Cauchy Integral Formula, Applications, Liouville's Theorem 
Cauchy Integral Formula, Applications, Liouville's Theorem
by Bethel / Bill Kinney
Video Lecture 25 of 35
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Date Added: July 29, 2017

Lecture Description

The lecture also includes a proof of a theorem that can be used to prove the generalized Cauchy integral formula (once you assume the truth of the Cauchy integral formula). (0:00) Mathematical beauty and Philippians 4:8. (0:54) Animation of the flow of a linear vector field (you could relate these ideas to vector fields generated by analytic functions). (3:23) Cauchy integral formula. (5:58) Generalized Cauchy integral formula. (7:06) Application to evaluation of integrals and check the answers with parameterizations on Mathematica. (19:20) Theorem statement about differentiating a special kind of integral (which can be used to prove the generalized Cauchy integral formula from the ordinary Cauchy integral formula) and an example on Mathematica. (30:47) Verbally describe Liouville's Theorem and its proof. (33:56) Liouville's Theorem can be used to prove the Fundamental Theorem of Algebra (and describe basic idea of proof). (36:12) Verbal description of (two forms of) the Maximum Modulus Principle. (37:44) Go back to some details of the proof of theorem about differentiating a special kind of integral.

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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