# Airy functions: Airy_Ai and Airy_Bi

### 5.8.19 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