Re: fourth degree equation

Axel Vogt wrote:
Gerry Myerson wrote:
In article <euu96396u145kld9mvbqctiib3vbu58b1r@xxxxxxx>,
quasi <quasi@xxxxxxxx> wrote:

On Mon, 04 Jun 2007 23:52:51 -0500, Robert Israel
<israel@xxxxxxxxxxxxxxxxxxxxxxxxxxxxx> wrote:

quasi <quasi@xxxxxxxx> writes:

On Mon, 04 Jun 2007 20:51:37 -0700, drc1@xxxxxxxxxxxx wrote:

What is the solution for x^4 - x + a = 0. (Solve for x in terms of a.)
It's not pretty.

Perhaps you'd find this series form pretty:

x = sum_{j=0}^infinity (4j choose j)/(3 j + 1) a^(3j+1)
Much better looking.

And possibly useful as well.

But does it work?

I tried it in Maple with a=-1 and also with a=-2. Neither value seemed
to work.

Can you give a numerical example?

Since 4 j choose j is going to infinity pretty fast, I'd say that at the very least you need |a| < 1 for convergence.

'yes', the radius of convergence in a is (3/8*2^(1/3)). But one can write
the series as h:= a*hypergeom([1/4, 1/2, 3/4],[2/3, 4/3],256/27*a^3) with
Maple's notation for a 3F2 hypergeometric function.

Then it can extended analytically beyond the disc and written in that form
Maple will evaluate it also in a=-1 or a=-2 (do it numerically).

Actually Maple 10 does compute an analytic solution as well (with a as a parameter).
.

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