In the real numbers the expression $x^x$ converges to 1 coming from "any" direction (which here means from the positive side or from the negative side). Now I feel to remember that this does not hold in the complex plane, because there the expression $z^z$ should be able to take any value (when $|z|$ goes to 0) depending on the direction/angle you're coming from.
Now I tried to make sense of that, but couldn't complete the picture. I will demonstrate how far I got. Let $z = r \cdot e^{i \phi}$ then
$z^z = (r \cdot e^{i \phi})^{r \cdot e^{i \phi}} = (r)^ {r \cdot e^{i \phi}} \cdot (e^{i \phi})^{r \cdot e^{i \phi}} = (r)^ {r \cdot e^{i \phi}} \cdot e^{i \phi \cdot (r \cdot e^{i \phi})}$.
With this parametrization, when we look at $|z| \rightarrow 0$, this just means $r \rightarrow 0$. My intuition (and Wolframalpha) would now say that the term with $r$ as the base would approach 1 as $r$ approaches 0. But now Wolframalpha also says that the second term with $e$ as the base also approaches 1 as $r$ approaches 0, which would mean the combined limit is 1, not depending on $\phi$. This would then mean that $z^z$ does indeed also approach 1 in the complex plane from whereever we are coming (as the limit is independent of $\phi$).
Now, where is the flaw?
- Does $z^z \rightarrow 1$ as $|z| \rightarrow 0$ for all $z \in \mathbb{C}$?
- Is there a flaw in my calculations? (either in an intermediate step or the reasoning at the end)