I've been messing around with this complex function visualizer: davidbau.com/conformal/
The picture it shows for $f(z) = z^z$ is pretty crazy, so I was wondering: is this function holomorphic?
In the real case we'd find the derivative of $f(x) = x^x$ by taking the log of both sides and applying the chain rule. Can we still do that here?
$z^z = \exp(z\log z)$ is multi-valued. Choosing any specific branch of $\log$ gives a branch of $z^z$ which is holomorphic where $\log$ is.
For example, taking $\log$ as the principal branch, we get a $z^z$ which is holomorphic everywhere except at the negative real axis.