(10/25/2009, 11:26 AM)bo198214 Wrote: I am still not really familiar with those disconnected Riemann surfaces.
Do these constants have a meaning without those limit formulas?
Something path related, infinite paths, fractal paths?
If one examines at the limit formulas, the meaning should be apparent. Each \( s_k \) represents a winding number about one of the singularities. \( s_0 \) would represent the winding number about the singularity at \( z = -2 \) (and any periodicity "clone" of it), \( s_1 \) would represent the winding number about hte singularity at \( z = -3 \), \( s_2 \), the singularity at \( z = -3 \), and so on. Note that \( \log_b(z) + \omega n \) is the branch of log obtained by winding \( n \) times counterclockwise around the singularity at \( z = 0 \) from the principal branch. I'd suppose that you could think of it as "run \( s_0 \) times about \( z = -2 \), then go and run \( s_1 \) times around \( z = -3 \), then go and run \( s_2 \) times around \( z = -4 \), then go and run \( s_3 \) times around \( z = -5 \), and so on, ad infinitum and take the limiting function which all the generated branches approach". For the given infinite winding sequences, the branches approach constant functions with the limits given. I suspect it approached a constant function for every infinitely long sequence of \( s_k \) (i.e. one that does not end in an infinite string of 0s). So an infinite path might be one way of imagining it, or the limit of infinitely many finite paths. In this way, meaning can be given without resorting to the limit formulas.