Airy functions: Airy_Ai and Airy_Bi

### 6.8.19 Airy functions: Airy_Ai and Airy_Bi

The Airy functions of the first and second kind are defined by

Ai(x) | = |
(1/π) | ∫ |
∞
0
| cos(t3/3 + x*t) dt
Bi(x) | = |
(1/π) | ∫ |
∞
0
| (e− t3/3 + sin( t3/3 + x*t)) dt

The have the properties that, if f and g are two entire series solutions of

w′′−x*w=0

then

Ai(x) | = | Ai(0)*f(x)+ Ai′(0)*g(x)
Bi(x) | = |
√ |
3
| (Ai(0)*f(x) −Ai′(0)*g(x) )

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)!

The Airy_Ai and Airy_Bi commands compute the Airy functions.

- Airy_Ai and Airy_Bi take one argument:
  x, a real number.
- Airy_Ai(x) and Airy_Bi(x) return the values of the Airy functions.

Examples.

- 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