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

[Gottfried]

@Gottfried
Also, it took me 3 months to understand your graphs, but I think I understand them now. It seems like you were trying to plot a graph like this:


which is the contour plot of \( \text{Im}(x^{1/x}) = 0 \). However, I still don't understand the questions your asking, which makes it hard to get through this 8-page thread. If someone could summarize the questions and results in this thread, I would be very thankful.

Andrew Robbins

Hi Andrew -
first please excuse my delay in answering your other post: I was struck in bed by sickness, and could not concentrate as I wanted to.

Now, you are correct - the first attempt was just an approach to find such a contour, and since at that time I had no versatile implementation of the h()-function in Pari I had to search for an idea about this contour at all, programming the graph in Delphi-code.

The second step was then, to find an analytical expression for the Im(z^(1/z))=0 in terms of one real parameter. The second type of plot is then a combined graph :

u = alpha + i*beta
t = exp(u) = a + i*b
s = exp(u/t); IM(s)=0

where beta is the single parameter and alpha,a,b,s are dependent on beta. Everything again handwaved; the Lambert-W and h()-implementation, which was available to me at this time was still not branch-enabled (I could finally implement one using Henryk's last hint)

For what this all was needed: I have these hypotheses about the composition of the eigensystem of the (infinite) Bell-matrix.
This includes a relation to the fixpoints, actually the t and u-values above are needed to construct the eigen-matrices, from which then the Bell-matrix (and its fractional or general powers) can be composed.
For bases s <e^(1/e) things are simple and numerical computations support my construction-hypothesis, but for s>e^(1/e) I needed a possibility to compute examples to work on the verification of the hypothese, too.
The found parameter-formula for s, based on the independent parameter beta, allowed then to find for an arbitrary s>e^(1/e) the according u and t (and even branches) employing regula falsi - just a poor-mans Lambert-W-replacement....

So, a sufficient (zip-) compression of this (part of the) thread would possibly be a one-liner in Maple/Mathematica & co after all ;-)

Gottfried