Daily Assignment for Friday, January 17

Due Wednesday, January 22 on Canvas.

• Make a post to Discord. Ask a question or answer someone else's question (or both).
• Optional reading : §1.3 of Needham. These sections are very nice, but we won't need them explicitly. You might want to pick one that looks interesting.
• Optional reading : §1.4.2–1.4.4 of Needham. This will anticipate Wednesday's class, but Needham's presentation is a little challenging, so the reading isn't required.
• More computational exercises (do some of these if you need more practice with complex arithmetic) : exercises 14, 15, 17, 18 on p. 53.
• More challenging exercises (do some of these if you are feeling confident with complex arithmetic) : 11, 13, 24, 32 on pp. 52–56.
• Find all complex solutions to the equation \( z^5 - 1 = 0 \). Use these to find the factorization of \( z^5 - 1 \) into irreducible polynomials with real coefficients. Now use this factorization to find formulas for \( \cos(\frac{2\pi}{5}) \) and for \( \cos( \frac{4\pi}{5}) \) using only integers, addition, subtraction, multiplication, division, and square roots. You may find it helpful to read §1.3.5 along with this problem.
• Finish the argument that we began in class, showing that \( \lim_{n \to \infty} (1 + \frac{it}{n})^n = \cos(t) + i \sin(t) \). Recall that we wrote \( 1 + \frac{it}{n} = ( 1 + \frac{t^2}{n^2} )^{1/2} \angle \arctan (\frac tn) \). In class, we showed that \( \lim_{n \to \infty} ( 1 + \frac{t^2}{n^2} )^n = 1 \). Complete the argument by showing that \( \lim_{n \to \infty} n \arctan(\frac tn) = t \).
• Review the "proofs" of \( e^{it} = \cos(t) + i \sin(t) \) from class and from the textbook. In both the textbook and the proofs from class, we often needed to differentiate a function \( f : \mathbb R \to \mathbb C \). What does it mean to differentiate such a function ?
• In class we used the product rule, \( \frac{d(fg)}{dt} = f \frac{dg}{dt} + \frac{df}{dt} g \) when \( f : \mathbb R \to \mathbb C \) and \( g : \mathbb R \to \mathbb C \) are differentiable functions. Where did we use the product rule ? And why does the product rule work here ? (Hint : deduce the complex product rule from the real one.)
• In class, we also used the chain rule, \( \frac{d f}{dt} = \frac{df}{dg} \frac{dg}{dt} \) for a function \( f : \mathbb R \to \mathbb C \) and \( g : \mathbb R \to \mathbb R \). Where did we use the chain rule ? And why does it work ? (Same hint : relate this chain rule to the familiar real chain rule.)
• 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.