# Mathematical and Computational Methods in Physics and Engineering (fall 2016)

## Daily outline for Fri, Sep 16, 2016 9:10 AM (8 months ago)

### Complex numbers 2

assigned today:
• standard-complex functions:

I can manipulate complex functions.

1. Show how a complex function can produce a vector field.
2. Derive the Cauchy-Riemann equations.
3. Problem 13.4.25

Due today:

# what we did in class

• quiz asking to find an expression for $$\sin(4\theta)$$ where I gave them Pascal's triangle.
• We talked briefly about how to study for these quizzes. I talked about how some students can comfortable get the "homework" done but still struggle on the quizzes because I always make changes. I said that if there's something I could change that makes the problem impossible (from the student's perspective) then that likely means they don't have a full grasp on the concept.
• I asked them to write down a complex function and evaluate it for 1, i, and 1+i
• We plotted the complex conjugate function as a vector field and realized that 0.1 z* is a better one to visualize.
• We talked about the fundamental definition of derivatives:

$$\frac{df}{dz}=\lim_{\Delta z\to 0}\frac{f(z+\Delta z)-f(z)}{\Delta z}$$

• and applied it to f(z)=z^2, showing that it works just fine
• then we did the complex conjugate function and showed that it has no derivatives!
• We were running out of time so I quickly waved my hands about the Cauchy-Riemann stuff and the Harmonic functions stuff.

Resources requested:

• example of mapping using Mathematica
• I expressed my preference (in this class) for only giving a video and not the mathematica code itself. That went over pretty well, in my opinion
• Derive the Cauchy-Riemann equations. We jokingly called this "ready 13.4 aloud to them."
• Do an example like 13.4.25 both on paper and in Mathematica

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