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Re: Frustration with opaque functions in Mathematica
*To*: mathgroup at smc.vnet.net
*Subject*: [mg93259] Re: Frustration with opaque functions in Mathematica
*From*: David Bailey <dave at Remove_Thisdbailey.co.uk>
*Date*: Sat, 1 Nov 2008 05:10:38 -0500 (EST)
*References*: <geeedj$agl$1@smc.vnet.net>
Aaron Fude wrote:
> Hi,
>
> This is a followup to a previous message, but I find that answers in
> terms of MeijerG and Hypergeometric* have limited utility.
>
> For example, even mathematica cannot simplify them satisfactorily. I
> find myself needing an answer in terms of elementary (or nearly
> elementary) functions and I find that MeijerG is largely useless. I
> understand that I can evaluate them, decompose in series, find limits,
> graph, etc., but what I need is analytical insight which they do not
> provide.
>
> eqn = r^2 y''[r] + r y'[r] + (r^2) y[r] == r BesselJ[1, r];
> soln = y[r] /. Assuming[Element[m, Integers] && m > 0, DSolve[eqn,
> y[r], r][[1]]] // FullSimplify
> r^2 D[soln, {r, 2}] + r D[soln, r] + r^2 soln - r BesselJ[1, r] //
> FullSimplify
>
> So how can I find out what MeijerG[{{1, 1}, {1/2}}, {{1, 1}, {0, 0,
> 0}}, r, 1/2] really is?
>
> Many thanks in advance,
>
> Aaron
>
First, you should understand that many differential equations/integrals
have no analytic solutions, and others only have solutions in terms of
MeijerG, Hypergeometric, or similar functions.
Second, Mathematica normally returns answers in terms of more familiar
functions if that is possible.
Therefore, unless you know that Mathematica has failed to simplify a
solution when it could have done, you have little to complain about.
Probably the best approach is to treat an answer in terms of these very
general functions as a no-solution, and switch to NDSolve to get a
numerical answer.
David Bailey
http://www.dbaileyconsultancy.co.uk
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