# Talk:Airy function

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Field: Analysis

Hi, I should say that in the integral formula for the Bi Airy function the exponential expression misses the zt component. See eg Abramowitz formula 10.4.33. regards, jan

Checked and fixed, thanks. -- Jitse Niesen 12:45, 25 Feb 2004 (UTC)

## Asymptotic formula

the asymptotic formula for Ai and Bi when x sto -infinite seems to be inverted regards

You're right. Thanks very much for letting us know. -- Jitse Niesen (talk) 13:17, 5 October 2006 (UTC)

## Solving the Airy equation

One should write down two ways to solve the Airy equation: 1. The series solution method and, 2. The Laplace transform. With regard to the Laplace transform, one should explain how to take the contour on the complex plane. Watson1905 (talk) 20:50, 11 February 2014 (UTC)

## Properties

As far as I can see, both Ai(x) and Bi(x) are convex for positiv x. Juergen.

Indeed. The differential equation immediately implies that ${\displaystyle y''>0}$ if ${\displaystyle x>0}$ and ${\displaystyle y>0}$. Thanks. -- Jitse Niesen (talk) 00:23, 9 November 2006 (UTC)

## Asymptotic formula again

"the asymptotic formula for Ai and Bi when x sto -infinite seems to be inverted" This is still not quite correct. If the asymptotic expressions for Ai(x) and Bi(x) at large negative real x, as given on the Article page, are interchanged, then a minus sign needs to be put in front of the new expression for Bi(x). Alternatively, the given expressions can be corrected without interchanging them: in the arguments of the sine and cosine terms, one changes the sign of the factor pi/4 from minus to plus. LROS 13:23, 28 November 2006 (UTC)

I don't know what it is; for some reason, this seems to be extremely confusing for me. I changed the sign of pi/4 and I hope everything is correct now, as this is all very embarrassing. -- Jitse Niesen (talk) 07:49, 29 November 2006 (UTC)

## Is it integrable?

The text says that cosine integral in the solution for Ai(x) is not integrable, although the integral converges everywhere. Does this even make sense?

Yes. For a function to be Lebesgue integrable, its absolute value must be integrable. This is not the case for Ai(x), because it doesn't decay. However ${\displaystyle \lim _{a\to \pm \infty }\int _{0}^{a}Ai(x)dx}$ does exist as an improper integral, due to the cancellation mentioned in the article. -- GWO

## Bairy function

Bi(x) is apparently colloquially known as the "Bairy" (or "bAiry") function. Unfortunately a Google search gives mainly references to function libraries using the term as an abbreviation, so I don't know how to back it up. Confusing Manifestation 06:15, 24 May 2007 (UTC)

## Web Calculators

Last night, Arthur Rubin removed a link to an web calculator providing useful free services to users of airy functions. Many wikipedia articles on topics that have calculational aspects provide links to web calculators. This was the only calculator link on the article, so the link to that useful service is now gone. Web calcualtors for such relative obscrure functions are rare. Please cite an official policy justifcation, explain your actions in light of these points, or engage in a conversation as to why you believe this information to be inappropriate. Otherwise I plan to revert the change. Ichbin-dcw (talk) 19:47, 25 May 2010 (UTC)

## Disambiguation: Fabry-Perot etalon Airy Function

In undergraduate optics classes, it's taught that the Airy function is what's listed on: http://www.phy.davidson.edu/stuhome/cabell_f/diffractionfinal/pages/fabry.htm#Characteristics or http://www.tecoptics.com/etalons/theory.htm . It would be great to update this page to include this popular equation. OpticalEngineer (talk) 02:56, 31 July 2012 (UTC) Went ahead and updated the page. OpticalEngineer (talk) 03:42, 31 July 2012 (UTC)

## Airy wavelength

One can define the Airy wavelength ${\displaystyle \lambda }$ in terms of a far-field distance ${\displaystyle X}$ and wavenumber ${\displaystyle k_{0}}$ at that distance (such that ${\displaystyle Ai(x/\lambda )}$ oscillates like ${\displaystyle \sin(k_{0}x+\phi )}$ at ${\displaystyle x\sim X}$). Can anyone provide a good reference for this usage?

One should explain the meaning of Airy transform and that it is a unitary transform.

Also one should PROOF the orthogonality of Airy functions. This is quite important in physical problems. — Preceding unsigned comment added by Watson1905 (talkcontribs) 22:06, 10 February 2014 (UTC)

## Orthogonality and unitarity properties

One should explain the meaning of Airy transform and that it is a unitary transform.

Also one should PROOF the orthogonality of Airy functions. This is quite important in physical problems. Watson1905 (talk) 20:42, 11 February 2014 (UTC)