06/26/2009, 10:51 PM

(08/19/2007, 09:50 AM)bo198214 Wrote: I just verified numerically that the superlog critical function (originally defined on ) for base satisfies

for all .

So it is quite sure that the piecewise defined slog is also analytic.

Congratulation Andrew!

(However once someone has to prove this rigorously and also compute the convergence radius.)

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)

maybe i missed it , but i didnt see that mentioned.

also this makes me doubt - especially considering that for every base b slog should 'also' ( together with the oo value at the fixed point A mentioned above ) have a period ( thus abs ( slog(A + period) ) = oo too ! ) - , however thats just an emotion and no math of course ...

( btw the video link mentioned in this thread doesnt work for me bo , maybe it isnt online anymore ? )