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 Universal uniqueness criterion? bo198214 Administrator Posts: 1,389 Threads: 90 Joined: Aug 2007 07/06/2009, 09:20 AM (This post was last modified: 07/06/2009, 09:30 AM by bo198214.) (07/06/2009, 08:56 AM)bo198214 Wrote: One needs to show that then there is always a pair of points $c_1=\gamma(t_1)$ and $c_2=\gamma(t_2)$ with equal real part and with $\Im(c_2)-\Im(c_1)=2\pi$. This is equivalent to that $\gamma$ and $\gamma+2\pi i$ intersect. If $\gamma$ only extend to the right, i.e. $\Re(\gamma(t))>\Re(a)\forall t\in (0,1)$, then this is a consequence of the Jordan curve theorem. We have $\Im(a) < \Im(a+2\pi i) < \Im(b) < \Im(b+2\pi i)$. The closed Jordan curve $[a,b]\cup \gamma$ has the point $\gamma(0+\epsilon)+2\pi i\approx a+2\pi i$ in its interior and the point $\gamma(1-\epsilon)+2\pi i\approx b+2\pi i$ in its exterior. Hence there must be an intersection of $\gamma+2\pi i$ and $\gamma$ (as $\gamma+2\pi i$ does not pass [a,b] for $t\in (0,1)$.) « 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

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