1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
|
@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
$$
\eqalign{
Ai(x) & = {1\over\pi} \int_0^\infty \cos(t^3/3 + xt ) \,dt, \cr
Bi(x) & = {1\over\pi} \int_0^\infty (e^{-t^3/3 + xt} + \sin(t^3/3 + xt)) \,dt.
}
$$
\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 + xt) + \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 scaled Airy function derivative
@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_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 scaled Airy function derivative
@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 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
|
