Q: Define a single-valued branch of the function f(z)=z^{z} on an open set U in \mathbb{C}, show f is analytic on U.

Here is my work


Let log(z) range over the principle branch. Thus, log(z) is holomorphic on \mathbb{C}-\{z+iy:y=0, x\leq\\0\}.

Now, if I let w(z)=log(z) we have f(z)=e^{zw(z)}. Now, e^{anything} is defined on all of \mathbb{C}, to make sure it's bijective though, we need to restric the domain to a 2\pi period strip. So, let e^{zw} be defined on A=\{z:y_{0}<Im(z)<y_{0}+2\pi\}.

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

zlog(z) on its own. Since this is multiplication by a complex number, I am streching and roatating whatever log(z) is, so when z 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