## Help choosing branch cuts

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

$f(z)=z^{z}=e^{zlog(z)}$

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}.

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