Showing Integral form satisfies the Airy function

In summary, the conversation discusses the integral form of the Airy function for real inputs and how it satisfies the Airy Differential Equation. The individual attempts to show this by differentiating and integrating the function, but it ultimately leads to a non-convergent integral. However, a change of variables can simplify the integral and show that it converges to zero.
  • #1
thrillhouse86
80
0
Hey All,

Can someone please give me the gist of how to show that the integral form of the Airy function for real inputs:
[tex]
Ai(x) = \frac{1}{\pi} \int_0^\infty \cos\left(\tfrac13t^3 + xt\right)\, dt,
[/tex]

satisfies the Airy Differential Equation: y'' - xy = 0

I tried differentiating twice wrt to the x variable (assuming I could just bring it inside the integration) and then subbing back into the ODE but that failed.

Regards,
Thrillhouse
 
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  • #2
Assume that
[tex]y = \frac{1}{\pi} \int_0^\infty \cos\left(\tfrac13t^3 + xt\right)\, dt,[/tex]

and the operations integrate and differentiate can be interchange (??), I obtain

[tex]y''-xy = -\frac{1}{\pi} \int_0^\infty (t^2+x)\cos\left(\tfrac13t^3 + xt\right)\, dt[/tex]

Why is RHS identically zero ? :cry:
 
Last edited:
  • #3
make a change of variables:

[tex]u=\frac{1}{3}t^{3}+xt => du=(t^{2}+x)dt [/tex]

you will simply have:

[tex]-\frac{1}{\pi}\int^{\infty}_{0}cos(u)du[/tex]

that technically doesn't converge to zero. So that's the most far you can get
 
  • #4
surely the positive and negative components of the cos function will add up to zero when you integrate ?
 

Related to Showing Integral form satisfies the Airy function

1. What is the integral form of the Airy function?

The integral form of the Airy function is given by:
Ai(x) = 1/π * ∫e^(-(t^3)/3 + xt) dt

2. How is the integral form derived?

The integral form of the Airy function can be derived using the Laplace transform of the differential equation satisfied by the Airy function. This transforms the differential equation into an integral equation, which can then be solved for the Airy function.

3. Why is it important to show that the integral form satisfies the Airy function?

Showing that the integral form satisfies the Airy function provides a way to compute the value of the Airy function at any point, which is useful in various mathematical and scientific applications. It also helps to establish the validity and accuracy of the integral form.

4. Are there any other forms of the Airy function?

Yes, there are other forms of the Airy function, such as the series and asymptotic forms. These forms may be more useful in certain situations, but the integral form is often preferred for its simplicity and ease of computation.

5. Can the integral form be used to evaluate complex values of the Airy function?

Yes, the integral form can be used to evaluate complex values of the Airy function. However, caution must be taken when dealing with complex integrals, and specialized techniques may be required in some cases.

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