Monday:
- Because we are not meeting Friday this week, our
"first week" consists of today's lecture along with our x-hour
Tuesday, Wednesday's lecture, and Monday's lecture! Hence the
homework for this four lectures is due Tuesday rather than the
usual Monday. Please keep up
or Monday night will be very unpleasant.
- Read:Sections 1.1 and 1.2.
- Do:
- In section 1.1 work problems: 4, 8, 12, 21 and
30.
- In section 1.2, work: 6,7dehi, 14 and 16.
- Suggested Only:
- In section 1.1 look at: 15, 19, 22, 24 and 28.
- In section 1.2: 8 and 17.
- Just for fun, suppose that $F$ is an ordered field as in problem 30.
- Show that $-x$ is unique; that is, show that if $x+y=0$, then $y=-x$.
- Show that $(-1)(x)=-x$.
- Conclude that $0<1$.
- Show that $(-1)(-1)=1$.
- Conclude that if $x<0$ and $y<0$, then $xy>0$.
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Tuesday (x-hour) and Wednesday:
- Files:The slides from Monday's lecture can be found
under the "Files" tab on our canvas page. I also posted a
solution to the ellipse example under the "Files" tab in the
Solutions folder.
- Read: Sections 1.3, 1.4 and 1.5.
- Do: In section 1.3, work: 7defg, 9, 11, 13, 16 and 23.
In section 1.4 work: 2, 4, 11 and 20.
- Hint: In section 1.4 number 20, I used the trigonometric
identity $\cos(y)-\cos(x)=2\sin\bigl(\frac{x+y}2\bigr) \sin\bigl
(\frac{x-y}2\bigr)$.
- Suggested Only: In section 1.3: 5 and 10. In section
1.4: 7,8 16 and 17.
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Monday (April 4th):
- Read: Sections 1.5 and 1.6. Skim 1.7. We won't
cover section 1.7 in class, but we'll come back to some of the
concepts later.
- Do: In section 1.5, work: 6b, 10, 11, 14, 15, 16 and 17.
In section 1.6: 1, 10, 15, 18 and 20.
- Suggested Only:In section 1.5: 5acf, 12 and 13. In section
1.6: 2-8 and 19.
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Tuesday (x-hour):
- Study: Sections 2.1 and 2.2
- Do:
- In Section 2.1: 10, 12, 13.
- In Section 2.2: 5, 11de, 15, 22, 25bde. (The answer in the
book to 11d is incorrect. In problem 15, we know from lecture that
a complex valued function is continuous if and only if its real and
imaginary parts are. Hence it is "legal" to use that in
homework. The authors had in mind you proving one direction of that
in this problem. So you can either cite that result, or try to
prove it from the definitions. Either way would be acceptable
here.)
- Suggested:
- In Section 2.1: 1ace, 3d, 5, 7, 8, 9.
- In Section 2.2: 4, 6, 12.
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Wednesday:
- Study: Read Sections 2.3 and 2.4. We will not finish
section 2.4 today so I moved some of the homework from section 2.4
to Friday.
- Do:
- In Section 2.3: 4a, 12, 16.
- In Section 2.4: 3, 5.
- Suggested:
- In Section 2.3: 1, 3, 8, 11egf, 13, 14.
- In Section 2.4: 1, 2, 4 (from lecture), 6.
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Friday:
- Study: Read Section 2.5 and 3.1. We will only cover a
small part of section 3.1 in lecture, but you should skip the
parts we don not talk about.
- Do:
- In Section 2.4: 8, 12, 14.
- In Section 2.5: 5, 6, 8, 18, 20*, 21*. (See below for hints
on 20 and 21.)
- Compare the asserions in problems 20 and 21. Why is there no
contradiction here?
- I didn't understand the author's hint for problem 20.
Instead, I used the Fundamental Theorem of Calculus. We want to
show a function $u$ harmonic in $D=\{z\in \mathbf{C}:|z-z_0|<
d\,\}$ has a harmonic conjugate in $D$. Then let $z_0=x_0+iy_0$.
Now if $a+ib\in D$, then the line seqments from $a+ib$ to $a+iy_0$
and from $x_0+iy_0$ to $a+iy_0$ are also in $D$. Define
$$v(a,b)=\int_{y_0}^b u_x(a,t)\,dt +\phi(a),$$ where $\phi$ is a
function to be defined by you later. You may assume that we know
from our calculus courses that this defines a continuous
function $v$ with continuous second partial derivatives. Note
that the second term in the displayed equation above depends only
on $a$ and not on $b$. You may also assume that
$$\frac{\partial}{\partial x}\int_{y_0}^b u_x(a,t)\,dt
=\int_{y_0}^b u_{xx}(a,t)\,dt.$$ (This is called "differentiating
under the intergral sign", and we will also assume this from
calculus.)
- For 21, the idea is that any two harmonic conjugates
in a domain must differ by a real constant. You may assume
without proof that $z\mapsto \ln(|z|)$ is harmonic on
$\mathbf{C}\setminus\{0\}$ and that $z\mapsto
\ln(|z|)+i\operatorname{Arg}(z)$ is analytic on the complement
$D^*$ of the nonpositive real axis. (If you wish, you can check
that $\ln(\sqrt{x^2+y^2})$ is harmonic on
$\mathbf{C}\setminus\{0\}$, and you can show $\operatorname{Arg}
(x+ i y)$ is harmonic by computing its partials using inverse trig
functions and taking care to note what quadrant you are in -- but
we will find a better way later. Then the analticity of
$\ln(|z|)+\operatorname{Arg}(z)$ follows from one of our
Cauchy-Riemann theorems. But we will agree to make this problem
less messy by making the above assumptions.)
- Suggested: In Section 2.5: 1b, 2, 3cd.
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Monday:
- Study: As announced in lecture, selected solutions for
the HW1 have been posted on our canvas page (under "Files"). The
solutions to HW2 will be posted once gradebook closes this
evening. For today, you should read Sections 3.1 and 3.2. We
covered only part of Section 3.1 in lecture but the reading may
help firming up that material. We may not finish all of 3.2 in
lecture on Monday.
- Do:
- In Section 3.1: 3c (see the first forumla in problem #20 in
section 1.4), 7, 10, 15ac.
- In Section 3.2: 18, 19, 23. (In problem 18, we can not apply
L'Hopital as we haven't proved it in the complex case.)
- Suggested:
- In section 3.1: 4 and 12.
- In section 3.2: 5de, 8, 9, 11.
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Wednesday:
- Study: Read Section 3.3. Some care is required as
students often find the complex logarithm confusing the first time
they see it.
- Do: In Section 3.3: 3, 4, 9, 14.
- Suggested: In Section 3.3: 5, 6.
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Friday:
- Study: Read Section 3.5. (We are not covering Section
3.4 although a quick skim of the material could still be helpful.)
We may start of Section 4.1
if there is time. The preliminary exam on Tuesday will cover
through Section 3.5.
- Do: In Section 3.5: 1ae, 5, 11, 12, and 19.
- Suggested: In Section 3.5: 3, 4, and 15a.
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Monday:
- Study: Read Section 4.1 and get started on Section
4.2. While this material is not on the prelminary exam, it is
fundamental for what is to come. In particular, so called "contour
integrals" will be crucial for our main results. These turn out to
just be suitably disguised versions of the line integrals we
studied in vector calculus.
- Do:
In Section 4.1: 3, 4, 8.
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Tuesday (x-hour):
- Preliminary Exam: There will be a 50 minute "in class"
portion in our x-hour. The "take-home" portion will be available
on gradescope must be uploaded prior to the start of class on Wednesday.
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Wednesday:
- Study: Finish Section 4.2 and read the first part of
section 4.3.
- Do:
- In Section 4.2: 5, 6a and 14.
- Recall from multivariable calculus that if
$\mathbf{F}(x,y)=(P(x,y),Q(x,y))$ is a vector field continuous on a
contour $\Gamma$ parameterized by $z(t)=(x(t),y(t))$ with $t\in
[a,b]$ (we would write $z(t) =x(t)+iy(t)$ in Math 43), then the
"line integral" is $$\int_\Gamma \mathbf{F}\cdot
d\mathbf{r}=\int_\Gamma P\, dx + Q\,dy,$$ where, for example,
$$\int_\Gamma P\,dx=\int_a^b P(x(t),y(t))x'(t)\,dt.$$ (If we think
of $\mathbf{F}$ as a force field, the line integral gives us the
work done in traversing $\Gamma$ through $\mathbf{F}$.) Now
suppose that $f(x+iy)=u(x,y)+iv(x,y)$ is continuous on
$\Gamma$. Find $P$, $Q$, $R$ and $T$ such that $$ \int_\Gamma
f(z)\,dz=\int_\Gamma P\,dx+Q\,dy +i \Bigl(\int_\Gamma R\,dx +
T\,dy\Bigr). $$
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Friday:
- Study: Finish Section 4.3.
- Do: In Section 4.3: 2, 3, 5.
- Suggested: In Section 4.3: 6, 7. Note that we
approached 6 (when $z_0=0$) differently in lecture.
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Monday:
- Study: Read Section 4.4a. We are not covering Section 4.4b.
- Do: In Section 4.4: 2, 15, 18, 19.
- Suggested: In Section 4.4: 1, 3, 5, 9, 11.
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Wednesday:
- Study: Read Section 4.5
- Do:
- In Section 4.5: 2, 16, 17.
- Recall from multivariable calculus that Green's Theorem says
that if $\Gamma$ is a positively oriented simple closed contour in a
simply connected domain $D$, then provided $P$ and $Q$ have continuous
partial derivatives, $$ \int_\Gamma P\,dx + Q\,dy =\iint_E
(Q_x-P_y)\,dA, $$ where $E$ is the interior of $\Gamma$.
Use Green's Theorem and your analysis of line integrals from last Wednesday's
(April 17th) assignment to prove (without using the Deformation
Invariance Theorem) a weak form of Cauchy's Integral Theorem which
says that if $f=u+iv$ is analytic is a simply connected domain $D$,
then $$ \int_\Gamma f(z)\,dz=0 $$ for any simple closed contour
$\Gamma$ in $D$. You may assume that $u$ and $v$ have continuous
partials.
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Friday:
- Study: Finish or re-read Section 4.5.
- Do: In Section 4.5: 6, 8, 13, 15.
- Suggested:In Section 4.5: 1, 3, 10.
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Monday:
- Study: Read Section 4.6.
- Do:
- In Section 4.6: 4, 5, 7, 11, 13, 14.
- Suppose that $f$ is entire and that $|f(z)|\ge 1$ for all
$z$. Prove that $f$ is constant.
- Chapter 5: Today and Wednesday, we start working with power
series. A review of series as well as both Talyor and MacLaurin
series would not go amiss.
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Wednesday:
- Study: Read Section 5.1
- Do: In Section 5.1: 5, 6, 16, 18, 20.
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Friday:
- Study: Read Section 5.2.
- Do: In Section 5.2: 4, 10, 11bc, 13.
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Monday:
- Study: Read Section 5.3.
- Do:
- In Section 5.3: 1, 6, 8.
- Prove the following result from lecture: Consider the
power series $$\sum_{n=0}^\infty a_n z^n .$$ We want to see that
there is an $R$ such that $0\le R\le \infty$ with the property
that the series converges absolutely if $|z|< R$ and diverges if
$|z|>R$. Furthermore, the convergence is uniform on any closed
subdisk $\overline{B_r(0)}$ provided $0< r < R$. I suggest the
following approach. (Also see Lemma 2 in Section 5.3 of the
text.)
- Show that if the series converges at $z_0$, then there is a
constant $M<\infty$ such that for all $n\ge0$ we have $|a_n
z_0^n|\le M$. (Consider problem 5 in section 5.1.)
- Suppose the series converges at $z_0$ with $M$ as above.
Show that if $|z| < |z_0|$ then $|a_n z^n| \le M \bigl |\frac
z{z_0}\bigr |^n$. Conclude from the Comparison Test that the
series converges absolutely if $|z|<|z_0|$.
- Let $A=\{\,|z|: \text{the series converges at $z$}\}$. Note
that $0\in A$ so that $A$ is not the empty set. If $A$ is bounded
above, let $R$ be the least upper bound of $A$. Otherwise, let
$R=\infty$. Show that $R$ has the required properties. (Hint:
you may want to use the fact (without proof) that if
$\sum_{n=0}^\infty c_n$ converges absolutely, then $|\sum
c_n|\le\sum|c_n|$.) Recall that if $R<\infty$, then for all
$x\in X$, $x\le R$ and if $x\le S$ for all $x\in A$, then $R\le
S$.
- (Optional) Show that
the convergence is uniform on $\overline{B_r(0)}$.
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Wednesday:
- Study: Read Section 5.5. We are not formally covering
Section 5.4.
- Do: In Section 5.5: 1ac, 6, 7ab, 9, 13.
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Friday:
- Study: Start Section 5.6. We probably will not cover
all of Section 5.6 today.
- Do: Let $$f(z)=\sum_{j=1}^\infty
\frac{b_j}{z^j} \quad\text{
for $|z|>r$. } $$ Show that we can differentiate $f$
term-by-term. That is, we want to show $$f'(z) =
\sum_{j=1}^\infty -j\frac{b_j}{z^{j+1}}.$$ Hint: I suggest
introducing the function $g(z)=\sum_{j=1}^\infty b_j z^j$ and
using the chain rule and what you know about differentiating a
Taylor series term-by-term.
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Monday:
- Study: Finish Section 5.6. Note that
some of the results proved in lecture are sharper (and more
useful) than the versions proved in the text.
- Do: In Section 5.6: 4, 6, 10, 12, 15. For problem 10,
what kind of singularities does
$g(z)=(z-z_1)^{-m_1}\cdots(z-z_n)^{-m_n}f(z)$ have?
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Wednesday:
- Study: Read Section 6.1.
- Do: In Section 6.1: 4, 5, 6.
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Friday:
- Study: Read Section 6.2 and start Section 6.3.
- Do:
- In Section 6.2: 3, 5, 9. (For problem 9, the
binomial theorem might be helpful.) I'm only assigning a few of
these as the answers are provided. Use your own judgement about
how much practice you need. (I previously suggested you try
number 7. This was a mistake. The problem is ridulously
complicated and I can get my answer to look like the texts only
with a ridiculous trick. I suggest you DON'T try problem 7.
- EP-1: We proved in lecture that $$ \int_\pi^{2\pi}
f(\cos(\theta))\,d\theta=\int_0^\pi f(\cos(\theta))\,d\theta.$$
Does the same hold if we replace $\cos(\theta)$ with
$\sin(\theta)$? What if $f(-z)=f(z)$?
- EP-2: Suppose that $f$ is analytic on and inside a positively
oriented simple closed contour $\Gamma$ and that $f$ does not
vanish on $\Gamma$. Assume that $f$ has finitely many
distinct zeros $z_1,\dots,z_n$ inside $\Gamma$ with orders
$m_1,\dots,m_n$. (If $f$ is nonconstant, then $f$ has at
most finitely many zeros inside $\Gamma$, but you are not
required to proved this). Use the Residue Theorem to show
that $$ \frac1{2\pi i}\int_\Gamma \frac{f'(z)}{f(z)}\,dz=
m_1+\cdots + m_n. $$ Thus, in English, the contour integral
counts the number of zeros, $N_f$ of $f$ inside $\Gamma$ up
to multiplicity. (Problem #6 from section 6.1 is helpful here.)
- For Fun Only: This material will never be tested in this
course. I just thought you
might be interested. I meantioned in lecture that the sorts of definite
trigometric integrals we can do using complex methods
would be tedious to do in the classical
fashion by finding an anti-derivative. But back in the day, when
calculus was hard, we learned that we could find anti-derivatives
of rational functions of $\sin (\theta)$ and $\cos (\theta)$ by
making the substition $z=\tan( \frac\theta 2)$. You can check the
following.
- Show that $$d\theta=\frac{2dz}{z^2+1}.$$
- Show off your trigonometry by showing that
$$\cos(\theta)=\frac{1-z^2}{1+z^2}\quad\hbox{and}\quad\sin(\theta)=
\frac{2z}{1+z^2}.$$
- Observe this transforms the integral of a rational function
of $\cos(\theta)$ and $\sin(\theta)$ into an integral of a
bonafide rational function in $z$ which we also knew how to do
back in the day.
- For example, our first example from lecture: $$
\int\frac1{2+\cos(\theta)}\,d\theta \to \int \frac
2{3+z^2}\,dz = \frac23\int \frac 1{1+ (\frac
z{\sqrt3})^2}\,dz\to \frac2{\sqrt3}\arctan \left(
\frac{\tan(\theta/2)}{\sqrt3}\right)+C.$$
- You get to decide whether complex theory makes it easier.
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Monday:
- Study: Read Section 6.3 and start Section 6.4.
- Do: Do:
- In Section 6.3: Questions 1-7 all have answers provided
and are good practice for applying our "Plus Two Residue
Theorem". I suggest looking at at least 1 and 5. Note that
the algebra in 7 is challenging. Questions 11 and 13 are more
interesting. You can look at Questions 14 to 19 out of
curiosity, but we will not cover that material nor will it
appear on any exam this term.
- (EP-3) Suppose $f$ has a pole of order k at $z_0$. What is
$\operatorname{Res}(\frac {f'}{f};z_0)$?
- (EP-4) Use the Residue Theorem to restate the conclusion to the
written problem (EP-2) to include the case where $\Gamma$ encloses
finitely many poles of $f$ as well as finitely many zeros: that is,
assume $f$ is analytic on and inside of a simply closed contour
$\Gamma$ except for
possibly finitely many poles inside of $\Gamma$ and that $f$ does
not vanish on $\gamma$. Suppose $f$ has finitely many zeros
inside of $\Gamma$. Show that $$ \frac1{2\pi i}\int_\Gamma
\frac{f'(z)}{f(z)}\,dz= N_f-P_f, $$ where $N_f$ is the number of
zeros of $f$ inside of $\Gamma$ counted up to multiplicty and $P_f$
is the number of poles of $f$ inside of $\Gamma$ counted up to
multiplicity. For fun, use this to answer #3 in Section 6.7.
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Wednesday:
- Study: Read Section 6.4
- Do: Do: In Section 6.4: I suggest at least 2, 3, and 7.
Note that our methods do not apply to 4, 9, and 10. If you choose
to use the techniques in the text and not our versions from
lecture, you must explain what you are doing.
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Friday:
- Study: Read Section 6.7. Our approach is more streamlined
than that of the text. You are only responsible for what we do in
lecture.
- Do: Do: In Section 6.7: 2, 4, 6. Note we already did
problem 3.
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