Polar Form, Principal Value of Arg, Basic Mappings 
Polar Form, Principal Value of Arg, Basic Mappings
by Bethel / Bill Kinney
Video Lecture 3 of 35
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Date Added: July 29, 2017

Lecture Description

(0:00) Moodle quiz due within two class periods. (0:34) Review rectangular and polar forms for complex numbers. (7:31) Finding the argument of a complex number on Mathematica. (11:02) Principal value of the argument, Arg(z) and Mathematica's extended arctangent function. (14:12) Plot the argument in three dimensions as a function of x and y. (17:34) arg(z) as a multi-valued function and the fact that the argument of a product is the sum or the arguments of the factors. (23:22) Describing circles and disks (both closed and open) in the complex plane. (29:43) Summarize the geometric interpretation of complex multiplication and division (and outline the proof for the moduli of the product, using trigonometric identities). (39:14) Viewing complex arithmetic in terms of mappings of the complex plane (as well as the corresponding real mapping). Examples: 1) Add a complex constant. 2) Multiply by a positive real constant. 3) Multiply by -1. 4) Multiply by a complex conjugate of modulus 1. (55:13) Writing z^2 in terms of real and imaginary parts.

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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