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

*To*: mathgroup at smc.vnet.net*Subject*: [mg43783] Airy's Gi(x) function; asymptotic matching and asymptotic limits*From*: "Curt Fischer" <crf3 at po.cwru.edu>*Date*: Sat, 4 Oct 2003 02:05:01 -0400 (EDT)*Sender*: owner-wri-mathgroup at wolfram.com

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