Airy functions

### 6.3.19 Airy functions

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.

Let f and g be 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)),

where f(x)=∑k=0∞3k(Γ(k+1/3)/Γ(1/3)) x3k/(3k)! and 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

Airy_Ai(1)

0.135292416313 |  |  |  |  |  |  |  |  |  |

Airy_Bi(1)

1.20742359495 |  |  |  |  |  |  |  |  |  |

Airy_Ai(0)

0.355028053888 |  |  |  |  |  |  |  |  |  |

Airy_Bi(0)

0.614926627446 |  |  |  |  |  |  |  |  |  |