This lecture is part of an online undergraduate course on complex analysis.
We discuss 3 notions of convergence for functions: pointwise convergence, uniform convergence, and locally uniform convergence, and explain why locally uniform convergence is the best one. As applications we show that power series and Dirichlet series converge to holomorphic functions in open sets.
(This is a replacement for a previous video that had a minor but embarrassing error pointed out by an alert viewer: I missed out a quantifier in the definitions of uniform and pointwise convergence. I managed to introduce some extra typos: at 33:03, the factors in front of the integral signs should be s, Re s, and not 1/s, 1/Re s.)
For the other lectures in the course see https://www.youtube.com/playlist?list=PL8yHsr3EFj537_iYA5QrvwhvMlpkJ1yGN
