2.9.11 Airy functions : Airy_Ai and Airy_Bi
Airy_Ai and Airy_Bi takes as argument a real x.
Airy_Ai and Airy_Bi are two independant solutions
of the equation
They are defined by :
| Airy_Ai(x) | = | | (1/π) | ∫ | | cos(t3/3 + x*t) dt |
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| Airy_Bi(x) | = | | (1/π) | ∫ | | (e− t3/3 + sin( t3/3 +
x*t)) dt
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Properties :
| Airy_Ai(x) | = | Airy_Ai(0)*f(x)+
Airy_Ai′(0)*g(x) |
| Airy_Bi(x) | = | | √ | | (Airy_Ai(0)*f(x)
−Airy_Ai′(0)*g(x) )
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where f and g are two entire series solutions of
more precisely :
| f(x) | = | | 3k | ⎛
⎜
⎜
⎜
⎜
⎜
⎝ | | ⎞
⎟
⎟
⎟
⎟
⎟
⎠ | |
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| g(x) | = | | 3k | ⎛
⎜
⎜
⎜
⎜
⎜
⎝ | | ⎞
⎟
⎟
⎟
⎟
⎟
⎠ |
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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