On 2019-06-23 14:40, Brad Klee wrote:
Hi Eugene and Dan,
In one sense, it is really a non-issue because we have three equations in three unknowns. In Mathematica:
Girth[n_] := With[{c1c2 = {c[Mod[n, 3]], c[Mod[n+1, 3]]} }, Pi Total[c1c2] Hypergeometric2F1[-1/2, -1/2, 1, 1 - 4 (Times @@ c1c2)/(Total[c1c2])^2]]
FindRoot[MapThread[ Subtract, {Girth /@ Range[3], {2 \[Pi], 2 \[Sqrt]2 EllipticE[-1], 4 EllipticE[1/2]}}], Transpose[{c /@ Range[0, 2], {1, 1, 1}}]]
Out[]:= {c[0] -> 0.707107, c[1] -> 1., c[2] -> 1.}
I'm sure you already know this, and can guess that your intrigue has more to do with the underlying function theory. If a "smart" solution exists, it has to get around a serious limitation that series inversion is not an operation that closes the set of D-finite functions back to itself.
I'm not sure what Gosper is getting at (other than to upset S.E.),
For the record, although he didn't publish, Bill Dubuque was probably the first to find the decision procedure for q-hypergeometric summation. As for the five who jumped me: "Off topic" my asymptote. And no one seems to have noticed that the spheroidal special case that I added as appeasement is very easily solved in terms of InverseFunction@EllipticE. Years ago I posted here several trivariate series reversions for the semiaxes in the general case, and expressed surprise at the apparent absence of such a technique from the literature. —rwg
but there is another similar, somewhat easier problem in semi-classical quantum mechanics. Given a D-finite action function and a quantized action value, we would like to find the associated quantum energy value. This can be done by series inversion, but coefficients of the inverse series are not expected to have any simple p-recurrence.
In this sort of situation I'm happy to accept numerical values for the quantized energy levels, and then move on to worrying about experimental spectra. I think this is a fairly common attitude in industry, but yes, I do also sometimes wonder if there is a stronger function theory that could do the inversion exactly?
I'm only self-taught in maths, so maybe someone else already knows an answer to the overarching question: what "smart" techniques do we have for inverting differentiably-finite functions? Any at all?
I'm only aware of the Faà di Bruno approach, which I find to be overly cumbersome.
Cheers,
Brad
On Sun, Jun 23, 2019 at 3:27 PM Eugene Salamin via math-fun <math-fun@mailman.xmission.com> wrote:
Has anyone solved the problem of determining the axes of an ellipsoid given the three girths? It has me intrigued. The mathematics is more interesting than the politics of math.stackexchange.
-- Gene _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun