Complex Analysis, Video #21 (Complex Arithmetic, Part 21).

Review polar form of positive integer powers. Review use of Manipulate and Locator on Mathematica to animate this process; use Graphics and Circle to create the unit circle and see how the unit circle forms a boundary between two types of behavior of powers of complex numbers (either going to infinity or going to zero). Start with a point on the unit circle (the point 1/2 + sqrt(3)/2*i) and see that the powers generate a periodic sequence of points. Use ComplexExpand to check it. Try it for 3/5 + (4/5)*i and see that it never comes back to itself. Find 1^(1/2) by thinking about polar form. Emphasize that this really represents all possible square roots (it's "multi-valued"). Get +1 and -1. Find 1^(1/3) by thinking about polar form as well. Get 1, e^(i*2pi/3) = -1/2 + i*sqrt(3)/2, and e^(i*4pi/3) = -1/2 - i*sqrt(3)/2. Check with ComplexExpand.

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

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.