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 base holomorphic tetration mike3 Long Time Fellow Posts: 368 Threads: 44 Joined: Sep 2009 11/06/2009, 04:15 AM I've also been toying with this, too. It appears, however, that it continues to real values, not complex values, for $b > e^{1/e}$. Consider the following “regular iteration” limit formula: http://math.eretrandre.org/tetrationforu...10#pid1610 $^{z} b = \exp_b^z(1) = \lim_{n \rightarrow \infty} \log_b^n \left(F\left(1 - \log(F)^z\right) + \log(F)^z \exp_b^n(1)\right)$ where $F = \frac{-W\left(-\log(b)\right)}{\log(b)}$ is the attracting fixed point, i.e. $^{\infty} b$. There's also a series formula, apparently (with respect to "w" in $\exp_b^z(w)$, so setting w = 1 yields the tetrational), but I haven't yet figured out how one is supposed to evaluate the general coefficient. How can that be done? To test the analytic continuation, we can use the Cauchy integral: if $f(z)$ is analytic, then we derive the powerseries coefficients at $z_0$ via $a_n = \frac{1}{2\pi i} \oint \frac{f(z)}{(z - z_0)^n} dz$ and $f(z) = \sum_{n=0}^{\infty} a_n (z - z_0)^n$. This seems to provide a more efficient algorithm for the recovery of the coefficients, than straight numerical differentiation from the difference quotient (which seems to require more rapidly-escalating levels of numerical precision). We can now choose some $z_0$ close to $e^{1/e}$, set $f(z) = {}^t z$ for some fractional tower $t$ where $^{t} z$ is obtained from the regular formula, and a path that encircles it, but does not leave the kidneybean ("Shell-Thron" region) of convergence, e.g. a small circle round the point. Then, by increasing n, we obtain the Taylor coefficients. For $f(z) = {}^{1/2} z$, expanded about $z_0 = 1.42$, using a circle of radius 0.01, we get the following estimates for the first 25 coefficients: Code:a_0 ~ 1.24622003310 a_1 ~ 0.447921100148 a_2 ~ -0.194428566238 a_3 ~ 0.143167873861 a_4 ~ -0.144967399774 a_5 ~ 0.182159301224 a_6 ~ -0.263407426426 a_7 ~ 0.417098477762 a_8 ~ -0.702204147888 a_9 ~ 1.23535078363 a_10 ~ -2.24705897139 a_11 ~ 4.19678269601 a_12 ~ -8.00913024091 a_13 ~ 15.5621057277 a_14 ~ -30.7029302261 a_15 ~ 61.3746543453 a_16 ~ -124.093757768 a_17 ~ 253.427621734 a_18 ~ -522.152429631 a_19 ~ 1084.31773542 a_20 ~ -2267.61147731 a_21 ~ 4772.09902388 a_22 ~ -10098.4528601 a_23 ~ 21464.8938685 a_24 ~ -45852.6753827 For $z = 1.5 = \frac{3}{2}$, we can use this get $^{\frac{1}{2}} \left(\frac{3}{2}\right) \approx 1.28087727794$, which is real, not complex. How does that agree with other methods of tetration for bases greater than $e^{1/e}$? This series should have radius of convergence 0.42, determined by the distance to the nearest singularity/branchpoint, which is at z = 1. I'm not sure of a formal proof of the "continuability", though one approach may be to try and differentiate the regular iteration formula, then prove that the limit of the derivative as $b \rightarrow e^{1/e}$ converges -- in order for it to switch to non-real complex values as $b = e^{1/e}$ is passed, that point would have to be some sort of singularity, like a branch point, and so the function would not be differentiable there, and if it is, then that is not the case. I'll see if maybe I can get some graphs on the complex plane but calculating the regular iteration is a bear as it requires lots of numerical precision, at least for the limit formula. Maybe that series formula would be better? « Next Oldest | Next Newest »

 Messages In This Thread base holomorphic tetration - by bo198214 - 11/05/2009, 02:12 PM RE: base holomorphic tetration - by Base-Acid Tetration - 11/05/2009, 10:50 PM RE: base holomorphic tetration - by mike3 - 11/06/2009, 04:15 AM RE: base holomorphic tetration - by mike3 - 11/06/2009, 11:58 AM RE: base holomorphic tetration - by bo198214 - 11/06/2009, 12:12 PM RE: base holomorphic tetration - by mike3 - 11/06/2009, 09:16 PM RE: base holomorphic tetration - by bo198214 - 11/06/2009, 11:29 PM RE: base holomorphic tetration - by mike3 - 11/07/2009, 12:23 AM RE: base holomorphic tetration - by bo198214 - 11/07/2009, 08:17 AM RE: base holomorphic tetration - by mike3 - 11/07/2009, 08:21 AM RE: base holomorphic tetration - by bo198214 - 11/07/2009, 09:55 AM RE: base holomorphic tetration - by bo198214 - 11/07/2009, 04:47 PM RE: base holomorphic tetration - by bo198214 - 11/08/2009, 05:39 PM RE: base holomorphic tetration - by mike3 - 11/08/2009, 08:27 PM RE: base holomorphic tetration - by mike3 - 11/08/2009, 08:25 PM RE: base holomorphic tetration - by bo198214 - 11/08/2009, 08:44 PM RE: base holomorphic tetration - by mike3 - 11/08/2009, 09:51 PM

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