When trying to understand better the question Eisenstein Series in Mathematica?, I stumbled on the following: issuing
Derivative[1][DedekindEta][.11 I]
gives
This by itself I cannot argue is strange (except that there is nothing about EllipticReducedHalfPeriods
in the documentation) - there seemingly is some internal computation mechanism that fails on this value. What I find strange is this:
Table[Derivative[1][DedekindEta][I t], {t, .01, .2, .01}] // TableForm
produces
so that this internal mechanism somehow oscillates between failing and not failing. It seems to exhibit quite complicated behavior: sort of zooming in,
Table[Derivative[1][DedekindEta][I t], {t, .099, .101, .0001}] // TableForm
results in
What is happening? How to compute the derivative of the Dedekind $\eta$ function for values with small imaginary part?
Update
Using the answer by Michael E2, I made a plot with
ContourPlot[Log[Abs[N@N[Derivative[1][DedekindEta][x + I y], 3 $MachinePrecision]]],
{x, -.5, .5}, {y, 0, .5},
PlotPoints -> 50, Contours -> 50, ImageSize -> Full, AspectRatio -> Automatic]
and it reveals a strange blind spot:
The white area along the real line is understandable - $\eta$ has singularities there - but where the white disk comes from, I cannot figure out.
What is also very strange (although this probably belongs to another question), I only managed to produce this picture by cropping a screenshot: when I tried to save the plot as an image (right-clicking and choosing "Save graphic as..." png, I got the following message:
(sorry for unreadable text, it can be enlarged one way or another).
Strange.
EllipticReducedHalfPeriods[]
effectively performs a modular transformation to return half-periods that are effectively equivalent to the input ones, for subsequent use with the Weierstrass and modular functions. $\endgroup$