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 complex base tetration program sheldonison Long Time Fellow Posts: 641 Threads: 22 Joined: Oct 2008 03/16/2012, 07:42 PM (This post was last modified: 03/19/2012, 10:52 PM by sheldonison.) Quote:... The experiment, is to develop taylor series for all of the coefficients of sexp_b(z), developed around complex sexp(z) in the neighborhood of base=2. The conundrum, which I'm beginning to be able to explain, is that the numerical results are way too good, and its hard to see the effects of the branch point at eta. In fact, numerically convergence seems limited by the more distant singularity for sexp_b(z), at b=1, and not at all by the closer singularity at eta.... I have a good way of explaining why the branch point at $\eta=\exp(1/e)$ is so slight. The graph below is for base $\eta-0.25\approx1.195$ Start with this formula for the merged sexp(z), after rotating counterclockwise around eta. Of course, there is a corresponding formula in terms of the lower repelling superfunction too, but the upper attracting superfunction is of primary interest here. Note that the attracting superfunction is real valued at the real axis, however, the theta(z) means that the sexp(z) has a small negative imaginary component at the real axis, for the counterclockwise sexp(z), and a small positive imaginary component for the clockwise sexp(z). $\text{sexp}_{+\pi}(z)=\text{superf_u(z+\theta_u(z))$ (counterclockwise) $\text{sexp}_{-\pi}(z)=\text{superf_u(z+\overline{\theta_u(z)})$ (clockwise)     Here, the graph on the left is for superf from the attracting fixed point, and the two graphs in the middle show the merged sexp(z) from both fixed points, rotating counter clockwise around eta, vs rotating clockwise around eta. The sexp(z) function looks much more like the attracting fixed point superfunction, than the repelling fixed point superfunction on the right. In fact, for $b=\eta-0.25$ at the real axis, $\theta_u(z)$ is an analytic function, dominated almost entirely by the first overtone, and very nearly equal to zero, with a peak negative imaginary component of about 7.06*10^-13. The graph below shows the 1-cyclic $\theta_u(z)$ function at the real axis, with its negative imaginary component in magenta, where the attracting superfunction(0) has been centered so that superfunction(0)=1.     $\theta_u(z)$ is very different than $\theta_l(z)$ which is for the repelling superfunction, which has a singularity at the real axis. So the clockwise versus counterclockwise functions are both nearly identical at the real axis, since both are very nearly identical to the attracting superfunction, depending on how small $\theta_u(z)$ is. First of all, $\theta_u(z)$ by definition converges to a constant as z goes to imag(infinity). $\theta_u(z)=\sum_{n=0}^{\infty}t_n\times\exp(z\times 2n\pi i)$ It seems logical to assume that somewhere between the Period of the attracting superfunction, and the period of the repelling superfunction, $\theta_u(z)$ will have its singularity, and this matches the computational results. Also, the period of the attracting and repelling superfunctions are both functions of L. $\text{period}=2\pi i / \log(\log(L))$. In the neighbordhood of $\eta$, L(b) itself is an analytic function (real valued for b

 Messages In This Thread complex base tetration program - by sheldonison - 02/29/2012, 10:28 PM RE: complex base tetration program - by Kouznetsov - 02/29/2012, 11:49 PM RE: complex base tetration program - by sheldonison - 03/01/2012, 12:09 AM RE: complex base tetration program - by sheldonison - 03/01/2012, 10:34 AM RE: complex base tetration program - by Kouznetsov - 03/01/2012, 12:04 PM RE: complex base tetration program - by sheldonison - 03/01/2012, 03:47 PM RE: complex base tetration program - by sheldonison - 03/02/2012, 09:20 AM RE: complex base tetration program - by sheldonison - 03/07/2012, 12:08 AM RE: complex base tetration program - by sheldonison - 03/07/2012, 06:08 PM RE: complex base tetration program - by sheldonison - 03/08/2012, 09:51 PM RE: complex base tetration program - by mike3 - 03/10/2012, 06:59 AM RE: complex base tetration program - by sheldonison - 03/10/2012, 08:53 PM RE: complex base tetration program - by mike3 - 03/11/2012, 12:27 AM RE: complex base tetration program - by sheldonison - 03/12/2012, 04:20 AM RE: complex base tetration program - by sheldonison - 03/16/2012, 07:42 PM RE: complex base tetration program - by Gottfried - 02/06/2016, 01:37 AM RE: complex base tetration program - by sheldonison - 02/06/2016, 04:04 PM RE: complex base tetration program - by Gottfried - 02/06/2016, 05:36 PM RE: complex base tetration program - by Gottfried - 02/07/2016, 05:27 AM RE: complex base tetration program - by sheldonison - 02/07/2016, 12:28 PM RE: complex base tetration program - by Gottfried - 02/07/2016, 01:25 PM RE: complex base tetration program - by Gottfried - 10/24/2016, 11:50 PM RE: complex base tetration program - by sheldonison - 10/26/2016, 08:01 AM RE: complex base tetration program - by Gottfried - 10/26/2016, 10:02 AM

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