The focusing gets better about 30 seconds in. (0:00) Lecture topics and Exam 2 date. (1:47) Summary of the definition of the complex logarithm as a multivalued function and corresponding single valued branches. (11:34) Properties of logarithms still hold, with a caveat that an appropriate branch must be chosen. (18:37) Functions as sets (subsets of Cartesian products...special relations). In effect, we are saying the graph of the function "is" the function (also includes a discussion of Naive Set Theory versus Axiomatic Set Theory).
Based on "Fundamentals of Complex Analysis, with Applications to Engineering and Science", by E.B. Saff and A.D. Snider (3rd Edition). "Visual Complex Analysis", by Tristan Needham, is also referred to a lot. Mathematica is often used, especially to visualize complex analytic (conformal) mappings.