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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:
(08/21/2009, 09:54 PM)bo198214 Wrote:
(08/21/2009, 09:11 PM)jaydfox Wrote: As you deform the paths to determine that they are the same, you may have to push one or more trivial singularities through a path. Each time you push a trivial singularity "through" a path, you keep of track of whether it crossed from the right or left side (relative to the direction of travel along the path from the starting point), and if the number of right and left singularities (corresponding to windings in opposite directions) are the same, then you can still manage to arrive at the same value.

Oh thats indeed interesting. So is the branch (say on the real axis to have a closed path) determined by the sum of the (oriented) winding numbers around each trivial singularity (assuming no windings around other singularities)?

This has me concerned that the base-change formula is truly undefined for non-real values, because does it even make any sense to say that all complex values are somehow mapped back to a real number (which would be the result as n goes to infinity)? Worse yet, this mapping is determined by the path taken.

well C and R have the same cardinality so there are functions from C to R.

But those are not analytic functions.

But why map to reals only ?

why not map non-real complex numbers to non-real complex numbers ?



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