(0:00) Mathematica project idea (the Riemann sphere and stereographic projection). (1:04) Quiz 2 possible due dates. (1:21) Enabling dynamic updating. (2:49) The real and imaginary parts of the complex exponential function, along with the corresponding real planar mapping. (4:40) Entering the function into Mathematica. (7:23) Animating the geometry of the mapping with Manipulate and ParametricPlot. (12:17) The image of a vertical line (at a fixed value of x) is a circle centered at the origin with radius e^(x). (17:45) The image of a horizontal line (at a fixed value of y) is a ray emanating from the origin (but not including the origin). (25:56) Other kinds of graphs that can be made in complex analysis. (32:44) Dynamic behavior of orbits under iteration of z^2 on Mathematica with NestList. (38:39) Preimages of mappings (of circles through the origin under z^2). (46:49) Modular surfaces of z^2 - 1 and z^2 + 1. (47:59) Overview of images of circles under the reciprocal map 1/z. (49:08) Quick introduction to the topology of the complex plane (focused on domains of typical functions being open and connected sets).
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.