Monday:
- Homework 0: Before class on Monday, you should
download a copy of the text from the "Files" tab on our canvas
page and read/skim sections 1.1 thru 1.4. Also download
homework-0 from the gradescope page and answer the
survey questions there. Then upload a PDF file with your
answers in gradescope. Part of the point of homework-0
is to make sure everyone is comforable with gradescope
and the process of uploading solutions and linking the problems
to their respective pages. Also, it's fun to tell your friends
that there was an assignment due the first day of class. (Note
that you can actually turn in homework-0 any time this
week.)
- Solutions: I have uploaded solutions to (most of) the
exercises in Sections 1.1 to 1.3 under "Files" on our canvas
page. Please let me know if you have trouble accessing them.
These are also fair game for the Typo competition.
- Do: Ex 1.4.1, Ex 1.4.7(a), Ex 1.4.8, Ex 1.4.12.
- Study: (Not to be turned in.) If you ever might teach
this stuff, you should have a look at most of the exercises in
Section 1.3 and 1.4. Especially Ex 1.4.3-5.
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Wednesday:
- Read Section 1.5.
- Do: Ex 1.5.2 and Ex 1.5.8.
- Study: Ex 1.5.9 and Ex 1.5.10.
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Friday:
- Read Sections 2.1 and 2.2. Since I've fallen a bit behind,
we probably won't finish section 2.2 today. But you shuold be
able to work on everything by Ex 2.2.3 and Ex 2.2.5. (Ex 2.2.5
is really untanglind definitions, but it will probably be tricky
for most.) Please turn in the first three assignments by
Tuesday evening. This will give you time to stop by office
hours if you need some help or hints.
- Do: Ex 2.1.8, Ex 2.2.3, Ex 2.2.5
- Study: Ex 2.1.4, Ex 2.1.6, Ex 2.2.4, Ex 2.2.7 (This is
tricky, but gives another version of Thm 2.6 without the
unmotivated step of the proof).
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Monday:
- Study: NO CLASS.
- Do: Use Eye Protection.
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Tuesday (x-hour):
- Study: Read Section 4.1
- Do: Ex 4.3.1
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Wednesday:
- Study: Read Section 4.2 and 4.3
- Do:
- Ex 4.2.1, Ex 4.3.2, Ex 4.3.4, and Ex 4.3.8. The challenge
of Ex 4.3.4 is the write up and elegant solution without hand
waving.
- Complete the proof of Cauchys Theorem for Convex Domains by
showing that if $f$ is continuous at $z_0$, then $$ \lim_{z\to
z_0}\frac1{z-z_0}\int_{[z_0,z]} \bigl(f(w)-f(z_0)\bigr) \,dw=0.$$
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Friday:
- Study: Read Section 5.1. I should not have assigned Ex
4.3.2 until we have covered Moreras Theorems in lecture. (I
suppose this is a hint as to how I would solve it.)
- Do: Ex 5.1.2 and Ex 5.1.3.
- Study: You should know how to do Ex 5.1.1. Notice that Ex
5.1.4 is supposed to be trivial since the identities hold in the
real case and you apply Corollary 5.3. You should at least think
about Ex 5.1.5 long enough to realize that it is not an immediate
consequence of Theorem 5.2.
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Monday:
- Study: Read section 5.2. Last week's homework is due
tomorrow.
- Do: Ex 5.2.4, Ex 5.2.10, and Ex 5.2.11
- Be Aware: Ex 5.2.3, Ex 5.2.5, Ex 5.2.7, Ex 5.2.12.
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Wednesday:
- Study: Read sections 5.3 and 5.4
- Do: Ex 5.3.6, Ex 5.3.9, Ex 5.3.12, Ex 5.4.1, and
5.4.2. Especially in Ex 5.3.6, you can use the result of Ex
5.3.5 without proof.
- Be Aware: You should be comfortable with Ex 5.3.5. If
fact you should consider it assigned but ungraded. If you like Ex
5.3.12, have a look at Ex 5.3.13.
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Friday:
- Study: Read section 5.5 up to the Open Mapping Theorem.
- Do: Try Ex 5.4.5. This is sort of a challenge problem
and no one should get stressed if they can't do it.
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Monday:
- Study: Read section 7.2. Recall that we are meeting
in our x-hour (aka tomorrow) this week.
- Do: Ex 7.2.2
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Tuesday (x-hour):
- Study: Start section 7.3. We will finish up (probably)
on Wednesday.
- Do: No new assignment
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Wednesday:
- Study: Finish section 7.3 and start section 7.4
- Do: Suppose that $D$ is an open connected
subset of $\mathbf C$.
- If $p\in D$, show that $D\setminus\{p\}$ is connected.
(You can assume that any deleted neighborhood,
$D_{\epsilon}'(p)$ is path connected and therefore
connected.)
- Show that if $D$ is a neighborhood of $\infty$, then $D\cup
\{\infty\}$ is connected in $\mathbf C^+$.
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Friday:
- Study: Finish up section 7.4. If there is time, we
will skip ahead to Chapter 10.
- Do: Ex 7.4.2, Ex 7.4.3, and as another challenge
problem try Ex 7.4.4.
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