09/04/2009, 06:42 AM
(08/26/2009, 04:01 PM)tommy1729 Wrote: The idea is to arrive at the same superlogaritms by choosing an appropriate ( of possibly many ) solution for development points\( x_0 \) and \( x_1 \) .
if that is possible for all the points on the (real) interval [\( x_0 \),\( x_1 \)]
then i assume THAT PARTICULAR SLOG has a radius at least \( x_1 - x_0 \) when developped at \( x_0 \).
I have thought about this, and I have a discussion about this too.
I'd like to write up the discussion here, but I'd rather just upload my notes. Mind you these notes are not final at all, but rather unsolicited ramblings of a curious mind. The section I'm talking about is draft/Robbins_paper3.pdf (pages 10-16), which is also what I was referring to here, when I said a "detailed investigation of the coefficients".
I noticed that if you tabulate the coefficients of z (Taylor coefficients of the super-logarithm) with the coefficients of ln(a) in the numerator and denominator of each coefficient, then the diagonals of this tabulation make interesting patterns. So I started finding expressions for the diagonals of this table, and got stuck on the 5th diagonal. That is why I haven't really developed this idea much farther.
If any of the expressions can be proven with induction or the like, then it would form the basis of a proof for the base-1 slog and base-\( \infty \) slog (which only require the first row and the last row). I thought that maybe if we had expressions for all coefficients as functions of n, then perhaps we could find a closed form for \( \text{slog}_e'(0) \) or something.
Andrew Robbins