Infinite tetration and superroot of infinitesimal

[Ivars]

Ops, sorry ... ! The first one, the second (e) is, in .... reality: (f) .

That's a very good one!

GFR

I also vote for now clearly defined e) but with additional imaginary dimension or "hyperdimensions " where it all happens. Cylinder. Any 2 points on real axis are connected by a line (curve) which leaves real axis and goes to infinity and returns on next point, infinitesimally close to previous. So a point on real axis is just such an infinite but connected line looked at from end. And, there are 2 possible rotation directions of these lines. And, what about third, x^(1/x) which tends to be continuous without leaving real xy plane? it seems like that one is rotating along its own axis, experiencing torsion.

I had a question which I did could not find a satisfactory answer-if n= 3 , x^n has 1 value, x^1/n = 3 values, 2 of them complex.

How many values has x^(1/x) when x= rational ( lime m/n m is not 1) irrational, transcendental, complex, i?

For me it is obvious that mathematics have to be defined in terms of i from which we can obtain via "backtetration" e^(pi/2) - not in terms of counting and integer numbers. From the other end, so to say, or rather, from both other ends simultaneously.



Ivars