06/27/2009, 11:50 PM
(This post was last modified: 06/29/2009, 08:24 AM by Kouznetsov.)
(06/27/2009, 09:55 AM)bo198214 Wrote: 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.
You have no need to deal with so exotic objects like sexp, in order to reproduce the same paradox. Consider the logatirhmic spiral x=log(t)cos(pt), y=log(t)sin(pt) .
For posiive p, at real t, this spiral winds infinitely around zero.
"If we slowly continuously transform" this spiral to the straight line (just reduce p to zero), then you may wander, how the infinite number of turns of this spiral arond zero becomes finite (no windings at all).