08/12/2009, 04:13 AM
(08/11/2009, 04:30 PM)jaydfox Wrote: I'm not sure if the limit exists in the normal sense (i.e., it's well-defined for real numbers, but as you say, not for non-real numbers).
However, rather than two distinct points x and x+d*i (where d goes to 0), consider a line segment L defined between those two points. As exp_b(x) is entire, the image exp_b(L) will be continuous, as will be exp_b(exp_b(L)), etc. Thus, no matter how many times this curve wraps around the origin in some bizarre fractal nature, there is always a well-defined "path" back to the real line.
We can then iteratively perform the logarithm log_a(x), which has branches. We start at the real endpoint of exp_b^[ok](L) as we're performing the logarithms, so that when we wrap around the origin, we will always "know" which branch of the logarithm to use.
When all is said and done, we will arrive at the correct location, with no ambiguity....
Agreed, for each value of "k", there is one and only one unambiguous solution, and that helps a lot... I'm working on some ideas, and calculations, as well as formalizing my thoughts on the limit equations, as k increases, in the complex plane.
- Shel