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 Real and complex behaviour of the base change function (was: The "cheta" function) bo198214 Administrator Posts: 1,386 Threads: 90 Joined: Aug 2007 08/23/2009, 12:17 PM (This post was last modified: 08/23/2009, 12:18 PM by bo198214.) (08/21/2009, 10:49 PM)jaydfox Wrote: Given a starting and ending point, it is possible to create a path between them that starts and ends in the principal branch, even if it must sometimes leave the principal branch. I'm working on pictures of what I mean, so in the meantime, hopefully you can picture what I mean.Normally not, but in this case, yes Quote:This has me concerned that the base-change formula is truly undefined for non-real values, I was thinking about a different aproach: I would say all singularities of all $f_n$ are bounded. So out there in the complex plane are points $p$ which are distant from all singularities of all $f_n$. Perhaps one can define the value of each $f_n$ at $p$ by using a path that has all singularities to the right side, respectively. Then one would show that $f_n$ converges uniformly in a neighborhood of $p$. The limit function $f$ in the neighborhood of $p$ is necesarily again holomorphic there. From *there* we continue the function $f$ to the real line. I guess $f$ has only singularities on the real line and at the fixed points of $\exp_b$. Then it would turn out that $f$ has an *asymptotic* power series development (for a certain sector of approach) at all points of the (upper) real axis. Which though of course has zero convergence radius. « 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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