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 base holomorphic tetration bo198214 Administrator Posts: 1,386 Threads: 90 Joined: Aug 2007 11/12/2009, 03:23 PM actually I think that the regular tetration at the lower fixed point is not continuable to bases $>e^{1/e}$. Actually I think the Shell-Thron region is the domain of holomorphy, every point on the boundary is singular. This became clear to me some days ago: 1. The regular iteration depends on the derivative of $\exp_b$ at the fixed point. 2. If $a$ is the fixed point then the derivation is $\log(a)$ 3. The fixed point is given by $a = \exp(-W(-\log(b)))$, or $\log(a)=-W(-\log(b))$. 4. The problematic value of the derivation at the fixed point is $|\log(a)|=1$. In this case $\exp_b^{\circ t}$ is not analytic at $a$. I guess this implies that $b\mapsto\exp_b^{\circ t}(1)$ is not analytic at $b$. 5. $|\log(a)|=1$ is exactly at those $b$ in the image of the unit circle under of the inverse of $b\mapsto -W(-\log(b))$, i.e. under $z\mapsto\exp(z \exp(-z))$ which is just the boundary of the Shell-Thron region. « 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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