(0:00) Quiz ready. (0:22) Purely algebraic proof of the Fundamental Theorem of Algebra (involves Field Theory, Galois Theory, Group Theory, and the Sylow Theorems). (2:32) Plan for the day. (3:11) Taylor series for an analytic function centered at a point z0 (at which it is analytic). (6:58) Find the Taylor series for the sine function centered at z0 = 0. (12:07) Show graphs of partial sums on the real line converging to the graph of the sin(x). (14:40) Write down the Taylor series for the cosine function centered at z0 = 0 and graph the partial sums on the real line. (16:36) Write down the Taylor series for the exponential function e^z centered at z0 = 0 and graph the partial sums on the real line. (18:47) Taylor series for 1/(1 - z) centered at z0 = 0 (geometric series), for z within 1 unit of 0. (21:18) Use the Ratio Test to prove the convergence of the Taylor series for e^z for all z (though this doesn't prove it converges to e^z) and graph the partial sums on the real line. (29:41) Taylor series for 1/(1+z^2). (32:21) Graph of partial sums on the real line. Why is the interval of convergence just the open interval from -1 to 1? The function has a pole at z = i and z = -i. (38:28) Integrate the Taylor series for 1/(1+z^2) term-by-term to get the Taylor series for the principal value of the arctangent function. (42:29) Approximations of pi with the Taylor series for arctangent (including Machin's formula). (46:56) Differentiate the series for 1/(1+z^2) term-by-term to get the Taylor series for -2z/((1+z^2)^2).

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.