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

[Ivars]

Hej BO,

Again, no! Try for example Maple (LambertW) or Mathematica (ProductLog), you can chose the branch there, for example

Code:

LambertW(-3,-Pi/2)=-2.198342630-13.98120831*I
LambertW(-2,-Pi/2)=-1.604290913-7.647192276*I
LambertW(-1,-Pi/2)=-1.570796327*I
LambertW(0,-Pi/2)=1.570796327*I
LambertW(1,-Pi/2)=-1.604290913+7.647192276*I
LambertW(2,-Pi/2)=-2.198342630+13.98120831*I

where the first argument denotes the branch.

I see, thanks. I appreciate a lot the lessons I get from You.

I wonder if there is analytical way to derive all branch values for W(-pi/2) as function of k and hence h(e^pi/2). What does Mathematica says about the values of h(e^pi/2) and h(e^-pi/2) along few first branches? (I have no access to such instruments...). Any kind of pattern?

From paper I have I see these formulas for Quadratix of Hippias but can not yet understand how these could be used to derive values of W(-pi/2) on other branches than 0 and -1.

I still think there are actually 4 values h(e^pi/2) and may be even in some more general divergent cases of real number tetration: 2 corresponding to -i, 2 to +i , but I do not know yet how to find out if that is true and what are these values. The only case when there are only 2 distinquishable values ar +-i bacause the real parts here are 0..

I also make a conjecture that as number of branch k-> infinity,
h(e^pi/2) = -1 independent of way it is reached.

Also, I think there must be some relation between all h(e^pi/2) and
h(e^-pi/2) branch values, possibly type of Mobius transform. Again , just a conjecture, to be proved wrong to make progress

Best regards,

Ivars