7.3.19 Airy functions
The Airy functions of the first and second kind are defined by
| Ai(x)= | | | ∫ | | cos | ⎛
⎝ | t3/3+x t | ⎞
⎠ | dt,
Bi(x)= | | | ∫ | | ⎛
⎝ | 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)= | √ | | (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