(10/24/2009, 01:23 PM)Base-Acid Tetration Wrote: Why does it have to be a curve from A to B? CAn't it go off to infinity, winding around the inifnite number of points along the way, in the infinite limit?

Why do the branches depend on the path? This is because you do an analytic continuation along the path. You start with some point where your function is defined, e.g. for tetration.

And then you choose a path (not crossing any singularity) from to where you want to determine the function's value at .

You continue the function along this path (i.e. cover it with powerseries expansions with overlapping disks of convergence) until you reach and then you have your value at .

One can show that if you choose to homotopic paths, i.e. two paths that you can continuously deform into each other without crossing singularities, then the resulting continuation value is the same.

E.g. if you continue the logarithm from to you can do this with a path through the northern halfplane and you arrive at or you can do it with a path through the southern halfplane and you arrive at . Any two paths through the northern halfplane would give you the same value .

You can not deform the northern path into the southern path as you would have to cross 0. Thatswhy both values may be different.

So the non-homotopic paths from to determine the possibly different values at and hence it branches.

Thatswhy you need paths between two points of the complex plane, and not some limit of paths or paths going to infinity or so.