Andrew Robbins' Tetration Extension

[tommy1729]

[quote='tommy1729' pid='3446' dateline='1246053079']
as for the radius of convergence :

let A be the smallest fixpoint => b^A = A

then ( andrew's ! ) slog(z) with base b should satisfy :

slog(z) = slog(b^z) - 1

=> slog(A) = slog(b^A) - 1

=> slog(A) = slog(A) - 1

=> abs ( slog(A) ) = oo

so the radius should be smaller or equal to abs(A)

Its not only valid for Andrew's slog but for every slog and also not only for the smallest but for every fixed point.
However not completely:
One can not expect the slog to satisfy slog(e^z)=slog(z)+1 *everywhere*.
Its a bit like with the logarithm, it does not satisfy log(ab)=log(a)+log(b) *everywhere*.
What we however can say is that log(ab)=log(a)+log(b) *up to branches*. I.e. for every occuring log in the equation there is a suitable branch such that the equation holds.
The same can be said about the slog equation.
So if we can show that Andrew's slog satisfies slog(e^z)=slog(z)+1 e.g. for \( z,e^z\in \{\zeta: |\zeta| <|A|\} \) then it must have a singularity at A.

---

of course for 'every' fixed point !

i know that silly :p

but the smallest is of course closest to the origin , so that is the one i considered , since i wanted the radius ( which is the distance to the origin )

i completely agree with you Sir Bo ( or whatever you like to be called :p )

but now seriously.

andrew nowhere mentioned " branches " or even the complex plane in his paper.

well , at least not in the pdf's of his website.

i personally feel like those branches are one of the most important topics in tetration debate.

*** warning : highly speculating below ***

as for your * everywhere * , in general i think you are correct , but maybe some bases do satisfy that ' almost everywhere ' ?

i think the only exceptions for some bases are numbers a_i

sexp(slog(a_i) + x ) = a_i for positive real x.

however thats quite ' alot ' ( uncountable and dense )

***

i often like to consider invariant and branches as ' inverses '

like

exp( x + 2pi i ) <-> log(x) + 2 pi i

following that ' philosophy '

the branch(es) of slog(z) we are looking for are the invariants of sexp(z)


just some quick musings , plz forgive any blunders , im an impulsive poster with little time

also forgive me if this has been discussed before , like eg many years ago , im only here since a few months.


regards

tommy1729