03/01/2012, 12:04 PM

Beautiful job, Sheldon!

If you put points uniformly at the circle around exp(1/e),

then with few points, you can get a good precision evaluating the derivatives of tet_b(z) with respect to b by the Cauchi contour integral. So. you can plot, for example,

d tet_b(z) / db,

d^2 tet_b(z) / db^2,

as functions of z, and make conclusion about the behavior in vicinity of b=exp(1/e).

I try to guess the result:

As b approaches exp(1/e), the quasiperiods become large; so, all the cut lines at the z plane are far away from zero (and out of the field of your pics). In this sense, in vicinity of b=exp(1/e), there should be no cutlines at all, so, b=exp(1/e) is regular point for the most of values of z. Do your evaluations allow to confirm or to refute such a guess?

Is your evaluation for fixed b and various z faster than evaluation for fixed z and varable b?

If you put points uniformly at the circle around exp(1/e),

then with few points, you can get a good precision evaluating the derivatives of tet_b(z) with respect to b by the Cauchi contour integral. So. you can plot, for example,

d tet_b(z) / db,

d^2 tet_b(z) / db^2,

as functions of z, and make conclusion about the behavior in vicinity of b=exp(1/e).

I try to guess the result:

As b approaches exp(1/e), the quasiperiods become large; so, all the cut lines at the z plane are far away from zero (and out of the field of your pics). In this sense, in vicinity of b=exp(1/e), there should be no cutlines at all, so, b=exp(1/e) is regular point for the most of values of z. Do your evaluations allow to confirm or to refute such a guess?

Is your evaluation for fixed b and various z faster than evaluation for fixed z and varable b?