Imaginary zeros of f(z)= z^(1/z) (real valued solutions f(z)>e^(1/e))

[Gottfried]

Dear Gottfried, Andrew and Henryk,

I was impressed by the last plot posted by you (Gottfried) and I am pleased to tell you that I agree completely with your conclusions. Please find attached some notes concerning that matter. Actually, the same result can be reached from another end, by using the two real branches of the product-logarithm (Lambert) function. Therefore, I presume that the plot is correct.

Nevertheless, by using the same method, it seems that for b -> oo the complex infinite towers don't vanish, as it could be felt from a first look at your graph. Is it correct? Can that be proved?

Best wishes.

Gianfranco

Dear Gianfranco -

thanks for that comment! I just read your pdf.
In fact I found the values being symmetric w.r. to the real line.
The first image adressing this was
[attachment=67]
in this thread, where the black lines are the different branches. Each of these branches seem to contain *all* real valued b's, so it shows for each b a multitude of solutions (possibly infinitely many).
The complex conjugacy can possibly better be seen in the other graph



which focuses the lines better.
The contour, which I used in my last post, is that one, which escapes in the above grey contourplot at the top of the image; it curves then to the right. It would be interesting to see its contour down to b->1.4446.... (eta). I don't think it escapes to infinity, but let's see...

It is also nice to see your graphs of the productlog; unfortunately I don't have the possibility to obtain them using Pari/Gp, so I had to hack my own procedure.
Tonight I'l read your post a second time, I just came back home from a walk.

Kind regards -

Gottfried