Monday March 28:
- Homework 0: Before class on Monday, you should
down load a copy of the text from our canvas page and read/skim
sections 1.1 thru 1.4. Also
download homework-0 and answer the
survey questions there.
- 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
- 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: 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
$$ \lim_{z\to z_0}\frac1{z-z_0}\int_{[z,z_0]}
\bigl(f(w)-f(z_0)\bigr) \,dw=0.$$
- Study: Ex 4.3.3 and Ex 4.3.6. Ex 4.3.6 is the
"solution" to proving a form of Cauchy Theorem is many elementary
texts.
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Wednesday:
- 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.
- Typo Competition: Here is the
current typo contest
(Last modified May 20, 2022) tally. Let us
not let Dylan run away with it.
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Friday:
- Study: Read Section 5.2.
- Do:
- It follows from Theorem 5.2 that $f\in H(D)$ has a zero of
order $m\in \mathbf N$ at $a\in D$ if there is a $g\in H(D)$ such
that $g(a) \not=0$ and $f(z)=(z-a)^m g(z)$ for all $z\in D$. Show
that $f$ has a zero of order $m\in \mathbf N$ if and only if
$0=f(a)=f'(a)=\cdots =f^{(m-1)}(a)$ and $f^{(m)}(a)\not=0$. (This is
probably how I should have defined zeros of order $m$ to begin
with.)
- Ex 5.2.4, Ex 5.2.10, Ex 5.2.11.
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Monday:
- Typo Competition: Here is the
current tally for the
typo contest (Last modified
May 20, 2022). I will keep
updating this link even if it does not appear each
week. I really appreciate the help. You should
also feel free to make suggestions on the exposition
if you like.
- Study: Read Sections 5.3 and 5.4.
- Do: Ex 5.3.6, Ex 5.3.9, 5.3.12, Ex 5.4.1, Ex 5.4.2, Ex
5.4.5.
- Study: I hope everyone is comfortable with Ex 5.3.5. If
you were amused by Ex 5.3.12, have a look at Ex 5.3.13.
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Wednesday:
- Study: Read Section 5.5 through the Schwarz Lemma.
We will finish 5.5 on Friday.
- Comments: As Steve pointed out, in Ex 5.3.6, it should
say "of degree at most $n\ge1$". Furthermore, you can use
Ex 5.3.5 here without proof. Also, Ex 5.4.2 was meant to be
easy: just note that any compact subset of an open disk is
contained in some closed subdisk.
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Friday:
- Study: Finish Section 5.5 and start on Section 6.1.
- Do: Ex 5.5.5 and Ex 5.5.6. (For Ex 5.5.6, you can use
Ex 2.2.4.)
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