- Another homework due today: pp. 74-, #1cd, 2bc, 5, 7b
- Today's activity, and new assignment:
Wed 3/3 29-31 - Read sections 32-34
- problems: pp. 94--, #2, 4, 5, 6, 11 (due Wed, 3/17)
- Definition -- the extension of
into the complex plane
- The old familiar properties hold:
- There are some new things under the sun:
- f can be negative
- f is periodic (with imaginary period)
- Examples
- 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
