(0:00) Goals and a demonstration of what happens near where the derivative of an analytic function is zero. (5:42) Visualizing the amplitwist based on the modulus of the derivative and the argument of the derivative. (9:35) Find a (finite) Taylor expansion of a polynomial function about z = 2 in two ways: 1) substitution, 2) Taylor's formula. (20:07) Comments about functions versus multi-valued functions. (21:20) The complex exponential function: formula, periodicity, mapping properties. (28:05) The complex cosine and sine functions: derivatives, identities, modulus of cosine along the imaginary axis. (39:34) The complex hyperbolic functions: identities, relationships to trigonometric functions, geometric/numerical demonstration of sin(iz) = i*sinh(z). (46:15) Introduction to the complex logarithm
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.