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

[Ivars]

Hello,

I am new here- I was wandering about the imaginary unit for some time to be able to understand physics better, but I was quite positively surprised that you have confirmed by rigorous analysis what I just yesterday found out by studying the possible content of i- a definition ( or perhaps one of the definitions of i) -of imaginary unit.

Perhaps there are many infinite values involved as well as this definition arises from Lambert function of a logarithm, and logarithm may take infinitely many values, but still I keep - i sign):


-i = power tower (e^pi/2) = h (e^pi/2)

i = - h(e^pi/2)

so here z = e^pi/2 = 4,81047738097.........

h(z) = - i

Derivation:

h(e^pi/2) = -W( -ln e^pi/2)/ ln e^pi/2 = -W(-pi/2) / pi/2

But W( -pi/2) = i*pi/2 so

h(e^pi/2) = - i*pi/2 / pi/2 = - i .

but - h(e^pi/2) = W(-pi/2) / pi/2 = i*pi/2/pi/2 = i


There must be many more such beauties outside radius of convergence of h(z).

Best regards,

Ivars Fabriciuss