Talk:Airy function

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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)

Contents

[edit] 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)

[edit] 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 y'' > 0 if x > 0 and y > 0. Thanks. -- Jitse Niesen (talk) 00:23, 9 November 2006 (UTC)

[edit] 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)

[edit] 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 \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

[edit] 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)