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) | ∫ | | cos(t3/3 + x*t) dt |
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| Bi(x) | = | | (1/π) | ∫ | | (e− t3/3 + sin( t3/3 + x*t)) dt
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The have the properties that, if f and g are two entire series
solutions of
then
| Ai(x) | = | Ai(0)*f(x)+ Ai′(0)*g(x) |
| Bi(x) | = | | √ | | (Ai(0)*f(x) −Ai′(0)*g(x) )
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more precisely:
| f(x) | = | | 3k | ⎛
⎜
⎜
⎜
⎜
⎜
⎝ | | ⎞
⎟
⎟
⎟
⎟
⎟
⎠ | |
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| g(x) | = | | 3k | ⎛
⎜
⎜
⎜
⎜
⎜
⎝ | | ⎞
⎟
⎟
⎟
⎟
⎟
⎠ |
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The Airy_Ai and Airy_Bi commands compute the Airy
functions.
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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:
- Input:
Airy_Bi(1)
Output:
- Input:
Airy_Ai(0)
Output:
- Input:
Airy_Bi(0)
Output: