(10/25/2009, 09:22 PM)bo198214 Wrote:(10/25/2009, 08:47 PM)mike3 Wrote: So an infinite path might be one way of imagining it, or the limit of infinitely many finite paths.

Ya, but there are no infinite paths. Only little before I discussed that in this post. If you want to come to a branch value you need a path between two points (or a closed path). And a path between two points or a closed path can not be infinite, can it?

So then it would be better to imagine it as a limiting value approached by an infinitely long sequence of results of of finite paths. Every value of , can be indexed by a finite sequence of integers relative to some principal branch, which I call a "branch code". This is what gets plugged into the limit formula I gave. This sequence means that to get to some non-principal value for a given , you start at the principal branch, wind times (positive = counterclockwise, negative = clockwise) around , then wind times around , then wind times around , then wind times around , and so on until you have wound times around , and then finally come back to (not winding around any more singularities!) the value that you wanted to evaluate at. However, we can also assign a sort of value to an infinitely long branch code by considering the of the branches obtained with the finite partial branch codes , , , , , and so on. Note that this is not a value of as an analytic multivalued function because, as you say, no finite path (hence none that ends at the point "z" we are evaluating at) gets you there.

That such limit values exist makes me wonder what the structure of the Riemann surface must look like.