Daily Assignment for Wednesday, February 12

Due Friday, February 14 on Canvas.

• Here are the Desmos demonstrations from class today :
  
• Suppose that \( p(z) \) is a nonconstant polynomial. How would you define \( p(\infty) \) to extend \( p \) to the Riemann sphere ? How would you extend the exponential function ?
• In class, we saw that the cross ratio \( [-, a, b, c] \) transforms the circle (or line) through \( a, b, c \) to the real line, sending \( a \) to \( 0 \), sending \( b \) to \( \infty \), and sending \( c \) to \( 1 \). Can you find a formula for the inverse transformation, which transforms \( 0 \) to \( a \), transforms \( \infty \) to \( b \), and transforms \( 1 \) to \( c \) ?
• Fix \( a, b, c \in \mathbb C \) and let \( f(z) = [ z, a, b, c ] \). Express \( f \) as a composition of a translation, a rotation, a dilation, and another translation. Conclude that \( f(z) \) transforms circles (and lines) to circles (and lines).
• Read §§4.3–4.5 of Needham's book. Optionally, also read §§4.1–2.
• 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.