Sequence of holomorphic functions that converge compactly in Upper Half Plane $\operatorname{Im}(x)>0$ but not in $\mathbb{C}$

Question

The question is to find a sequence of holomorphic functions that converge compactly in Upper Half Plane $\operatorname{Im}(z)>0$, but not in $\mathbb{C}$. I'm guessing that the solution may contain something to do with $\bar{z}$ but I'm unsure.

Answer 1

For real variables, $a^n\to0$ if $a<1$ and $a^n\to\infty$ if $a>1$ as $n\to\infty$. Through an exponential change of variables, we can write $\exp(n x)\to 0$ for $x<0$ and $\exp(nx)\to\infty$ for $x>0$. Can you turn this into something that converges to $0$ on the upper half-plane but diverges on the lower half in $\mathbb{C}$?