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 Real and complex behaviour of the base change function (was: The "cheta" function) sheldonison Long Time Fellow Posts: 631 Threads: 22 Joined: Oct 2008 08/20/2009, 01:44 PM (This post was last modified: 08/20/2009, 06:12 PM by sheldonison.) (08/20/2009, 10:28 AM)bo198214 Wrote: Though currently I wonder whether these arbitrary close singularities indeed imply that the function is not analytic in any point. I mean there is a theorem that if a holomorphic function sequence converges locally uniformly (i.e. for each point there is a neighborhood where it converges uniformly) then the limit is again a holomorphic function (which is not true for just differentiable functions). However I dont think that the inverse statement is also true, that if a function sequence does not converge locally uniformly that then resulting function can not be holomorphic. For example a sequence of non-continuous functions can have a continuous function as a limit. Also Jay showed that the singularities gets milder with increasing n. So there maybe a very little tiny hope that the resulting function is analytic despite.Henryk, I finally caught up with your formula for the singularities -- its very helpful. I have a pretty good intuitive feel for the singularities for small values of k. $\log_\eta^{[n]}(e(1+2\pi i k))$ It appears there are about 500,000 or so n=3 singularities in the critical strip used by the base change equation (from 5.016 to 6.330). I would assume the n=4 singularities would be super-exponentially denser yet. k=1, 4.8688+0.5713i k=2, 5.0732+0.4586i k=3, 5.1734+0.4068i k=500,000 6.3301+0.0706i In looking at the singularities for small values of k, It seems that the function becomes undefined (or multi-valued?), once passing the neighborhood of the singularity. Each singularity is associated with a particular increment of the windings. Even ignoring the singularities associated with very large values of k, values of n>4, (which approach arbitrarily close to the real axis), can we continue the function for smaller of k, where n=4 as opposed to n=infinity? Typo correction: Actually I used n=3 in Henryk's equation, but to see the singularities in the "f" base change equation below requires using n=4. $f_n = \log_e^{[n]}\circ \exp_\eta^{[n]}$ I have not yet made the leap to understanding the singularities associated with larger values of k, and how they change the behavior of f, but I hope to do so. - Sheldon « Next Oldest | Next Newest »

 Messages In This Thread Real and complex behaviour of the base change function (was: The "cheta" function) - by bo198214 - 08/12/2009, 08:59 PM RE: Real and complex behaviour of the base change function (was: The "cheta" function) - by jaydfox - 08/15/2009, 12:54 AM RE: Real and complex behaviour of the base change function (was: The "cheta" function - by jaydfox - 08/15/2009, 05:00 PM RE: Real and complex behaviour of the base change function (was: The "cheta" function - by bo198214 - 08/15/2009, 05:36 PM RE: Real and complex behaviour of the base change function (was: The "cheta" function - by jaydfox - 08/15/2009, 06:40 PM RE: Real and complex behaviour of the base change function (was: The "cheta" function - by jaydfox - 08/15/2009, 07:13 PM RE: Real and complex behaviour of the base change function (was: The "cheta" function - by bo198214 - 08/15/2009, 09:44 PM RE: Real and complex behaviour of the base change function (was: The "cheta" function - by jaydfox - 08/15/2009, 10:40 PM RE: Real and complex behaviour of the base change function (was: The "cheta" function - by jaydfox - 08/15/2009, 10:46 PM RE: Real and complex behaviour of the base change function (was: The "cheta" function - by jaydfox - 08/15/2009, 11:02 PM RE: Real and complex behaviour of the base change function (was: The "cheta" function - by jaydfox - 08/15/2009, 11:20 PM RE: Real and complex behaviour of the base change function (was: The "cheta" function - by bo198214 - 08/16/2009, 11:15 PM RE: Real and complex behaviour of the base change function (was: The "cheta" function - by jaydfox - 08/16/2009, 11:38 PM RE: Real and complex behaviour of the base change function (was: The "cheta" function - by bo198214 - 08/17/2009, 08:50 AM RE: Real and complex behaviour of the base change function (was: The "cheta" function - by jaydfox - 08/17/2009, 12:07 PM RE: Real and complex behaviour of the base change function (was: The "cheta" function - by sheldonison - 08/17/2009, 04:01 PM RE: Real and complex behaviour of the base change function (was: The "cheta" function - by bo198214 - 08/17/2009, 04:30 PM RE: Real and complex behaviour of the base change function (was: The "cheta" function - by jaydfox - 08/17/2009, 05:26 PM RE: Real and complex behaviour of the base change function (was: The "cheta" function - by sheldonison - 08/18/2009, 04:37 AM RE: Real and complex behaviour of the base change function (was: The "cheta" function - by jaydfox - 08/17/2009, 05:47 PM RE: Real and complex behaviour of the base change function (was: The "cheta" function) - by bo198214 - 08/17/2009, 02:40 PM base change with decremented exponential - by bo198214 - 08/18/2009, 08:47 AM singularities of base change eta -> e - by bo198214 - 08/18/2009, 06:51 PM RE: singularities of base change eta -> e - by bo198214 - 08/20/2009, 10:28 AM RE: Does the limit converge in the complex plane? - by sheldonison - 08/13/2009, 12:49 AM RE: Does the limit converge in the complex plane? - by bo198214 - 08/13/2009, 07:17 AM RE: Does the limit converge in the complex plane? - by sheldonison - 08/13/2011, 10:32 AM RE: Does the limit converge in the complex plane? - by bo198214 - 08/13/2011, 06:33 PM RE: Does the limit converge in the complex plane? - by sheldonison - 08/13/2009, 06:48 PM

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