(09/17/2010, 10:32 PM)sheldonison Wrote: Mike, thanks for your detailed description. So, am I correct that your approximating the sexp(z) with a long periodic function, to approximate switching from the space domain to the frequency domain. Then, to get a more accurate version of the sexp(z), you take sexp(z+1)=e^sexp(z), using Faà di Bruno's formula? I mean, high level overview, is that more or less correct?
Yes on the approximation, no on the iteration. Rather, I use the periodic approximation because it is possible to solve its continuum sum
\( f(z) = \sum_{n=0}^{z-1} \mathrm{TetApprox}_b(n) \)
in a way that has a wider regime of convergence than Faulhaber's formula does (I describe this in the initial post.) -- Fourier series are more amenable to being continuum summed than power series.
Then the iteration is
\( \mathrm{NewTetApprox}_b(z) = \frac{1}{\mathrm{TetApprox}_b(0)} \int_{-1}^{p_P(z)} \log(b)^w \exp_b\left(\sum_{n=0}^{w-1} \mathrm{TetApprox}_b(n)\right) dw \)
where \( p_P(z) \) is the periodizing function (see the initial post again.).
(09/17/2010, 10:32 PM)sheldonison Wrote: So, as to the general applicability to complex domains, and getting other solutions then we're used to seeing -- here's my intuitive feeling. Any analytic solution, especially one with limiting behavior matching the super function, is a 1-cyclic transformation, via theta(z), of the superfunction.
f(z)=regularsuper(z+theta(z)).
To preserve the behavior at either +I*infinity or -I*infinity, then theta(z) must go to zero at +I or -I infinity. Then, I believe it is most likely that there will be a singularity in theta(z), and we usually get around the singularity with a schwarz transformation about the real axis. Maybe something similar applies in the complex domain. The other possibility, is that theta(z), when wrapped around the unit circle, is an annular Laurent series, with convergence between the two radius's of singularities. This hasn't been explored (to my knowledge), but I think that's what would be going on when converting between the regularsuperf_sqrt(2)(z) and the "regular iteration of log" developed from the other fixed point for sqrt(2). That's what I'd like to explore when I have time.
Yes, in theory one can turn any solution to any other by a 1-cyclic transform, though due to multivaluedness of the necessary transforms, spurious branches may be generated. E.g. if you consider the 1-cyclic transform for taking the regular iteration at one of the conjugate fixed points for a real base \( b > e^{1/e} \) to turn it into the real-valued tetrational function for that base, every integer is a branch point, I think.