### 5.9.11 Airy functions : Airy_Ai and Airy_Bi

Airy_Ai and Airy_Bi take as arguments a real x.
Airy_Ai and Airy_Bi are two independent solutions of the equation

y′′−x*y=0

They are defined by :

Airy_Ai(x)

=

(1/π)

∫

∞
0

cos(t3/3 + x*t) dt

Airy_Bi(x)

=

(1/π)

∫

∞
0

(e− t3/3 + sin( t3/3 + x*t)) dt

Properties :

Airy_Ai(x) | = | Airy_Ai(0)*f(x)+ Airy_Ai′(0)*g(x)

Airy_Bi(x)

=

√

3

(Airy_Ai(0)*f(x) −Airy_Ai′(0)*g(x) )

where f and g are two entire series solutions of

w′′−x*w=0

more precisely :

f(x)

=

∞
∑
k=0

3k

⎛
⎜
⎜
⎜
⎜
⎜
⎝

Γ(k+

1
3

)

Γ(

1
3

)

⎞
⎟
⎟
⎟
⎟
⎟
⎠

x3k
(3k)!

g(x)

=

∞
∑
k=0

3k

⎛
⎜
⎜
⎜
⎜
⎜
⎝

Γ(k+

2
3

)

Γ(

2
3

)

⎞
⎟
⎟
⎟
⎟
⎟
⎠

x3k+1
(3k+1)!

Input :

Airy_Ai(1)

Output :

0.135292416313

Input :

Airy_Bi(1)

Output :

1.20742359495

Input :

Airy_Ai(0)

Output :

0.355028053888

Input :

Airy_Bi(0)

Output :

0.614926627446