Let $f(z) = z^{z}$ where $z \in \mathbb{C}$. Are there are any points where this function is analytic, given that Log($z$) is the principle branch? If it is analytic, how would I go about finding it's derivative?
So far, I've tried to find an explicit representation of $z^{z}$ by representing it as $e^{zLog \left( z \right)}$ and then taking the derivative using the chain rule. Thus, I have $e^{zLog\left(z\right)}Log(z) + e^{zLog\left(z\right)}$. This function is analytic everywhere but $z = 0$.
If anybody could shed some light on this, that would be greatly appreciated. Thank you!