Oh foo, they're obvious roots with closed forms. E.g, Out[11]= (-1)^(1/4)/Sqrt[2 Pi] In[13]:= E^-%11^-2 Out[13]= 1 In fact, In[14]:= Solve[1==E^-z^-2,z] Out[14]= {{z->ConditionalExpression[-(1/(Sqrt[2 \[Pi]] Sqrt[I C[1]])),C[1]\[Element]Integers]}, {z->ConditionalExpression[1/(Sqrt[2 \[Pi]] Sqrt[I C[1]]),C[1]\[Element]Integers]}} But strangely, In[15]:= Solve[1 == E^-z^-2] Out[15]= {} --rwg Correction: (Trying to save selection as [png] crashes 9.0.1 for PlotPoints->666. And the plotting artifacts are worse.) On Thu, Apr 18, 2013 at 8:10 PM, Bill Gosper <billgosper@gmail.com> wrote:
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