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

[Gottfried]

You can not see what I can not express properly

(...)

ln sqrt2=1/2 ln 2

when k=1 -> 4 values

ln sqrt2 = 1/2 ln 2 +- 2pi I -> 2 values, but + 2pi *I and - 2pi*I is the same
ln -sqrt2 = 1/2 ln2 + - pi*I -> 2 values and +pi and - pi which does not coincide with any other values

I have a feeling that tetration is the field where reducing this ambiquity early on and reaching definitions of tetration via convergent series or functions ( like W(z) ) looses important information on the way.

When You see e^i*pi/2 You think about rotation of 90 degrees. When i = h(e^pi/2) then e^h(e^pi/2) * pi/2 is also a rotation by 90 degress but involves infinite operation on real numbers. NO i. It is an identity- You can replace i with h(z) and -i with another branch of h(z) . And depending on the sign of h(z) it will be either rotation anticlockwise, or clockwise.

so h(e^pi/2)^2 = -1 but we clearly know that each h(e^pi/2) which is root of -1 is found via different branch, so they can not be replacable so easily .

May be I am terribly wrong, as I really can not nail the place where it really matters.

Hmm, surely there are more possibilities to express a value not only by convergent, but also by divergent series. May be your idea was already covered with my observation, that seemingly the same matrix Bs containing the needed coefficients for tetration can be seen as compsed of different versions of eigen-matrices
Bs = W0 * D0 * W0^-1
= W1 * D1 * W1^-1
etc
where W0, W1, etc are constructed by different powerseries based on the multitude of different fixpoints. So the different solutions you talk about are possibly reflected in the different versions of the eigenmatrices?

I'll read your post again later and try myself to eliminate further misunderstandings, if there is still one.

Gottfried