From: Alan Jones <asj@maths.uq.edu.au>
Subject: Re: Special functions Ai and Bi
Date: Thu, 23 Dec 1999 10:10:37 +1000
Newsgroups: sci.math.research
Keywords: characterization of Airy functions

Franck MICHEL wrote:
> 
> In classical books about special functions, for example the "Handbook of
> mathematical functions" of Abramowitz and Stegun, the Airy equation w''(z) =
> z w(z) is considered.
> 
> A pair of linearly independant solutions are given: the Airy function Ai(z)
> and the function Bi(z).
> 
> I have a little idea of the interest of the function Ai(z); in particular,
> it seems to be the unique solution that tends to zero when Z tends to
> +infinity along the real axis.
> 
> But I don't see why the function Bi(z) has been chosen as a
> "special function"; I don't see what "special"properties this solution has.
> Perhaps it has none in particular (they had to choose a solution linearly
> independent from Ai, the function Bi was chosen randomly) but I think it has
> something special. But I don't know what (I looked at books as Abramowitz and
> Stegun, Whittaker and Watson, or Ince, and I did not find any information that
> answer to this question).
> 
> I would be interested to know:
> - if the Bi function has no particular property, it has only been chosen to
> obtain a basis of solutions with the Airy function
> - or if Bi has a particular interesting property, something "special" that
> justify to distinguish this solution of the Airy equation
> 
> Thanks in advance.
> 
>       Franck Michel
The function Bi is chosen to complement Ai along the negative real
axis. Where Ai is asymptotically a sin function, Bi is the equivalent
expression in terms of cos.

Alan Jones