Day 14 in Mat430

Last time: Exam 1

Next time: Derivative formulas

Today:

Announcements
- Sorry about the glitches in the exam(s) -- damn that copy/paste!;)
- Today's activity, and new assignment:
  Fri 2/12 17-18
  
  Read 18 and 19
  Exercises to be determined.
- Let me collect your take-home component -- I'll hope to have these graded by Monday.
Let's look at the (Wrong!) Limit problem from the in-class exam 1 (Sorry about that....)
- Thanks to Jeremy for bringing this to my attention
- Jeremy did exactly the right thing: he gave a counter-example.
Section 16: The point at infinity
- We extend the complex plane by adding the point at infinity!?
- Riemann sphere (p. 49), and the stereographic projection between the complex plane and the Riemann sphere.
- An ${\epsilon}$ neighborhood of infinity is the set of points ${|z|>1/\epsilon}$
- Theorem (p. 49):
  
  ${\lim_{z\rightarrow{z_0}}f(z)=\infty}$ ${\Longleftrightarrow}$ ${\lim_{z\rightarrow{z_0}}\frac{1}{f(z)}=0}$
  
  ${\lim_{z\rightarrow{\infty}}f(z)=w_0}$ ${\Longleftrightarrow}$ ${\lim_{z\rightarrow{0}}f(\frac{1}{z})=w_0}$
  
  ${\lim_{z\rightarrow{\infty}}f(z)=\infty}$ ${\Longleftrightarrow}$ ${\lim_{z\rightarrow{0}}\frac{1}{f(1/z)}=0}$
- Last time we made the correspondence between points and the plane and points on the sphere explicit, by coming up with an equation for the one-to-one correspondence: Stereographic.nb
- Examples: Let's look at #10 from your homework (which was originally scheduled to be part of your exam)
  - #10, p. 54
Section 17: Continuity
- I don't want to dwell on these issues: they're intuitively the same as we're accustomed to.
- Continuity of Polynomials
- Continuity of Rational functions
- Compositions of continuous functions are continuous.
- etc....
Section 18: Derivatives
- Definition
- Example: #3, p. 60
- A bizarre example: Example 2, p. 55

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${\lim_{z\rightarrow{z_0}}f(z)=\infty}$	${\Longleftrightarrow}$	${\lim_{z\rightarrow{z_0}}\frac{1}{f(z)}=0}$
${\lim_{z\rightarrow{\infty}}f(z)=w_0}$	${\Longleftrightarrow}$	${\lim_{z\rightarrow{0}}f(\frac{1}{z})=w_0}$
${\lim_{z\rightarrow{\infty}}f(z)=\infty}$	${\Longleftrightarrow}$	${\lim_{z\rightarrow{0}}\frac{1}{f(1/z)}=0}$