Complex Analysis -- Uniform Convergence on Compact Sets

I've taken a course Complex Analysis, but I don't understand why the phrase "For Uniform Convergence on Compact Sets" was used all the time. I always got the impression this was "good enough" for something. But for what? Why is it enough? Why do we care? When is it enough? When is it not enough?

Did this same idea pop up somewhere in real analysis/measure theory?

I really appreciate your exposition on this matter!

Solution 1:

In essence, the key idea is that given an open region $D$, a sequence of functions $\{f_n\}$ analytic on $D$, and a function $f$ such that $\{f_n\}$ converges to $f$ uniformly on every compact subset of $D$, then the function $f$ is itself analytic.

Sometimes we say "$\{f_n\} \to f$ on compacta," though I hate this phrasing.

Ok, that's a lot of hypotheses. Why does it matter?

Convergence on compacta is necessary to set up some key results in complex analysis, for instance, Hurwitz' theorem.

In the end, uniform convergence on compacta shows that analyticity is preserved under uniform limits, which underscores the idea that a function $f:\mathbb{C} \to \mathbb{C}$ that is differentiable once is differentiable infinitely many times. This is one of the key differences between analysis on $\mathbb{C}$ and $\mathbb{R}^2$.