### Lecture Description

Complex Analysis, Video #17 (Complex Arithmetic, Part 17). Applications of De Moivre's and Euler's Formulas to Trigonometric Identities and Calculation of Integrals.

Details: The trig identities derived in previous videos are true, but perhaps not very useful. In this video, we do an application that is useful for (real) integration. Integrate (cos(theta))^3 by using the fact that cos(theta) = (e^(i*theta)+e(-i*theta))/2, the binomial theorem, and then Euler's formula (in your head). Do the integral. Check it graphically on Mathematica (use Mathematica's "Plot" command...as well as the formatting option "PlotStyle").

### Course Index

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