Lecture Description
(0:00) Exam 1 in two class periods. (0:36) Modulus taking is operation preserving with respect to multiplication and division. (1:20) Proof that open disks are open sets. (13:23) Possible proofs on Exam 1. (14:29) Differentiability at a point. (17:16) Calculate the derivative of f(z) = z^3 from the limit definition. (24:21) Be able to derive basic derivative rules (linearity & product rule especially...look up the derivation of the product rule) using the field properties ("ordinary" algebraic properties) of C. (30:57) Even simple complex functions can fail to be differentiable (such as the function that returns the complex conjugate of a given number) which is an example you should study. (32:53) Definition of what it means for a complex function to be analytic at a point. (36:04) Analyticity of a function defined on an open set. (39:19) Polynomials are analytic everywhere on C (they are "entire") and rational functions are analytic everywhere that they are defined. (42:01) This is related to the Cauchy-Riemann equations. (42:34) Linear approximation for real differentiable planar mappings (equations, approximations, and animation of geometric meaning). (53:27) Linear approximation for complex differentiable mappings.
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.