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 \( z,e^z\in \{\zeta: |\zeta| <|A|\} \) 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