


Airy Functions and Derivatives
==============================

The Airy functions :math:`\operatorname{Ai}(x)` and :math:`\operatorname{Bi}(x)` are
defined by the integral representations,

.. math::
  \operatorname{Ai}(x) = \frac{1}{\pi} \int_0^\infty
    \cos(\frac{1}{3} t^3 + xt) dt \\
  \operatorname{Bi}(x) = \frac{1}{\pi} \int_0^\infty
    (e^{-\frac{1}{3} t^3 + xt} + \sin(\frac{1}{3} t^3 + xt)) dt

.. index:: Airy function

For further information see Abramowitz & Stegun, Section 10.4.

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

.. function:: gsl_sf_airy_Ai(x)

 This routine computes the Airy function :math:`\operatorname{Ai}(x)`.

.. function:: gsl_sf_airy_Bi(x)

 This routine computes the Airy function :math:`\operatorname{Bi}(x)`.

.. function:: gsl_sf_airy_Ai_scaled(x)

 This routine computes a scaled version of the Airy function
 :math:`\operatorname{S_A}(x) \operatorname{Ai}(x)`. For :math:`x > 0` the scaling
 factor :math:`\operatorname{S_A}(x)` is :math:`\exp(+(2/3) x^{3/2})`, and is :math:`1`
 for :math:`x < 0`.

.. function:: gsl_sf_airy_Bi_scaled(x)

 This routine computes a scaled version of the Airy function
 :math:`\operatorname{S_B}(x) \operatorname{Bi}(x)`. For :math:`x > 0` the scaling
 factor :math:`\operatorname{S_B}(x)` is :math:`\exp(-(2/3) x^{3/2})`, and is :math:`1`
 for :math:`x < 0`.

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

.. function:: gsl_sf_airy_zero_Ai(s)

 This routine computes the location of the :math:`s`-th zero of the Airy
 function :math:`\operatorname{Ai}(x)`.

.. function:: gsl_sf_airy_zero_Bi(s)

 This routine computes the location of the :math:`s`-th zero of the Airy
 function :math:`\operatorname{Bi}(x)`.

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

.. function:: gsl_sf_airy_zero_Ai_deriv(s)

 This routine computes the location of the :math:`s`-th zero of the Airy
 function derivative :math:`\operatorname{Ai}'(x)`.

.. function:: gsl_sf_airy_zero_Bi_deriv(s)

 This routine computes the location of the :math:`s`-th zero of the Airy
 function derivative :math:`\operatorname{Bi}'(x)`.
