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

• To: mathgroup at smc.vnet.net
• Subject: [mg43796] Re: Airy's Gi(x) function; asymptotic matching and asymptotic limits
• From: "Kevin J. McCann" <kjm at KevinMcCann.com>
• Date: Mon, 6 Oct 2003 02:07:49 -0400 (EDT)
• References: <bllok7\$bus\$1@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

```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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