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MathGroup Archive 2003

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Re: Airy's Gi(x) function; asymptotic matching and asymptotic limits

  • To: mathgroup at smc.vnet.net
  • Subject: [mg43823] Re: Airy's Gi(x) function; asymptotic matching and asymptotic limits
  • From: "Curt Fischer" <crf3 at po.cwru.edu>
  • Date: Tue, 7 Oct 2003 02:41:01 -0400 (EDT)
  • References: <bllok7$bus$1@smc.vnet.net> <blr15g$3f6$1@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

Hi Kevin,

Thanks for your reply.  I noticed the Hypergeometric answer given by
Mathematica in my original post.  I guess my question was unclear...

How can I relate the messy hypergeometics to Airy's Gi (not Ai nor Bi)
function?  I know that the two expressions are not already the same because

f[x_]:=(hypergeometric stuff from below)
g[x_]:=1/3 AiryBi[x] + Integrate[AiryAi[x] AiryBi[t] + AiryAi[t]
AiryBi[x],{t,0,x}]

Plot[f[x]-g[x],{x,0,10}]

gives a function with huge spike at about x = 7 or so.  If the hypergeometic
expression and Airy's Gi were one and the same, I would not expect this
behavior.  (Note my definition of Airy's Gi given below contains typos and
is computationally more difficult for Mathematica than the one given above,
but both are equivalent [except for the typos].)

So, the relationship between Airy's Gi[x] and this hypergeometric business
is still not clear to me.

Thanks again,

Curt Fischer


I think that
"Kevin J. McCann" <kjm at KevinMcCann.com> wrote in message
news:blr15g$3f6$1 at smc.vnet.net...
> Mathematica gives the result in terms of Airy's and Hypergeometrics. Copy
these into
> Mathematica to see what's going on.
>
> Cheers,
>
> Kevin
>
>
>
> Problem:
>
> \!\(\*
> RowBox[{"DSolve", "[",
> RowBox[{
> RowBox[{"{",
> RowBox[{
> RowBox[{
> RowBox[{
> RowBox[{
> SuperscriptBox["y", "\[Prime]\[Prime]",
> MultilineFunction->None], "[", "x", "]"}],
> "-", \(x\ y[x]\)}], "==", \(-1\)}], ",",
> RowBox[{
> RowBox[{
> SuperscriptBox["y", "\[Prime]",
> MultilineFunction->None], "[", "0", "]"}], "==", "0"}],
> ",", \(y[0] == \[ScriptCapitalC]\)}], "}"}], ",", \(y[x]\), ",",
> "x"}], "]"}]\)
>
>
>
> Solution:
>
> \!\({{y[x] ->
> 1\/2\ \((3\^\(2/3\)\ \[ScriptCapitalC]\ AiryAi[x]\ Gamma[2\/3] +
> 3\^\(1/6\)\ \[ScriptCapitalC]\ AiryBi[x]\ Gamma[2\/3] -
> 2\ x\^2\ Hypergeometric0F1[4\/3,
> x\^3\/9]\ HypergeometricPFQ[{1\/3}, {2\/3, 4\/3},
> x\^3\/9] +
> x\^2\ Hypergeometric0F1[2\/3,
> x\^3\/9]\ HypergeometricPFQ[{2\/3}, {4\/3, 5\/3},
> x\^3\/9])\)}}\)
>
> "Curt Fischer" <crf3 at po.cwru.edu> wrote in message
> news:bllok7$bus$1 at smc.vnet.net...
> > Dear Group,
> >
> > Recently I had to solve the differential equation y''[x] - x y[x] == -1,
> > with one known boundary condition y'[0]==0.
> >
> > The general solution is conveniently written as C[1] AiryAi[x] + C[2]
> > AiryBi[x] + Pi airyGi[x].  One of the constants can be solved for with
> > respect to the other
> >
> > Question 1:
> >
> > Mathematica does not have the airyGi[x] function built-in.  It returns
the
> > solution
> > \!\(1\/2\ \((2\ \@3\ AiryAi[x]\ C[2] + 2\ AiryBi[x]\ C[2] -
> > 2\ x\^2\ Hypergeometric0F1[4\/3,
> > x\^3\/9]\ HypergeometricPFQ[{1\/3}, {2\/3, 4\/3}, x\^3\/9] +
> > x\^2\ Hypergeometric0F1[2\/3,
> > x\^3\/9]\ HypergeometricPFQ[{2\/3}, {4\/3, 5\/3}, x\^3\/9])\)\)
> >
> > which is a big messy expression involving AiryAi[x], AiryBi[x], and
> > hypergeometric functions.  Does anyone know how I can relate this
> > hypergeometric stuff is equal to airyGi[x] == Integrate[Sin[t^3 + z t]
> > dz,{t,0,infinity}] ?
> >
> > Question 2: When I solved my problem analytically, I was interested in
> > evaluating the unknown integration constant by asymptotic matching to
> > another function which I knew.  This worked great on paper, but
> Mathematica
> > could not take the limit of
> >
> > 2\ x\^2\ Hypergeometric0F1[4\/3,
> > x\^3\/9]\ HypergeometricPFQ[{1\/3}, {2\/3, 4\/3}, x\^3\/9] +
> > x\^2\ Hypergeometric0F1[2\/3,
> > x\^3\/9]\ HypergeometricPFQ[{2\/3}, {4\/3, 5\/3}, x\^3\/9])\)\)
> >
> > Is there a way to evaluate this limit in Mathematica?  Also, in general
is
> > there anyway to get the an "asymptotic limit" of a function in
> Mathematica?
> > For example, airyGi[x] -> 1/(Pi x) for large x.  Is there any way to
> elicit
> > this type of info about a function from Mathematica?
> >
> > (See Abramowitz and Stegun, 1974, Handbook of Mathematical Functions,
for
> > this and other info on Airy functions.)
> >
> > thanks for any help anyone can provide,
> >
> >
> >
> > Curt Fischer
> >
> >
> >
> >
>
>



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