Open Disks are Open, Derivatives, Analyticity, Linear Approximations 
Open Disks are Open, Derivatives, Analyticity, Linear Approximations by Bethel / Bill Kinney
Video Lecture 10 of 35
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Date Added: July 29, 2017

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

  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

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.

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