(08/15/2009, 12:54 AM)jaydfox Wrote: I think a similar effect is at work with the base change formula. For any finite n, we could in principle locate singularities by careful analysis, and as n increases, these singularities will get closer and closer to the real line. Yet they will also be less and less perceptible, unless you can manage to get really, really close to one.In thinking about it, the singularities are trivial to find. For a=eta, b=e, anywhere that the exp^[n-2](x) is equal to -1, we will have a singularity. The double logarithm of the double exponentiation, in the respective bases, will be 0. Regardless of branching, log(0) is always a singularity. And notice that, after getting this logarithmic singularity, we then proceed to take several more iterated logarithms, which would effectively diminish these singularities. As n increases, they would become very "skinny", almost undetectable when numerical precision is taken into account.
Furthermore, if we increase n by 1, then this singularity disappears, because exponentiating -1 gives 1/e, and the double logarithm of the double exponentiation gives e+1. Iterated logarithms in base eta should now resolve properly, avoiding a singularity.
This makes me wonder, then: for any given n, there are singularities near the real line, and as n increases, these singularities get arbitrarily close. On the other hand, there aren't any "fixed" singularities, at least not of this trivial variety. The singularities themselves become more and more "insubstantial", so I'm wondering if perhaps convergence in the complex plane is indeed possible?
~ Jay Daniel Fox