05/14/2015, 12:37 PM
(This post was last modified: 05/14/2015, 01:29 PM by sheldonison.)
(05/13/2015, 11:49 PM)tommy1729 Wrote: If slog was periodic even in the limit, then
1) sexp(slog(x)+1/2) is periodic too.
2) the cutlines and singularities due to L,L* are COPIED infinitely often because of the periodicity. Resulting in slog having infinitely many singularities.
By definition; slog(z) = slog(exp(z))-1. So if exp(z) is a small number, close to zero, than the behavior of slog(x) is governed by he behavior of slog(z) near zero, where slog is a nicely defined analytic function in the neighborhood of z=0; where slog(z)~=-1+0.915946x. So now consider a path for slog(z) for z=-5 to z=-5+pi i. slog(-5+pi i) ~= -2.0062. Notice that the pi i values for slog(z) are real valued again, numbers between -2 and -3. So we can do a Schwarz reflection about the pi i line as well....
If you look in the right places, than exp^{1/2}=sexp(slog(z)+1/2) has 2pi i periodicity as well, and exp^{1/2}(z+pi*I) for real(z)~<-0.3624, than exp^{1/2}(z) is real valued, and tends to sexp(-1.5)~=-0.696 as z goes to minus infinity; There is a singularity for slog(z)=-2.5. And exp^{1/2} has singularities at L,L*
Based on these arguments, additional singularities would be expected for slog(z) for z=L+2n pi i, L*+2n pi i
- Sheldon
