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

[Ivars]

Ivars -

Hmm, I've not been in your discussion with Henryk (have my mind elsewhere these days) and I don't know, perhaps I'm missing the point.
What is the point of your question - besides, that we have conjugate solutions? If I recall my above graphs, for instance, then they show obviously conjugacy. And my graph for complex fixpoints for real bases b>e^(1/e)

can be continued for any number of solutions t and u, where imag(u)=some selection +- 2*k*pi (= beta in the graph). (though changing the k-factor only does not lead exactly to the same b, but this should not be seen as a problem). So there may be some problem, that I'm missing when I read your posts?

You can not see what I can not express properly

Did You succeed in explaining why sqrt 2 tetrated might lead to both 2 and 4? Euler seems to have had an idea, and he also bothered with r=e^pi/2 in his article, but I can not read Latin good enough still.

May be what I am trying to say is:

We should define tetration generally with a help of some divergent series Euler style and then investigate all possible analytical developments of these series, including convergent ones in the region of convergence of h(z).

I have a gut feeling I can not get rid of that somewhere the +i and - i difference= ambiquity will pop out in such case clearly. And z=e^pi/2 should be the easiest place to investigate it properly. There has to be not 2 , but 4 values at all k which sometimes overlap, but in some places adding adding pik to i and pik to - i leads to different values.E.g.

ln(sqrt2) gives : ln sgrt2 +- 2 pi k =1/2ln2 +- 2 pi I k
ln(-sqrt 2) gives: ln sqrt 2+-pi k= 1/2ln 2 +- pi I k but k is not 0 as negative logarithms can only be complex, never real.

So totally you have : when k= 0-> 1 value

ln sqrt2=1/2 ln 2

when k=1 -> 4 values

ln sqrt2 = 1/2 ln 2 +- 2pi I -> 2 values
ln -sqrt2 = 1/2 ln 2 + - pi*I -> 2 different 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.