.. index:: Airy functions

.. index:: Ai(x)

.. index:: Bi(x)

The Airy functions :math:`Ai(x)` and :math:`Bi(x)` are defined by the

integral representations,

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   .. math::

      Ai(x) & = {1\over\pi} \int_0^\infty \cos(t^3/3 + xt ) \,dt \\

      Bi(x) & = {1\over\pi} \int_0^\infty (e^{-t^3/3 + xt} + \sin(t^3/3 + xt)) \,dt

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   | 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 + xt) + \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 :file:`gsl_sf_airy.h`.

Airy Functions

--------------

.. function:: double gsl_sf_airy_Ai (double x, gsl_mode_t mode)

              int gsl_sf_airy_Ai_e (double x, gsl_mode_t mode, gsl_sf_result * result)

   These routines compute the Airy function :math:`Ai(x)` with an accuracy

   specified by :data:`mode`.

.. function:: double gsl_sf_airy_Bi (double x, gsl_mode_t mode)

              int gsl_sf_airy_Bi_e (double x, gsl_mode_t mode, gsl_sf_result * result)

   These routines compute the Airy function :math:`Bi(x)` with an accuracy

   specified by :data:`mode`.

.. function:: double gsl_sf_airy_Ai_scaled (double x, gsl_mode_t mode)

              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

   :math:`S_A(x) Ai(x)`.  For :math:`x > 0` the scaling factor :math:`S_A(x)` is

   :math:`\exp(+(2/3) x^{3/2})`, 

   and is 1 for :math:`x < 0`.

.. function:: double gsl_sf_airy_Bi_scaled (double x, gsl_mode_t mode)

              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

   :math:`S_B(x) Bi(x)`.  For :math:`x > 0` the scaling factor :math:`S_B(x)` is

   :math:`exp(-(2/3) x^{3/2})`, and is 1 for :math:`x < 0`.

Derivatives of Airy Functions

-----------------------------

.. function:: double gsl_sf_airy_Ai_deriv (double x, gsl_mode_t mode)

              int gsl_sf_airy_Ai_deriv_e (double x, gsl_mode_t mode, gsl_sf_result * result)

   These routines compute the Airy function derivative :math:`Ai'(x)` with

   an accuracy specified by :data:`mode`.

.. function:: double gsl_sf_airy_Bi_deriv (double x, gsl_mode_t mode)

              int gsl_sf_airy_Bi_deriv_e (double x, gsl_mode_t mode, gsl_sf_result * result)

   These routines compute the Airy function derivative :math:`Bi'(x)` with

   an accuracy specified by :data:`mode`.

.. function:: double gsl_sf_airy_Ai_deriv_scaled (double x, gsl_mode_t mode)

              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 

   :math:`S_A(x) Ai'(x)`.  

   For :math:`x > 0` the scaling factor :math:`S_A(x)` is

   :math:`\exp(+(2/3) x^{3/2})`, and is 1 for :math:`x < 0`.

.. function:: double gsl_sf_airy_Bi_deriv_scaled (double x, gsl_mode_t mode)

              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 

   :math:`S_B(x) Bi'(x)`.

   For :math:`x > 0` the scaling factor :math:`S_B(x)` is

   :math:`exp(-(2/3) x^{3/2})`, and is 1 for :math:`x < 0`.

Zeros of Airy Functions

-----------------------

.. function:: double gsl_sf_airy_zero_Ai (unsigned int s)

              int gsl_sf_airy_zero_Ai_e (unsigned int s, gsl_sf_result * result)

   These routines compute the location of the :data:`s`-th zero of the Airy

   function :math:`Ai(x)`.

.. function:: double gsl_sf_airy_zero_Bi (unsigned int s)

              int gsl_sf_airy_zero_Bi_e (unsigned int s, gsl_sf_result * result)

   These routines compute the location of the :data:`s`-th zero of the Airy

   function :math:`Bi(x)`.

Zeros of Derivatives of Airy Functions

--------------------------------------

.. function:: double gsl_sf_airy_zero_Ai_deriv (unsigned int s)

              int gsl_sf_airy_zero_Ai_deriv_e (unsigned int s, gsl_sf_result * result)

   These routines compute the location of the :data:`s`-th zero of the Airy

   function derivative :math:`Ai'(x)`.

.. function:: double gsl_sf_airy_zero_Bi_deriv (unsigned int s)

              int gsl_sf_airy_zero_Bi_deriv_e (unsigned int s, gsl_sf_result * result)

   These routines compute the location of the :data:`s`-th zero of the Airy

   function derivative :math:`Bi'(x)`.