(0:00) Exam 2 next class period. (1:26) Various ways to integrate z^2 over a parabolic path, done on Mathematica, first in terms of complex quantities. (5:02) Relate to real line integrals from Multivariable Calculus and calculate with Mathematica. (14:10) Confirm the answer with the Fundamental Theorem of Calculus. (15:45) Independence of path, including calculation over alternative oriented contours. (22:45) Integrals around closed loops. (24:57) Cauchy's Integral Theorem (Cauchy-Goursat Theorem), including a description of a simply connected domain. (30:02) Outline of proof of weaker version with Green's Theorem and the Cauchy-Riemann equations (give statement of Green's Theorem as well). (40:48) Analytic functions are infinitely differentiable, which doesn't happen for real functions when you have differentiability over a neighborhood of a point. (43:09) Review for Exam 2 (including a review of the Chain Rule in polar coordinates).
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.