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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) 


0
cos(t3/3 + x*tdt 
Bi(x)=
(1/π) 


0
(e− t3/3 + sin( t3/3 + x*t)) dt

The have the properties that, if f and g are 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) )

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)!

The Airy_Ai and Airy_Bi commands compute the Airy functions.


Examples.


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