11/12/2009, 08:49 PM
(11/12/2009, 07:08 PM)mike3 Wrote: Then again, the hypothesis could also be incorrect. I'm a little dubious about point #4: how does \( \exp^{t}_b(w) \) being non-analytic at w = a imply it non-analytic in *b* when w = *1*?
Ya indeed, perhaps I get some stronger arguments later.
For illustration: I was considering the limit formula. Only if b is inside the STR the fixed point is attracting and hence reachable from 1 by repeated iteration.
If the base crossed the boundary one has to apply the limit formula for the inverse, and on the boundary the functions by approximating the fixed point from inside and from outside are different (no continuations of each other).
Quote: Can you do a graph of the derivative \( \frac{\partial}{\partial b} \exp^{t}_b(1) \) for the regular iteration, in the same way that you did the graph of the regular iteration you posted here earlier?
On the real axis you will not see anything special about the derivative I guess.
(11/12/2009, 07:37 PM)mike3 Wrote: I also noticed something in that paper where you describe the \( g \)-coefficients for the regular iteration. You have the iterating formula
\( g_n = \frac{1}{{f_1}^n - f_1} \left(f_n {g_1}^n - {g_1}{f_n} + \sum_{m=2}^{n-1} f_m {g^m}_n - g_m {f^m}_n\right) \).
where \( g = reg \)
But at n = 2, we have a sum from "2 to 1". How do you do that? Is it equal to the sum from 2 to 2, minus the value of the summand evaluated at 2 (like how the sum from 2 to 3 is that from 2 to 4 minus the value of the summand at 4)? Wouldn't that just be 0? If I assume so, then I get
yes, 0 is just the classical sense.
Quote:\( g_2 = \frac{\log(a)^2 \left(\log(a)^t\right)^2 - \log(a)^t \log(a)^2}{\log(a)^2 - \log(a)} \)
which disagrees with what you gave in your earlier post here.
No, it agrees. \( f_2 = \log(a)/2 \).
