Day 8 in Mat430

Last time: Topology/Functions

Next time: Mappings

Today:

Announcements
- You have an assignment due today. For #18, p. 22,
  - a couple of additional trig identities would be useful:
    - ${\frac{1}{2}-\frac{1}{2}\cos(\theta)=\sin^2(\frac{\theta}{2})}$
    - ${\frac{1}{2}+\frac{1}{2}\cos(\theta)=\cos^2(\frac{\theta}{2})}$
  - It often helps to work backwards: for example
    - ${\sin[(2n+1)\theta/2]=\sin[n\theta+\theta/2]=\sin(n\theta)\cos(\theta/2)+\sin(\theta/2)\cos(n\theta)}$
- You have an assignment due next time.
- Today's activity, and new assignment:
  Fri 1/29 11-12 (7th)
  
  Now using reference only to the 7th edition: Read 13, 14; problems to come!
Section 11 (7th edition; beginning of chapter 2): Functions of a complex variable
- Definition
- Examples
- Components as functions of two variables (multivariate!)
- Usual classes
- Multi-valued functions and branches
Section 12: Mappings
- Visualizing functions of a complex variable generally requires 4-D glasses!;)
  - Plotting Moduli cuts it to three
  - Alternatively, consider plotting
    - The Principal Argument of the image
    - The real part of the image
    - The imaginary part of the image
- Rotations, translations, scalings,....
  - Mathematica demo
- Mapping regions
- Mapping special curves, and straight lines
- Mapping boundaries

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