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 Henry may be the only funster who figured out that the url should really be http://gosper.org/flattop2.png <http://gosper.org/.flattop2.png> . (No period before flattop. It looks right in this compose buffer, but still 404s with a phantom period when I click it!) I don't see how they could be Gibbs, which is an artifact of series truncation. And I can't imagine that the pits aren't roots. Apropos Gibbs, http://www.tweedledum.com/rwg/gibbs.htm --rwg 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: On Wed, Apr 10, 2013 at 1:09 AM, Bill Gosper <billgosper@gmail.com> wrote: 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/ Sincerely, Adam P. Goucher