On pages 238-239 of his book An Introduction to Classical Real Analysis Stromberg gives a proof of the following Theorem:
I understand all the proof except for the statement (near the end) that we can write any $z \in C$ as $z=a^b e^{2\pi i t}$ for some $0\leq t <1$.
I can only show that it holds for some $-1/2<t\leq1/2$. Can you help me figure out why this is the case?
Thanks a lot for your help.
As noted by Conrad the periodicity of $e^{2\pi i t}$ is $1$ so if $t\in (-1/2,0)$ then I can simply take $t'=t+1 \in (1/2,1)$.