Further observations on fractional calc solution to tetration

[mike3]

I understand the circularity ^_^, it was worded a little weird. I got excited when I thought of applying these methods to tetration and I didn't consider too much about the erradic behaviour of the tetration function in the imaginary plane. As mike pointed out it blows up as n grows in 1/{n-s}^e which is not what I would've expected :\.

However! using the slogarithm makes a lot more sense.

NOW I have something to say. Using fractional calculus, (and even carlson's theorem if you want), if a slogarithm satisfies the exponential bounds in a half plane it is UNIQUELY determined by the values it takes on at integers. This implies, that the tetration it provides is fully determined by the sequence of numbers \( a_n \) such that \( (^{a_n} e) = n \).

Now that's gotta be something interesting.

Particularly:

\( \int_0 ^\infty t^{z-1} \sum_{n=0}^\infty a_n (-t)^n/n!\,dt = \G(z)slog(-z) \) for \( 0 < \Re(z) < \Re(L) \)

that's pretty interesting.

I'll have to mull on what we can accomplish with this but a very good idea I'm thinking about is determining these a_n (or a criterion they all satisfy) and seeing what else we can do with them.



FURTHERMORE

find me a function that (a) satisfies f(e^n) = f(n)+1 (b) holomorphic in a half plane and satisfies our exponential bounds then DA DUM DA DUM f is a slogarithm.

I just refreshed my memory on what Trappmann's uniqueness condition was. It turns out it's a little more complicated.

I mentioned about holomorphism on the "sickle" between the fixed points. The condition is actually stronger: the function must actually be biholomorphic (holomorphic and injective with holomorphic inverse) on that sickle, not just holomorphic.

In addition the image of the sickle under the function must be unbounded in the imaginary direction (both directions).

The second criterion seems related to the notion of tetration "approaching the fixed points of the logarithm" at \( \pm i\infty \) (which gives the super-logarithm singularities at the fixed points).

The sickle region is defined as the region bounded by the straight line connecting \( L \) and \( \bar{L} \) together with its image (a curve) under the exponential \( \exp \). It is a subset of the half-plane \( \Re(z) > \Re(L) \) if you don't include the boundary.

So if you can find coefficients which will make your super-logarithm function satisfy these criteria, then you will have the Kneser super-logarithm.