(08/30/2009, 04:32 AM)bo198214 Wrote: Why does a non-real development point not work (according to Jay) for b>eta?
It does work. But like the (0 < b < 1) case, the result is a complex-valued tetrational along the real axis. This is a huge difference between regular iteration and intuitive iteration. I would like to refer to this thread in which ONLY the purple region is where regular iteration and intuitive iteration would appear numerically equal. Outside the blue region, I believe intuitive iteration is undefined. Outside the red region, regular iteration still works, but its "justification" does not hold any more.
The "justification" is this: in order to express the coefficients of regular iteration in terms of \( f_k \) (the Taylor coefficients), then you must use finite geometric sums (as indicated in the thread above), however, to express the fixed points, then you must use infinite geometric sums, which have a different domain. The finite domain corresponds to the complex plane, and the infinite domain corresponds to \( |\log({}^{\infty}b)| < 1 \).