(0:00) Comments on ungraded exercises to do. (0:30) This lecture will focus on the geometry of complex arithmetic. (1:01) Pre-class and post-class lecture Mathematica notebooks. (1:57) The complex numbers form a field, with field properties of addition and multiplication, such as the commutative and associative properties. (5:03) The natural one-to-one correspondences (associations) between the set of complex numbers, points in the plane, and vectors in the plane. (14:07) Polar coordinates and PolarPlot on Mathematica. (23:19) Exercise to find the polar coordinates of a product in terms of the polar coordinates of the factors. (36:04) Modulus and argument and the general fact relating these things for the product of two complex numbers. (38:27) Mathematica demonstration for this fact. (41:22) Defining the Euclidean (distance) metric using the modulus and its geometric interpretations. (43:50) Parallelogram law for addition and the triangle inequality. (50:12) Complex conjugate and its relation to complex division (in particular, to interpreting 1/z geometrically). (57:48) Complex addition as a mapping that is a translation of the complex plane.

1. Complex Arithmetic, Cardano's Formula
2. Geometric Interpretations of Complex Arithmetic, Triangle Inequality
3. Polar Form, Principal Value of Arg, Basic Mappings
4. Mappings, Linear Mappings, Squaring Map, Euler's Identity
5. Squaring Mapping, Euler's Identity & Trigonometry, 5th Roots Example
6. Exponential Map on Mathematica, Squaring Map, Intro to Topology
7. Exponential & Reciprocal Maps, Domains, Derivative Limit Calculations
8. Topological Definitions, Limits, Continuity, Linear Approximation
9. Facts to Recall, Animations, Continuity Proofs (z^2 and 1/z)
10. Open Disks are Open, Derivatives, Analyticity, Linear Approximations
11. Areas of Images, Differentiability, Analyticity, Cauchy-Riemann Eqs
12. Cauchy-Riemann Eqs (Rectangular & Polar), Intro Harmonic Functions
13. Preimages, Laplace's Equation, Harmonic and Analytic Functions
14. Preimages, Mathematica, Maximum Principle (Harmonic), Polynomials
15. Review Analytic Functions, Amplitwist Concept, Harmonic Functions
16. Taylor Polynomials, Complex Exponential, Trig & Hyperbolic Functions
17. Complex Logarithm, Functions as Sets, Multivalued Functions
18. Branches of Arg, Harmonic Functions over Washers, Wedges and Walls
19. Complex Powers, Inverse Trigonometric Functions, Branch Cuts
20. Invariance of Laplace's Eq, Real & Im Parts of Complex Integrals
21. Conformality, Riemann Mapping Theorem, Vector Fields, Integration
22. Complex Integrals, Cauchy-Goursat Theorem, Quick Exam 2 Review
23. Real Line Integrals and Applications, Complex Integration
24. Integration, Cauchy-Goursat Theorem, Cauchy Integral Formula
25. Cauchy Integral Formula, Applications, Liouville's Theorem
26. Sequences and Series of Functions, Maximum Modulus on Mathematica
27. Review Cauchy's Theorem, Cauchy Integral Formulas, and Corollaries
28. Taylor Series Computations, Graphs of Partial Sums, Ratio Test
29. Uniform Convergence, Taylor Series Facts
30. Laurent Series Calculations, Visualize Convergence on Mathematica
31. Laurent Series, Poles, Essential Singularities
32. More Laurent Series, Review Integrals & Cauchy Integral Formula
33. Integrating 1/(1+z^2), Mathematica programming, Residues
34. Series, Zeros, Isolated Singularities, Residues, Residue Theorem
35. Residue Theorem Examples, Principal Values of Improper Integrals

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.