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 Carlson's theorem and tetration mike3 Long Time Fellow Posts: 368 Threads: 44 Joined: Sep 2009 08/19/2010, 08:43 AM (This post was last modified: 08/19/2010, 08:44 AM by mike3.) Hi. I found the following easy uniqueness theorem that characterizes the regular tetrational of the base $b = \eta = e^{1/e}$, and perhaps also the whole regular tetrational (with attracting fixed point) (though base-$\eta$ is particularly interesting since it seems that both the regular and non-regular (i.e. Kneser's, etc.) method approach the same tetrational at this base.), with some modification (of condition 4). The conditions are very simple and easy. Theorem: There is a unique complex function $F(z)$ satisfying 1. $F(z + 1) = \eta^{F(z)}$ 2. $F(0) = 1$ 3. $F(z)$ is holomorphic in the entire cut plane with real $z \le -2$ removed, 4. $\lim_{r \rightarrow \infty} F(re^{i\theta}) = e$ for all $\theta \ne (2n+1)\pi, n \in \mathbb{Z}$ (could be weakened simply to saying the limit exists) Proof: We use what is called Carlson's theorem. Assume $F(z)$ and $G(z)$ are two different solutions of the above conditions. Then consider $H(z) = F(z) - G(z)$. Carlson's theorem says if this function, in the right half-plane, vanishes at every nonnegative integer, and is bounded asymptotically by $O(e^{u|z|})$ for some $u < \pi$, then it vanishes everywhere. Conditions 1 and 2 imply that $F$ and $G$ are equal at every nonnegative integer, thus $H$ is zero there, and condition 4 implies the asymptotic bounding (if two functions $f(x)$ and $g(x)$ have a limit at a given point, then the difference $f(x) - g(x)$ does as well) because every function decaying to a fixed value will be bounded in the asymptotic by any exponential (can give proof here if needed to fill this out.). Thus $H(z) = 0$, so $F(z) = G(z)$ and we are done. QED. See: http://mathworld.wolfram.com/CarlsonsTheorem.html Indeed, this says that condition 3 can be weakened to just holomorphism in the right half-plane ($\Re(z) \ge 0$) and condition 4 to the function being of exponential type of at most $\pi$ in that same right half-plane. The modifications also provide the theorem characterizing the regular tetrationals for $1 < b < \eta$. « Next Oldest | Next Newest »

 Messages In This Thread Carlson's theorem and tetration - by mike3 - 08/19/2010, 08:43 AM RE: Carlson's theorem and tetration - by tommy1729 - 08/19/2010, 10:56 PM RE: Carlson's theorem and tetration - by bo198214 - 08/20/2010, 12:21 PM RE: Carlson's theorem and tetration - by mike3 - 08/20/2010, 08:35 PM RE: Carlson's theorem and tetration - by tommy1729 - 08/21/2010, 08:36 AM RE: Carlson's theorem and tetration - by mike3 - 08/21/2010, 08:08 PM RE: Carlson's theorem and tetration - by bo198214 - 08/22/2010, 05:12 AM RE: Carlson's theorem and tetration - by sheldonison - 08/20/2010, 05:08 PM RE: Carlson's theorem and tetration - by mike3 - 08/20/2010, 08:26 PM

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