Day 17 in Mat430

Last time: Necessary conditions

Next time: Analytic functions

Today:

Announcements
- Reminder: If you'll come and see me to discuss your test, you'll get 30% of your missed points back.
- Today's activity, and new assignment:
  Mon 2/22 21-22
  
  Read 23-25
  pp. 68-, #1, 2, 3b, 4a, 6, 9 (due Monday, 3/1)
Section 16: A final word on the world:
Section 20: Cauchy-Riemann Equations: necessary conditions for differentiability
- Well, we can get necessary conditions, by considering the limit definition of the derivative and then two different approaches: in order for the derivative to exist, we must get the same values along the two different approaches.
- In the end, we find that the following is necessary for a function to be differentiable at a point: the partial derivatives of the component functions must satisfy ${u_x=v_y}$ ${u_y=-v_x}$ at the point.
- Check:
  - A polynomial (e.g. ${f(z)=z^3}$
  - That troublesome function ${f(z)=2x+i3y}$
Section 21: sufficient conditions for differentiability
- Thm: Let the function ${f(z)=u(x,y)+iv(x,y)}$ be defined throughout some neighborhood of a point ${z_0=x_0+iy_0}$ , and suppose that the first-order partials of u and v exist everywhere in that neighborhood. If those partial derivatives are continuous at ${(x_0,y_0)}$ , and satisfy the Cauchy-Riemann equations at ${(x_0,y_0)}$ , then ${f'(z_0)}$ exists.
- Proof
- Check:
  - A polynomial (e.g. ${f(z)=z^3}$
  - The exponential function ${f(z)=e^z}$
  - The curious function ${f(z)=|z|^2}$
- What could go wrong if the partials weren't continuous?
Section 22: let's do it in polar coordinates....
- Note: it's useful if you remember your linear algebra!

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