Further observations on fractional calc solution to tetration

[JmsNxn]

It seems you are once again trying to use Ramanujan's master theorem.

As for the remark 3) :

Let N,M,i,j be integers.

we have slog(z+ 2 pi j i) = slog(z).

if tet (a_i) = b_i and tet ' (a_i) = 0

then slog(z) is not analytic at b_i.

it follows slog(z) is also not analytic at b_i + N 2pi i.

also tet ' (a_i + M) = 0

tet(a_i + M) = exp^[M](b_i) which is chaotic most of the time !!

SO your halfplane cannot contain all those points nor the fixpoints L.
Also none of the complex conjugates of those !!!

hence your half-plane is parallel to Re(z) > Q if it exists at all.

That was the reason for my comment about 3).

Maybe a link to H trapmann's paper is usefull.

I hope I have not offended you and you understand my viewpoint (now).

regards

tommy1729

Really? Hmm. I was under the impression it would be analytic in a half plane. If it's not then it fails completely. Sorry if I was rude ^_^, I was just excited. The power of using the weyl differintegral for recursion just gets me giddy.

I am using ramanujan's master theorem, but I've expanded on it more and reworded the result as a result in fractional calculus, I also proved it using a different method.


Luckily, if slog is non zero in a half plane but with poles, we can do the exact same procedure with \( 1 / slog(z) \) instead, if it has no poles and is exponentially bounded.