09/18/2010, 01:38 AM
mike, did not you try a periodic function not with arbitrary large period, but with the exact period of tetration, i.e.
Numerical algorithm for Fourier continuum sum tetration theory

09/18/2010, 01:38 AM
mike, did not you try a periodic function not with arbitrary large period, but with the exact period of tetration, i.e.
(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 FaĆ 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 \( f(z) = \sum_{n=0}^{z1} \mathrm{TetApprox}_b(n) \) 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 \( \mathrm{NewTetApprox}_b(z) = \frac{1}{\mathrm{TetApprox}_b(0)} \int_{1}^{p_P(z)} \log(b)^w \exp_b\left(\sum_{n=0}^{w1} \mathrm{TetApprox}_b(n)\right) dw \) 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 1cyclic transformation, via theta(z), of the superfunction. Yes, in theory one can turn any solution to any other by a 1cyclic transform, though due to multivaluedness of the necessary transforms, spurious branches may be generated. E.g. if you consider the 1cyclic 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 realvalued tetrational function for that base, every integer is a branch point, I think.
09/18/2010, 04:12 AM

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