### Lecture Description

(0:00) Celebrate pi day tomorrow (since spring break next week). (0:25) Do the quiz and search for project ideas in coming days. (2:41) Today will be a review day. (3:21) Abstract differentiation rules. (4:57) Local linearity, including a numerical animation made with Manipulate, and a reminder of the distinction between linear approximation for complex differentiable mappings and differentiable real planar mappings. (12:23) Analyticity, lack of analyticity (for functions involving conjugation, taking modulus, real and imaginary parts), polynomials are entire, and rational functions are analytic wherever they are defined. (15:45) Fundamental Theorem of Algebra. (17:55) Graph the modulus of f(z) as a 3-dimensional graph and a contour map to see where the roots are (and where the poles are for a rational function). (24:25) Amplitwist concept (from Tristan Needham's "Visual Complex Analysis") in terms of linear approximation. (32:02) Amplitwist concept in terms of infinitesimals. (35:19) Cauchy-Riemann equations in rectangular and polar coordinates, relationship to gradient vector fields and level curves for u and v. (37:42) Analytic functions are conformal (angle-preserving when the derivative is non-zero. (38:22) Representation of the derivative f' in terms of partial derivatives of u and v and application to the derivative of the complex exponential function being itself. (41:02) The constant function theorem over a (open and connected) domain. (42:12) Finding areas of images of regions under complex analytic mappings (writing the Jacobian determinant in terms of the derivative f'). Application to f(z) = z^2 for small disks near the origin. (52:14) Harmonic functions and Laplace's equation, including an example with discontinuous boundary values. (54:47) Relationship between analytic and harmonic functions. (55:39) Optimizing functions and harmonic functions over compact (closed and bounded) regions (Maximum principle), also look at the form of the determinant of the Hessian matrix. (59:04) Analytic and harmonic functions turn out to be infinitely differentiable. (1:00:12) Partial derivatives of harmonic functions are harmonic.

### Course Index

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

### Course Description

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.