Open Sets in the Complex Plane and illustrating the definition with Mathematica 
Open Sets in the Complex Plane and illustrating the definition with Mathematica by Bethel / Bill Kinney
Video Lecture 26 of 26
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Date Added: July 29, 2017

Lecture Description

Complex Analysis, Video #26 (Complex Arithmetic, Part 26.

Main Topics: definition of an open disk ("circular neighborhood"), interior point, and open set. Example: the open right half plane is an open set. Visualize why it is with Mathematica.

Review the definition of an open disk of radius "rho" centered at z0 and the set-builder notation used to define it. It's also called a circular neighborhood (I spelled it wrong in the video) of radius "rho" centered at z0. Define "interior point" of a set S. Discuss the visual interpretation of an interior point. Define a set S to be open if every point of S is an interior point of S. Use RegionPlot to illustrate for the (open) "right half plane". Show how to find a radius that proves that 2+i, 1+i, 0.5+i, and 0.1+i are interior point of the right half plane. Use Manipulate and Locator on Mma to make an animation that shows how the radius of the disk needs to decrease as we approach the boundary (in order to prove points are interior points). If we included the imaginary axis to create the closed right half plane, it would no longer be an open set because the points on that boundary would not be interior points of the set.

Course Index

  1. The imaginary unit and how to add complex numbers
  2. Complex Addition and the Parallelogram Law. Use ListPlot on Mathematica to make the plot.
  3. Complex Number Addition and the Parallelogram Law. Use of Mathematica to create vectors.
  4. Complex Number Addition, Parallelogram Law, Triangle Inequality, and Manipulate on Mathematica
  5. Modulus of a Complex Number, Triangle Inequality, Manipulate and Locator on Mathematica
  6. Complex Number Subtraction in terms of Vectors, Manipulate and Locator on Mathematica
  7. Introduction to Multiplying Complex Numbers and Geometrically Interpreting the Product
  8. Complex Multiplication in terms of Moduli and Arguments. Use Mathematica to illustrate.
  9. Confirm the Geometry of Complex Number Multiplication with Manipulate and Locator. Principal Value.
  10. Complex Number Reciprocals (Multiplicative Inverses), approached Algebraically
  11. Complex Multiplicative Inverses, Complex Division, and Complex Conjugates
  12. Complex Conjugates, Complex Division, and Visualization on Mathematica.
  13. Introduction to the Polar Form of a Complex Number and Complex Multiplication
  14. Polar Form of Complex Numbers, both with "Cis" & with "e" (Euler's Formula)
  15. De Moivre's Formula and Trigonometric Identities (mistake at the end...see description below)
  16. De Moivre, Trig Identities, Sine and Cosine in Terms of Exponentials
  17. A Real Integral done using Complex Arithmetic (Euler's Formula)
  18. Check the use of Cosine as an Exponential to the Evaluation of an Integral.
  19. Powers of Complex Numbers (and an intro to "Table" on Mathematica).
  20. Using Mathematica to Visualize Powers of Complex Numbers
  21. Dynamic Behavior of Powers of Complex Numbers, Intro to Roots and Multi-Valued Functions
  22. Deriving and Graphing Complex Roots of Unity
  23. Graphing Complex Roots with Mathematica
  24. More on Visualizing Complex Roots with Mathematica
  25. Introduction to Basic Topology of the Complex Plane (Define an Open Disk)
  26. Open Sets in the Complex Plane and illustrating the definition with Mathematica

Course Description

This is a mini crash course providing all you need to know to understand complex numbers, and study Complex Analysis. Mathematica is used to help visualize the complex plane.

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