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

[Gottfried]

Hej Gottfried,

I am not trying to show You have done something wrong , I think Your graph and approach is great.

I only want to make sure that reducing complex numbers to one root of sqrt(-1) = i so that a+ib work also in tetration and h(z) without loss of information. There fore I am trying to ask You since I can not do it- I understand You may feel annoyed - > if You feel any interest , of course, to carry the ambiquity of sign of sqrt(-1) through all derivation before getting rid of it.

No, there is no problem wth it. But without a workout with an example I seem to have persistent problems to get your problem right.

But exp(alpha+I*beta) is not equal to exp (alpha-I*beta)?
so if we consistently keep u=alpha+-i beta as definition of u as complex number, than t has 2 values = e^alpha*e^(+- I*beta).

If You continue from that and find that this other value t=e^alpha*e^-I*beta does not add any new information about the whole thing, than let it be.

Ivars

Well in this case, for the result s being real only, alpha is a unique function of beta. If you take +beta you need one alpha, and if you take -beta, you need another alpha adapted from this; in this case, the two alphas are even equal (the graph looks symmetrical w.r.t. y-axis).
So no ambiguity is here, just use the formula of my paper, by which the alpha is determined by beta to get a real s.

Gottfried