(0:00) Moodle quiz due within two class periods. (0:34) Review rectangular and polar forms for complex numbers. (7:31) Finding the argument of a complex number on Mathematica. (11:02) Principal value of the argument, Arg(z) and Mathematica's extended arctangent function. (14:12) Plot the argument in three dimensions as a function of x and y. (17:34) arg(z) as a multi-valued function and the fact that the argument of a product is the sum or the arguments of the factors. (23:22) Describing circles and disks (both closed and open) in the complex plane. (29:43) Summarize the geometric interpretation of complex multiplication and division (and outline the proof for the moduli of the product, using trigonometric identities). (39:14) Viewing complex arithmetic in terms of mappings of the complex plane (as well as the corresponding real mapping). Examples: 1) Add a complex constant. 2) Multiply by a positive real constant. 3) Multiply by -1. 4) Multiply by a complex conjugate of modulus 1. (55:13) Writing z^2 in terms of real and imaginary parts.
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.