### Lecture Description

(0:00) Exam information. (1:28) "Holomorphic function" means "analytic function"...we'll look at "meromorphic functions" as well. (2:26) A function that is analytic at a point has a Taylor series that converges to it in the largest open disk centered at the point over which the function is analytic. (5:32) Given a power series with a positive radius of convergence, it defines an analytic function whose Taylor series is the given power series. (7:30) Some people define analyticity in terms of power series. This also can be done to define real analytic functions, though that is not equivalent to being infinitely differentiable. (9:04) Laurent series representation in an annulus.

### 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.