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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

y′′x*y=0 

They are defined by :

Airy_Ai(x)=
(1/π) 


0
cos(t3/3 + x*tdt 
Airy_Bi(x)=
(1/π) 


0
(e− t3/3 + sin( t3/3 + x*t)) dt

Properties :

 Airy_Ai(x)=Airy_Ai(0)*f(x)+ Airy_Ai(0)*g(x
Airy_Bi(x)=
3
(Airy_Ai(0)*f(x) −Airy_Ai(0)*g(x) )

where f and g are two entire series solutions of

w′′x*w=0 

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)!

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

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