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

Here is my work

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

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

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

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

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