
30.8. Airy Functions 1529
30.8 Airy Functions
30.8.1 Definitio n and Basic Formulas
◮ Airy functions of the first and th e second kinds.
The Airy function of the first kind , Ai(x), and the Airy function of the second kind, Bi(x),
are solutions of the Airy equation
y
′′
xx
− xy = 0
and are defined by the formulas
Ai(x) =
1
π
Z
∞
0
cos
1
3
t
3
+ xt
dt,
Bi(x) =
1
π
Z
∞
0
exp
−
1
3
t
3
+ xt
+ sin
1
3
t
3
+ xt
dt.
Wronskian: W {Ai(x), Bi(x)} = 1/π.
◮ Relation to the Bessel functions and the modified Bessel functions.
Ai(x) =
1
3
√
x
I
−1/3
(z) − I
1/3
(z)
= π
−1
q
1
3
x K
1/3
(z), z =
2
3
x
3/2
,
Ai(−x) =
1
3
√
x
J
−1/3
(z) + J
1/3
(z)
,
Bi(x) =
q
1
3
x
I
−1/3
(z) + I
1/3
(z)
,
Bi(−x) =
q
1
3
x
J
−1/3
(z) − J
1/3
(z)
.
30.8.2 Power