06/28/2009, 12:08 AM

(06/27/2009, 09:39 AM)bo198214 Wrote: [ -> ][quote='tommy1729' pid='3446' dateline='1246053079']Its not only valid for Andrew's slog but for every slog and also not only for the smallest but for every fixed point.

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)

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 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