### Lecture Description

Pictures of the class were being taken the first few minutes. Also sorry about the clicking noises from the camera tripod. (0:00) Upcoming schedule and project information. (1:11) Amplitwist and Chain-Rule related reasoning for why analytic mappings are conformal when the derivative is nonzero. (7:10) Riemann mapping theorem statement. (10:43) Prelude to complex integration: complex-valued functions as vector fields and their graphs. (17:03) The vector field "rotates" around circles, and this can be used to define the index of a vector field at a singularity (which can be taken to be a zero or a pole). (25:36) It will be more natural to view complex functions as vector fields for the purposes of integration and the applications will switch over to work done by a force, flux of a (2-dimensional) fluid flow across a curve (1-dimensional) membrane. (26:56) The squaring function as a vector field. (28:41) Integrating a complex function over a (parabolic) contour via a parameterization. (37:43) Doing the same integral with the complex version of the Fundamental Theorem of Calculus. (42:23) Approximating the answer via a Riemann sum and a start at visualizing it.

### 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.