- Another homework due today: pp. 78-, #1c, 2, 5, 6, 8
- One homework returned, and a look at problem 5
- We need to schedule a test (Monday, 3/22?)
- Today's activity, and new assignment:
Fri 3/5 31-34 - I would like you to finish off chapter 3 on your own. Look over the last few sections in the spirit of what I say below.
- We're jumping now to Chapter 8 (Mapping by Elementary Functions), on our way to conformal mapping. Read sections 83-85 (should go quickly -- we've already seen much of this); we'll review that, then jump into section 86 (mobius transformation)
- Problems: pp. 99, #1a, 2, 5; pp. 103, #1, 2, 13 (due Fri, 3/19)
- Section 29: The log(s)
- Definition -- a multi-valued function (no surprise, really)
- Branches
- Even so, many of the old familiar properties hold.
- Section 30: Branches of log
- log is multi-valued ((here's a famous image of the multi-valued imaginary part -- the real part doesn't change!)
- Branch
- points
- cuts
- Log is single-valued, the Principal branch
- Section 31: Some identities
- We propose that
(for example). How can this be? In what sense is the first valid, but not the second?
- For every integer n we have
for any choice of the log function.
- Furthermore, for every natural number n we have that
for any choice of the log function. There will be n distinct values of this function, the nth roots of z.
- We propose that
- Complex powers
- Other exponentials (different bases)
- Trig functions (sines and cosines become unbounded!)
- Hyperbolic trig functions
- Inverses of the trig functions