai(x) with an accuracy specified by MODE.

 -- Function: double gsl_sf_airy_Bi (double X, gsl_mode_t MODE)
 -- Function: int gsl_sf_airy_Bi_e (double X, gsl_mode_t MODE, gsl_sf_result * RESULT)
     These routines compute the Airy function Bi(x) with an accuracy specified by MODE.

 -- Function: double gsl_sf_airy_Ai_scaled (double X, gsl_mode_t MODE)
 -- Function: int gsl_sf_airy_Ai_scaled_e (double X, gsl_mode_t MODE, gsl_sf_result * RESULT)
     These routines compute a scaled version of the Airy function S_A(x) Ai(x).  For x>0 the scaling factor S_A(x) is $\exp(+(2/3) x^3/2)$ \exp(+(2/3) x^(3/2)), and is 1 for x<0.

 -- Function: double gsl_sf_airy_Bi_scaled (double X, gsl_mode_t MODE)
 -- Function: int gsl_sf_airy_Bi_scaled_e (double X, gsl_mode_t MODE, gsl_sf_result * RESULT)
     These routines compute a scaled version of the Airy function S_B(x) Bi(x).  For x>0 the scaling factor S_B(x) is $\exp(-(2/3) x^3/2)$ exp(-(2/3) x^(3/2)), and is 1 for x<0.

Derivatives of Airy Functions
-----------------------------

 -- Function: double gsl_sf_airy_Ai_deriv (double X, gsl_mode_t MODE)
 -- Function: int gsl_sf_airy_Ai_deriv_e (double X, gsl_mode_t MODE, gsl_sf_result * RESULT)
     These routines compute the Airy function derivative Ai'(x) with an accuracy specified by MODE.

 -- Function: double gsl_sf_airy_Bi_deriv (double X, gsl_mode_t MODE)
 -- Function: int gsl_sf_airy_Bi_deriv_e (double X, gsl_mode_t MODE, gsl_sf_result * RESULT)
     These routines compute the Airy function derivative Bi'(x) with an accuracy specified by MODE.

 -- Function: double gsl_sf_airy_Ai_deriv_scaled (double X, gsl_mode_t MODE)
 -- Function: int gsl_sf_airy_Ai_deriv_scaled_e (double X, gsl_mode_t MODE, gsl_sf_result * RESULT)
     These routines compute the scaled Airy function derivative S_A(x) Ai'(x). For x>0 the scaling factor S_A(x) is $\exp(+(2/3) x^3/2)$ \exp(+(2/3) x^(3/2)), and is 1 for x<0.

 -- Function: double gsl_sf_airy_Bi_deriv_scaled (double X, gsl_mode_t MODE)
 -- Function: int gsl_sf_airy_Bi_deriv_scaled_e (double X, gsl_mode_t MODE, gsl_sf_result * RESULT)
     These routines compute the scaled Airy function derivative S_B(x) Bi'(x). For x>0 the scaling factor S_B(x) is $\exp(-(2/3) x^3/2)$ exp(-(2/3) x^(3/2)), and is 1 for x<0.

Zeros of Airy Functions
-----------------------

 -- Function: double gsl_sf_airy_zero_Ai (unsigned int S)
 -- Function: int gsl_sf_airy_zero_Ai_e (unsigned int S, gsl_sf_result * RESULT)
     These routines compute the location of the S-th zero of the Airy function Ai(x).

 -- Function: double gsl_sf_airy_zero_Bi (unsigned int S)
 -- Function: int gsl_sf_airy_zero_Bi_e (unsigned int S, gsl_sf_result * RESULT)
     These routines compute the location of the S-th zero of the Airy function Bi(x).

Zeros of Derivatives of Airy Functions
--------------------------------------

 -- Function: double gsl_sf_airy_zero_Ai_deriv (unsigned int S)
 -- Function: int gsl_sf_airy_zero_Ai_deriv_e (unsigned int S, gsl_sf_result * RESULT)
     These routines compute the location of the S-th zero of the Airy function derivative Ai'(x).

 -- Function: double gsl_sf_airy_zero_Bi_deriv (unsigned int S)
 -- Function: int gsl_sf_airy_zero_Bi_deriv_e (unsigned int S, gsl_sf_result * RESULT)
     These routines compute the location of the S-th zero of the Airy function derivative Bi'(x).