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Real and complex behaviour of the base change function (was: The "cheta" function)
(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 Wink

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 are bounded. So out there in the complex plane are points which are distant from all singularities of all . Perhaps one can define the value of each at by using a path that has all singularities to the right side, respectively.
Then one would show that converges uniformly in a neighborhood of . The limit function in the neighborhood of is necesarily again holomorphic there.
From *there* we continue the function to the real line. I guess has only singularities on the real line and at the fixed points of .
Then it would turn out that 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.

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