Daily Assignment for Monday, February 24

Due Wednesday, February 26 on Canvas.

• Do exercise 15 on pp. 240–241.
• In class, we considered 6 related properties of a \( \mathbb R \)-differentiable function \( f = u + iv : \mathbb C \to \mathbb C \) :
  1. \( f \) is \( \mathbb C \)-differentiable at \( z_0 \) ;
  2. \( \partial f / \partial \bar z (z_0) = 0 \) ;
  3. \( \frac{\partial f}{\partial x}(z_0) + i \frac{\partial f}{\partial y}(z_0) = 0 \) ;
  4. \( f \) is conformal at \( z_0 \) ;
  5. \( Df(z_0) \) is an orthogonal matrix with positive determinant ;
  6. \( \frac{\partial u}{\partial x}(z_0) = \frac{\partial v}{\partial y}{z_0} \) and \( \frac{\partial u}{\partial y}(z_0) = - \frac{\partial v}{\partial x}(z_0) \) .
  Determine exactly how all of these properties are related : which ones imply which others, and under what circumstances ?
• Read §§5.1.2, 5.4, 5.5. Compare Needham's §5.5.2 to Theorem 4.19 of Howie.
• Assign yourself a grade for this assignment :
  1. At least 60 minutes.
  2. About 45 minutes.
  3. About 30 minutes.
  4. About 15 minutes.
  Submit your grade as a submission comment with your assignment. This comment must be the first comment and it must be exactly 1 character long or the grade won't be registered.