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 Extension of tetration to other branches mike3 Long Time Fellow Posts: 368 Threads: 44 Joined: Sep 2009 10/25/2009, 02:14 AM (This post was last modified: 10/25/2009, 03:29 AM by mike3.) (10/24/2009, 08:13 PM)bo198214 Wrote: (10/24/2009, 08:01 PM)mike3 Wrote: Actually it does seem to converge. The problem is that it seems to converge to the same value for every z in such cases. I.e., converging to a constant function. Interesting! Hmm. Sounds like it's time for a graph... I'll see if I can prepare a 2D one looking at the values of various branches on the real axis (can't do 3D with anything I've got). (10/24/2009, 08:13 PM)bo198214 Wrote: Quote: There are uncountably many such limit values, yet as constant functions they are "analytically incompatible" (is that a real term?) with the function (you can't analytically continue a constant function to tetration!) Well, each constant fixed point of b^x is a tetration! I.e. it satisfies c(z+1)=b^c(z). Does it converge to fixed points? But it's a constant function, so it cannot be interpreted as analytic continuation of the specific function $\mathrm{tet}_b(z)$ to another branch (think about the problem in "reverse": how would you analytically continue from this constant function to a non-constant one? You can't). And not all branches of $\mathrm{tet}_b(z)$ satisfy $\mathrm{tet}_b(z+1) = b^{\mathrm{tet}_b(z)}$. Consider the branch $\mathrm{tet}_b_{[1]}(z) = \mathrm{tet}_b(z) + \omega$. We get $b^{\mathrm{tet}_b_{[1]}(z)} = b^{\mathrm{tet}_b(z) + \omega} = b^{\mathrm{tet}_b(z)} * b^{\omega} = b^{\mathrm{tet}_b(z)} * 1 = b^{\mathrm{tet}_b(z)} = \mathrm{tet}_b(z+1) \ne \mathrm{tet}_b(z+1) + \omega = \mathrm{tet}_b_{[1]}(z+1)$. Note that it takes us back to the principal branch. The equation $\mathrm{tet}_b(z+1) = b^{\mathrm{tet}_b(z)}$ seems to only hold for all $z$ when using the principal branch (though there may be some freedom in the choice of cut of course), if we are interpreting the symbol $\mathrm{tet}_b(z)$ as a specific single-valued branch. As you can see above, though, if we interpret it in a multivalued sense, that values on some branches of $\mathrm{tet}_b(z+1)$ equal $b^{\mathrm{tet}_b(z)}$ for values on some branches of $\mathrm{tet}_b(z)$, then it is true, as can be seen from the example I just showed. The situation is similar to that with $\log(z)$ and $\exp(z)$: $\log(\exp(z)) = z$ for some branch of log for any given $z$ but which branch that is will depend on what $z$ is. "Multivalued functions" are funny things, you know? « Next Oldest | Next Newest »

 Messages In This Thread Extension of tetration to other branches - by mike3 - 10/24/2009, 08:27 AM RE: Extension of tetration to other branches - by bo198214 - 10/24/2009, 09:54 AM RE: Extension of tetration to other branches - by mike3 - 10/24/2009, 08:01 PM RE: Extension of tetration to other branches - by bo198214 - 10/24/2009, 08:13 PM RE: Extension of tetration to other branches - by mike3 - 10/25/2009, 02:14 AM RE: Extension of tetration to other branches - by bo198214 - 10/25/2009, 07:43 AM RE: Extension of tetration to other branches - by mike3 - 10/25/2009, 09:33 AM RE: Extension of tetration to other branches - by bo198214 - 10/25/2009, 11:26 AM RE: Extension of tetration to other branches - by mike3 - 10/25/2009, 08:47 PM RE: Extension of tetration to other branches - by bo198214 - 10/25/2009, 09:22 PM RE: Extension of tetration to other branches - by mike3 - 10/25/2009, 10:56 PM RE: Extension of tetration to other branches - by bo198214 - 10/26/2009, 04:28 AM RE: Extension of tetration to other branches - by mike3 - 10/26/2009, 04:41 AM RE: Extension of tetration to other branches - by bo198214 - 10/26/2009, 05:07 PM RE: Extension of tetration to other branches - by mike3 - 10/27/2009, 08:59 PM RE: Extension of tetration to other branches - by bo198214 - 10/28/2009, 07:42 AM

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