(0:00) Taking (single-valued) branches of multivalued functions. (0:35) Two-parameter plot of the "full graph" of the multivalued argument function arg(z) with ParametricPlot3D. (4:10) What does it mean to take a branch? There's a lot of flexibility. The branch cuts don't even have to be straight lines.(17:08) Examples of harmonic functions (satisfying Laplace's equation) over various domains. (24:11) These examples can help us solve Laplace's equation on washers, wedges, and walls, which can ultimately help us solve Laplace's equation on more general regions. (36:40) Project ideas and readings. (40:53) Introduction to power functions and inverse trigonometric functions.
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.