• 1 Vote(s) - 5 Average
• 1
• 2
• 3
• 4
• 5
 Universal uniqueness criterion? Kouznetsov Fellow Posts: 151 Threads: 9 Joined: Apr 2008 06/24/2009, 05:02 AM (This post was last modified: 06/24/2009, 05:07 AM by Kouznetsov.) (06/23/2009, 09:28 PM)Tetratophile Wrote: @Kouznetsov: how are the other tetrations relevant to our discussion of uniqueness?I expect, all "other" tetrations have additional singularities and cutlines. Quote:@Kouznetsov: Also I just have a quick question to ask about tetration: What is the mathematical reason that the tetrational approaches the fixed points $L$ or $L*$ as you go to $\pm i \infty$? I think instead, that $\lim_{x \rightarrow -\infty} x\pm ci = L\mathrm$ or $L*$, where c is a nonzero real number, should be the limit that corresponds to the fixed point. You have no need to type "instead", both should be correct. The reason in not so mathematical, but computational: it is easier to work with a function, holomorphic in wide range, and create all other tetrations, just modifying the argument, if necessary. In principle, "all animals are equal"; but as soon as you begin to plot graphics, it happens, that "some animals are more equal than others". For me, "more equal" are animals with wide range of holomorphizm. Quote: .. to get from 1 to a non-real number by iteration of exp, you need complex iteration (real iterations always give real numbers) Yes. Quote:But then since $L,L*$ are repelling with respect to the exponential, you need infinite negative iterations of exp (positive iterations of log). Yes. Quote:If you infinitely iterate log on a nonreal number, you get closer to L, why isn't this reflected in the tetrational graph? ? In the left hand side of the plot of tetration, values approach $L$ in the upper halfplane and $L^*$ in the lower halfplane; this corresponds to the graphic you supply. If you mean our discussion with Bo about a super-exponential that approaches $L_1$ instead of $L=L_0$, then it is not correct: In some region, at the iteration of logarithm, we have to add $2 \pi i$ in the upper halfplane and $-2 \pi i$ in the lower halfplane. The question is, wether we can match them for large enough real part of the argument. I am not yet successful to construct or plot such a function. I expect such function to have two additional horizontal cutlines, going to -infinity. Quote:Bo said something like log is not in the initial region anymore, could you clarify that for me? As I understand, the initial region is $ C=\{z \in \mathbf{C} : \Re(z)\ge 1 , \|z|\le |L| \}$ if some point $z$ is inside the initial region (not at the margin), then $\log(z)$ in outside the initial region. The same about $\log(z)+2 \pi i$ . Is this that you were asking for? Quote:I have attached a schematic diagram to represent my reasoning: how does the tetrational for base b>exp(1/e) actually behave at large values of real part of non-real arguments? There are pictures at http://www.ils.uec.ac.jp/~dima/PAPERS/2009fractae.pdf The tetrational shows complicated quasi-periodic behavior. « Next Oldest | Next Newest »

 Messages In This Thread Universal uniqueness criterion? - by bo198214 - 05/21/2008, 06:24 PM RE: Universal uniqueness criterion? - by andydude - 05/22/2008, 05:19 AM RE: Universal uniqueness criterion? - by andydude - 05/22/2008, 06:42 AM RE: Universal uniqueness criterion? - by bo198214 - 05/22/2008, 11:25 AM RE: Universal uniqueness criterion? - by andydude - 05/22/2008, 03:11 PM RE: Universal uniqueness criterion? - by bo198214 - 05/22/2008, 05:55 PM RE: Universal uniqueness criterion? - by bo198214 - 05/23/2008, 12:07 PM RE: Universal uniqueness criterion? - by Gottfried - 06/25/2008, 06:15 AM Uniqueness of analytic tetration - by Kouznetsov - 09/30/2008, 07:58 AM RE: Uniqueness of analytic tetration - by bo198214 - 09/30/2008, 08:17 AM RE: Universal uniqueness criterion? - by bo198214 - 10/04/2008, 11:19 PM RE: Universal uniqueness criterion? - by Kouznetsov - 10/05/2008, 12:22 AM RE: Universal uniqueness criterion? - by Kouznetsov - 06/19/2009, 08:45 AM RE: Universal uniqueness criterion? - by Base-Acid Tetration - 06/19/2009, 02:04 PM RE: Universal uniqueness criterion? - by bo198214 - 06/19/2009, 02:51 PM RE: Universal uniqueness criterion? - by Base-Acid Tetration - 06/19/2009, 04:19 PM RE: miner error found in paper - by bo198214 - 06/19/2009, 04:53 PM i don't think it will work - by Base-Acid Tetration - 06/19/2009, 05:17 PM RE: Universal uniqueness criterion? - by bo198214 - 06/19/2009, 06:25 PM RE: Universal uniqueness criterion? - by Base-Acid Tetration - 06/19/2009, 06:27 PM RE: Universal uniqueness criterion? - by bo198214 - 06/19/2009, 07:59 PM RE: Universal uniqueness criterion? - by Base-Acid Tetration - 06/20/2009, 02:01 PM RE: Universal uniqueness criterion? - by bo198214 - 06/20/2009, 02:10 PM RE: Universal uniqueness criterion? - by Base-Acid Tetration - 06/23/2009, 02:39 PM RE: Universal uniqueness criterion? - by Kouznetsov - 06/23/2009, 05:46 PM RE: Universal uniqueness criterion? - by Base-Acid Tetration - 06/23/2009, 09:28 PM RE: Universal uniqueness criterion? - by Kouznetsov - 06/24/2009, 05:02 AM RE: Universal uniqueness criterion? - by Base-Acid Tetration - 07/04/2009, 11:17 PM RE: Universal uniqueness criterion? - by Kouznetsov - 07/05/2009, 08:28 AM RE: Universal uniqueness criterion? - by bo198214 - 07/05/2009, 06:54 PM

 Possibly Related Threads... Thread Author Replies Views Last Post Semi-exp and the geometric derivative. A criterion. tommy1729 0 1,415 09/19/2017, 09:45 PM Last Post: tommy1729 A conjectured uniqueness criteria for analytic tetration Vladimir Reshetnikov 13 12,028 02/17/2017, 05:21 AM Last Post: JmsNxn Uniqueness of half-iterate of exp(x) ? tommy1729 14 15,959 01/09/2017, 02:41 AM Last Post: Gottfried Removing the branch points in the base: a uniqueness condition? fivexthethird 0 1,641 03/19/2016, 10:44 AM Last Post: fivexthethird [2014] Uniqueness of periodic superfunction tommy1729 0 2,053 11/09/2014, 10:20 PM Last Post: tommy1729 Real-analytic tetration uniqueness criterion? mike3 25 22,753 06/15/2014, 10:17 PM Last Post: tommy1729 exp^[1/2](x) uniqueness from 2sinh ? tommy1729 1 2,358 06/03/2014, 09:58 PM Last Post: tommy1729 Uniqueness Criterion for Tetration jaydfox 9 11,660 05/01/2014, 10:21 PM Last Post: tommy1729 Uniqueness of Ansus' extended sum superfunction bo198214 4 6,970 10/25/2013, 11:27 PM Last Post: tommy1729 A question concerning uniqueness JmsNxn 3 5,807 10/06/2011, 04:32 AM Last Post: sheldonison

Users browsing this thread: 1 Guest(s)