Airy functions : Airy_Ai and Airy_Bi

### 5.9.19 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