The lecture also includes a proof of a theorem that can be used to prove the generalized Cauchy integral formula (once you assume the truth of the Cauchy integral formula). (0:00) Mathematical beauty and Philippians 4:8. (0:54) Animation of the flow of a linear vector field (you could relate these ideas to vector fields generated by analytic functions). (3:23) Cauchy integral formula. (5:58) Generalized Cauchy integral formula. (7:06) Application to evaluation of integrals and check the answers with parameterizations on Mathematica. (19:20) Theorem statement about differentiating a special kind of integral (which can be used to prove the generalized Cauchy integral formula from the ordinary Cauchy integral formula) and an example on Mathematica. (30:47) Verbally describe Liouville's Theorem and its proof. (33:56) Liouville's Theorem can be used to prove the Fundamental Theorem of Algebra (and describe basic idea of proof). (36:12) Verbal description of (two forms of) the Maximum Modulus Principle. (37:44) Go back to some details of the proof of theorem about differentiating a special kind of integral.
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.