1.. index:: Airy functions
2.. index:: Ai(x)
3.. index:: Bi(x)
4
5The Airy functions :math:`Ai(x)` and :math:`Bi(x)` are defined by the
6integral representations,
7
8.. only:: not texinfo
9
10   .. math::
11
12      Ai(x) & = {1\over\pi} \int_0^\infty \cos(t^3/3 + xt ) \,dt \\
13      Bi(x) & = {1\over\pi} \int_0^\infty (e^{-t^3/3 + xt} + \sin(t^3/3 + xt)) \,dt
14
15.. only:: texinfo
16
17   | Ai(x) = (1/\pi) \int_0^\infty \cos((1/3) t^3 + xt) dt
18   | Bi(x) = (1/\pi) \int_0^\infty (e^(-(1/3) t^3 + xt) + \sin((1/3) t^3 + xt)) dt
19
20For further information see Abramowitz & Stegun, Section 10.4. The Airy
21functions are defined in the header file :file:`gsl_sf_airy.h`.
22
23Airy Functions
24--------------
25
26.. function:: double gsl_sf_airy_Ai (double x, gsl_mode_t mode)
27              int gsl_sf_airy_Ai_e (double x, gsl_mode_t mode, gsl_sf_result * result)
28
29   These routines compute the Airy function :math:`Ai(x)` with an accuracy
30   specified by :data:`mode`.
31
32.. function:: double gsl_sf_airy_Bi (double x, gsl_mode_t mode)
33              int gsl_sf_airy_Bi_e (double x, gsl_mode_t mode, gsl_sf_result * result)
34
35   These routines compute the Airy function :math:`Bi(x)` with an accuracy
36   specified by :data:`mode`.
37
38.. function:: double gsl_sf_airy_Ai_scaled (double x, gsl_mode_t mode)
39              int gsl_sf_airy_Ai_scaled_e (double x, gsl_mode_t mode, gsl_sf_result * result)
40
41   These routines compute a scaled version of the Airy function
42   :math:`S_A(x) Ai(x)`.  For :math:`x > 0` the scaling factor :math:`S_A(x)` is
43   :math:`\exp(+(2/3) x^{3/2})`,
44   and is 1 for :math:`x < 0`.
45
46.. function:: double gsl_sf_airy_Bi_scaled (double x, gsl_mode_t mode)
47              int gsl_sf_airy_Bi_scaled_e (double x, gsl_mode_t mode, gsl_sf_result * result)
48
49   These routines compute a scaled version of the Airy function
50   :math:`S_B(x) Bi(x)`.  For :math:`x > 0` the scaling factor :math:`S_B(x)` is
51   :math:`exp(-(2/3) x^{3/2})`, and is 1 for :math:`x < 0`.
52
53
54Derivatives of Airy Functions
55-----------------------------
56
57.. function:: double gsl_sf_airy_Ai_deriv (double x, gsl_mode_t mode)
58              int gsl_sf_airy_Ai_deriv_e (double x, gsl_mode_t mode, gsl_sf_result * result)
59
60   These routines compute the Airy function derivative :math:`Ai'(x)` with
61   an accuracy specified by :data:`mode`.
62
63.. function:: double gsl_sf_airy_Bi_deriv (double x, gsl_mode_t mode)
64              int gsl_sf_airy_Bi_deriv_e (double x, gsl_mode_t mode, gsl_sf_result * result)
65
66   These routines compute the Airy function derivative :math:`Bi'(x)` with
67   an accuracy specified by :data:`mode`.
68
69.. function:: double gsl_sf_airy_Ai_deriv_scaled (double x, gsl_mode_t mode)
70              int gsl_sf_airy_Ai_deriv_scaled_e (double x, gsl_mode_t mode, gsl_sf_result * result)
71
72   These routines compute the scaled Airy function derivative
73   :math:`S_A(x) Ai'(x)`.
74   For :math:`x > 0` the scaling factor :math:`S_A(x)` is
75   :math:`\exp(+(2/3) x^{3/2})`, and is 1 for :math:`x < 0`.
76
77.. function:: double gsl_sf_airy_Bi_deriv_scaled (double x, gsl_mode_t mode)
78              int gsl_sf_airy_Bi_deriv_scaled_e (double x, gsl_mode_t mode, gsl_sf_result * result)
79
80   These routines compute the scaled Airy function derivative
81   :math:`S_B(x) Bi'(x)`.
82   For :math:`x > 0` the scaling factor :math:`S_B(x)` is
83   :math:`exp(-(2/3) x^{3/2})`, and is 1 for :math:`x < 0`.
84
85Zeros of Airy Functions
86-----------------------
87
88.. function:: double gsl_sf_airy_zero_Ai (unsigned int s)
89              int gsl_sf_airy_zero_Ai_e (unsigned int s, gsl_sf_result * result)
90
91   These routines compute the location of the :data:`s`-th zero of the Airy
92   function :math:`Ai(x)`.
93
94.. function:: double gsl_sf_airy_zero_Bi (unsigned int s)
95              int gsl_sf_airy_zero_Bi_e (unsigned int s, gsl_sf_result * result)
96
97   These routines compute the location of the :data:`s`-th zero of the Airy
98   function :math:`Bi(x)`.
99
100Zeros of Derivatives of Airy Functions
101--------------------------------------
102
103.. function:: double gsl_sf_airy_zero_Ai_deriv (unsigned int s)
104              int gsl_sf_airy_zero_Ai_deriv_e (unsigned int s, gsl_sf_result * result)
105
106   These routines compute the location of the :data:`s`-th zero of the Airy
107   function derivative :math:`Ai'(x)`.
108
109.. function:: double gsl_sf_airy_zero_Bi_deriv (unsigned int s)
110              int gsl_sf_airy_zero_Bi_deriv_e (unsigned int s, gsl_sf_result * result)
111
112   These routines compute the location of the :data:`s`-th zero of the Airy
113   function derivative :math:`Bi'(x)`.
114