06/15/2014, 07:35 PM
(This post was last modified: 06/15/2014, 07:38 PM by sheldonison.)
(06/15/2014, 07:09 PM)tommy1729 Wrote: z + theta(z) takes on all values in the strip -1=<Re(z)=<1 apart from possibly one value.
This follows from picard's little theorem and the periodicity of theta(z).
So since in the strip we take on all complex values (apart from 1 possible value) it follows that the range of sexp in that strip is the same range as sexp.
since the range of sexp is unbounded , than so is the range of sexp(strip).
Q.e.d.
The similarity with the unboundedness of the theta in the OP is striking.
Hope that clarifies.
regards
tommy1729
I'm aware that an entire 1-cyclic \( \theta(z) \) will take on all values in a unit strip, but I'm unsure of how to prove that \( z+\theta(z) \) will take on all values in the strip. I've been struggling with how to show that, forgive my ignorance here... It would also suffice to show that in the strip where \( \Im(z+\theta(z))=0, \; \Re(z+\theta(z)) \) is unbounded.
Second, how do we prove that \( \theta(z) \) has to be entire? Is there the possibility that theta(z) has singularities, but sexp(z+theta(z)) is still analytic? Those are the two issues I've struggled whenever I think about theta(z) mappings and uniqueness.
- Sheldon