(0:00) Topics for the lecture. (0:32) Level curves of the real and imaginary parts of a complex-valued function are preimages of horizontal and vertical lines under the mapping. (3:26) Symbolically confirm that hyperbolic functions parameterize hyperbolas. (6:33) Animate hyperbolas being mapped under f(z) = z^2. (12:13) Review Cauchy-Riemann equations in polar coordinates and check them for f(z) = z^3 in both rectangular and polar coordinates. (18:21) Laplace's equation (a partial differential equation), notation, overview of applications, harmonic functions, and connections to analytic functions. (27:49) Generate a harmonic function by taking the real part of an analytic function. (30:56) Find the harmonic conjugate by integration and the Cauchy-Riemann equations. (37:34) Example to illustrate a solution to Laplace's equation along the closed unit disk, using polar coordinates to describe the boundary conditions, and then imagine it to be a temperature distribution. (49:41) Infinitesimals, complex derivatives, and the amplitwist concept (due to Tristan Needham).
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.