Introduction to Basic Topology of the Complex Plane (Define an Open Disk) 
Introduction to Basic Topology of the Complex Plane (Define an Open Disk) by Bethel / Bill Kinney
Video Lecture 25 of 26
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Date Added: July 29, 2017

Lecture Description

Complex Analysis, Video #25 (Complex Arithmetic, Part 25)

Main Topics: numerically check 5th roots from previous video. Define an open disk and visualize it using RegionPlot on Mathematica.

Do a quick numerical check of the 5th roots of -1 + sqrt(3)*I from the previous video (use "N" on Mathematica). Use the percent sign % to refer back to the preceeding output (as a list) and raise every number in the list to the 5th power. Definition of an "open disk" of radius positive radius "rho", centered at a given complex number z0. Use set-builder notation to define, as well as the modulus of the difference z - z0, which represents the distance between the points representing z and z0 in the complex plane. Also write in terms of rectangular coordinates x and y by using the distance formula (Pythagorean Theorem). To be an open disk, we must use a strict inequality so that we do NOT include the boundary points of the disk. Use RegionPlot to graph a disk of radius 2 centered at the point 3 + 4*i. RegionPlot graphs it as if the boundary circle is included, but it is not. We can use Show, Graphics, Dashed, and Circle to emphasize that the boundary is not included.

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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