(0:00) Quiz 2 due date and topics. (0:30) Mathematica demonstration of the stereographic projection onto the Riemann sphere. (4:09) Review the formula for the exponential mapping and animate images of rectangles on Mathematica. (12:28) e^(z) is periodic with period 2*pi*i. (13:51) Making sense of the image from the formula. (15:04) Reciprocal mapping and algebraically verifying images of lines and circles. (28:51) Definition of a "domain" in complex analysis as an open connected subset of the complex plane (a typical "domain of definition" for functions in complex analysis). (30:21) What does it mean for a set to be (path) connected? (32:33) What does it mean for a set to be open (in terms of interior points)? Demonstrate idea with Mathematica. (39:52) Derivative definition and limit calculations for f(z) = z^2 and f(z) = 1/z.
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.