Wow! That's an amazing picture! Is there any way to estimate the height/depth of the pits? Are they analogous to Gibbs's ears on square waves? If so, perhaps they have a finite size. http://en.wikipedia.org/wiki/Gibbs_phenomenon At 06:20 PM 4/18/2013, Bill Gosper wrote:

Whoa, those pits in the leaves <http://gosper.org/.flattop2.png> go deep.

Roots?

--rwg

(I omitted the /2 in the exponent, since I wasn't fudging for √2√π.)

(Trying to save selection as pdf crashes 9.0.1 for PlotPoints->666. And the plotting artifacts are worse.)

On Wed, Apr 10, 2013 at 1:58 AM, Bill Gosper <billgosper@gmail.com> wrote:

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On Wed, Apr 10, 2013 at 1:09 AM, Bill Gosper <billgosper@gmail.com> wrote:

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Here <http://gosper.org/flattop.png>'s a more traditional plot near z=0. --rwg

Oops, and here's A. Goucher's missing antecedent:

When you said `ultraflat', I thought you were referring to the property of all derivatives being zero at that point. Obviously, complex-differentiable functions (such as yours) cannot have this property (except for constant functions);

however, there are infinitely differentiable examples over the reals such as f(x) = exp(-1/x^2). A function with this property is considered here: http://cp4space.wordpress.com/2013/02/28/radical-tauism/

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Sincerely, Adam P. Goucher