06/27/2009, 09:55 AM
I have new doubts that sexp(S) (where S is the strip: 0<Re(z)<1) winds infinitely around L.
When we consider T=slog(G) (where G is the crescent: |z|<|L| and Re(z)>Re(L)) then T is also a strip of width 1.
Let the z-plane contain T and S, and the w-plane being the image under sexp, i.e. containing L, G and sexp(S).
If we slowly continuously transform T into S in the z-plane, then G slowly transforms into sexp(S) in the w-plane.
But continuous deformations of G always wind finitely many around L.
So we would never reach sexp(S) if it winds infinitely around L.
When we consider T=slog(G) (where G is the crescent: |z|<|L| and Re(z)>Re(L)) then T is also a strip of width 1.
Let the z-plane contain T and S, and the w-plane being the image under sexp, i.e. containing L, G and sexp(S).
If we slowly continuously transform T into S in the z-plane, then G slowly transforms into sexp(S) in the w-plane.
But continuous deformations of G always wind finitely many around L.
So we would never reach sexp(S) if it winds infinitely around L.