Q: Define a single-valued branch of the function on an open set in , show is analytic on .

Here is my work

Let range over the principle branch. Thus, is holomorphic on .

Now, if I let we have . Now, e^{anything} is defined on all of , to make sure it's bijective though, we need to restric the domain to a period strip. So, let be defined on .

Now, I am sure if I am going in the right direction, because I am stuck. I am not sure if I need to consider a period strip like I said above, I was also thinking I may just have to consider

on its own. Since this is multiplication by a complex number, I am streching and roatating whatever is, so when is positive, I am moving a faster around, I would think. I am not sure how to handle this though.

I have a hard time visualizing this stuff, any help would be appreciated.

Thanks you