Monday:
- Study: Sections 1.1 and 1.2
- Do:
- If you haven't already,
complete "Homework Zero" on the Canvas
Website.
- Would you like to learn how to use a word processor that
is far superior to Word and is designed to process
mathematics such as $$\oint_{\delta D} P\,dx +Q\,dy =
\iint_D (\frac{\partial Q}{\partial x} - \frac{\partial
P}{\partial y}) \,dA$$, then you might want to attend a
workshop on using
$\LaTeX$. See this
flyer for details. (What is that formula called?)
- In section 1.1 work problems: 4, 8, 12, 21 and 30.
- Suggested Only:
- In section 1.1 look at: 15, 19, 22, 24 and 28.
- 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 $(-1)(x)=-x$.
- Show that $(-1)(-1)=1$.
- Conclude that $0<1$.
- Conclude that if $x<0$ and $y<0$, then $xy>0$.
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Wednesday:
- Study: Read section 1.2
- Do: In section 1.2, work: 6,7dehi, 14 and 16.
- Suggested Only: In section 1.2: 8 and 17.
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Thursday (x-hour):
- Study: 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.
- Suggested Only: In section 1.3: 5 and 10. In section
1.4: 7,8 16 and 17.
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Friday:
- Study: Read section 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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Monday:
- Study: Read sections 2.1 and 2.1
- Do:
- To save me some typing, I am no longer
breaking out the "suggested" problems. Instead, interesting problems
that you should look at but not turn in will be listed in
parentheses.
- Section 2.1: (1ace, 3d, 5, 6ab, 7, 8, 9), 10, 12, 13.
- Section 2.2: (2,4), 5, (6), 11de, (12), 15, (18), 22, 25bde.
- Comment on 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 author 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 is
acceptable here.
- Comment on section 2.2, \#11b. The answer in the back of
the book is incorrect.
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Wednesday:
- Study: No lecture on Wednesday. We'll meet Thursday in
our x-hour.
- Do: Catch up on your reading
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Thursday:
- Study: Read sections 2.3 and 2.4
- Do:
- Section 2.3: (1), (3), 4a, (8, 11efg), 12, (13, 14), 16.
- Section 2.4: (1, 2), 3, (4 mentioned in lecture), 5, (6), 8,
12, 14.
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Friday:
- Study: Read sections 2.5 and 3.1. Section 3.1 is faily
long and we'll only briefly discuss it in lecture. So, you'll be
on your own there and should read that section carefully.
- Do:
- Section 2.5: (1b, 2, 3cd), 5, 6, 8, (10,) 18, 20* and 21*.
- Compare 20 and 21! Why is there no contradiction there?
- Problems 20 and 21 are a bit harder than usual. I've included
some hints below. But while I wouldn't call then "extra credit",
don't waste too much time on them if you're stuck.
- 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'll 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 $z\mapsto
\ln(|z|)+\operatorname{Arg}(z)$ 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 $Arg (x+ i y)$ is harmonic by computing its
partials using inverse trig functions and taking care to note what
quadrant you're 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 let's make this problem less
messy by making the above assumptions.)
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Monday:
- Study: Read Section 3.2. We've skipped the majority of
section 3.1 in lecture. You'll want to study the section
none-the-less.
- Do: Section 3.1: 3c (see the formula in problem 20 of
section 1.4), (4,) 7, 10, (12) and 15ac.
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Wednesday:
- Study: Read section 3.3.
- Do:
- Section 3.2: (5de, 8, 9, 11), 18 (we haven't proved L'Hopital's
rule, so don't use it -- unless you prove it), 19 and 23. (Note
that 23 is a nice way to establish equation (8) in the text without
undue algebra. Later, when we've proved Corollary 3 in section
5.6, we'll see that we can verify equations (6) to (11) simply by
observing they hold for all real $z$.)
- Section 3.3: 3, 4, (5, 6), 9 and 14.
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Friday:
- Study: Read section 3.5. (We are skipping section 3.4.)
- Do:
- Section 3.5: 1ae, (3, 4,) 5, 11, 12, (15a,) and 19.
- Please also work this problem: Is there a branch of
$\log z$ defined in the annulus
$D=\{\,z\in\mathbf{C}:1<|z|<2\,\}$?
- Recall that our preliminary midterm is Thursday, April 23rd. It
will cover thru and including todays lecture -- that is, up to and
including section 3.5.
- Be aware that it is not likely that this assignment will be
returned prior to the exam.
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Monday:
- Study: Read sections 4.1 and 4.2. We are going to make
significant use of "contour integrals" in Math 43. They are just a
suitably disguised version of the line integrals we studied in
multi-variable calculus. Section 4.1 is mostly a tedious
collection of, unfortunately very important, definitions.
Fortunately, they are essentially the same that we used in
multivariable calculus but using our complex formalisim.
- Do: Section 4.1: 3, 4, and 8.
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Wednesday:
- Study: Study for the exam. Remember that this week's
homework is due Wednesday, April 29th.
- Do: No new assignment.
- The Exam: The exam will cover through Friday's lecture.
That means up to section 3.5 in the text and nothing from Chapter 4.
The in-class portion will be objective and closed book. On the
take-home you can use your text and class notes, but nothing else. For
example, no googling for the answers or other internet searches.
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Thursday:
- In Class (x-hour): In class portion
of the exam. No external sources allowed and you must work
alone.
- Due Friday: The take-home part of the exam will be due
at the beginning of class on Friday.
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Friday:
- Study: Review sections 4.1 and 4.2. Read section 4.3
- Do:
- 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 $$ \int_\Gamma
f(z)\,dz=\int_\Gamma P\,dx+Q\,dy +i \Bigl(\int_\Gamma R\,dx +
T\,dy\Bigr). $$
- Section 4.2: 5, 6a and 14.
- Section 4.3: 2, 3, 5.
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Monday:
- Study: Last weeks assignments are due Wednesday. Today
and the rest of this weeks assignments will be due Monday the 4th
of May.
For today, you should read section 4.4a. Section 4.4 has two
approaches and the one you are primarily responsible for, and the
one we'll cover in class, is part a. We are getting to the meat of
the matter. But it is subtle stuff, so please ask questions in
class and/or office hours.
- Do: Section 4.4: (1), 2, (3, 5, 9, 11), 15, 18, 19.
- Preliminary Exam: I have posted solutions to the exam as
well as information about the distribution of scores on
the Selected Solutions page. There
are also some estimates of current grades based on the exam and
homework.
- Honor Code: Please note that the solutions for exams and homework posted on the Selected Solutions page are for students enrolled in Math 43 in the spring of 2015 only. Sharing these solutions with students outside of the course is an honors violation.
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Wednesday:
- Study: Read section 4.5. Note that we will meet in our
x-hour tomorrow and that we are unlikely to finish all of section
4.5 today.
- Do:
- Section 4.5: (1), 2, (3), 6, 8, (10, 13), 15 and 16.
- Problem 16 is pretty cool. In particular, it implies that
there is a branch of $\log z$ in any simply connected domain $D$
provided $0\notin D$.
- 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 Friday's
(April 24th) 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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Thursday (x-hour):
- Study: Read section 4.6 up to Lemma 1.
- Do: Why does the result of problem 16 in section 4.5 imply that
$\mathbf{C}\setminus\{0\}$ and the annulus
$A=\{\,z\in\mathbf{C}:1<|z|<2\,\}$ are not simply connected?
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Friday:
- Study: Read section 4.6
- Do: Section 4.6: 4, 5, and 7.
- Also:Suppose $f$ is an entire function such that
$|f(z)|\ge1$ for all $z$. Show that $f$ is constant.
- Next week: We start working with power series on
Monday. Reviewing power series as well as MacLaurin and Taylor
series would not go amiss.
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Monday:
- Study: Read section 5.1. Review power series as necessary.
- Do:
- Section 4.6: 11, 13, and 14.
- Section 5.1: (3, 4), 5, 6, and 10.
- SKIP: THIS IS PROBLEM #20 FOR WEDNESDAY!
Don't include it there. Show that the series
$\sum_{n=0}^\infty z^n $ does not converge uniformly to $1/(1-z)$ on
the unit disk $D=\{\,z:|z|<1\}$.
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Wednesday:
- Study: Read section 5.2
- Do:
- Section 5.1: 16, 18, 20 (this is the problem skipped above) and 21.
- Section 5.2: (1), 4, and 10 .
|
Friday:
- Study: Read section 5.3.
- Do:
- Section 5.2: 13.
- Section 5.3: 1, 6, and 8.
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Monday:
- Study: Because of the exam on Thursday, this week's
assignments will be due Wednesday, May 20th. The exam will cover
through section 5.3 in the text. We finished most of 5.3 on
Friday. We'll fill in a few loose ends today (Monday).
- We will not cover section 5.4 at
all. Today, you should read section 5.5.
- Do: In section 5.5: 1ac.
- 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.
- 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 $|\sum c_n|\le\sum|c_n|$.)
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Wednesday:
- Study: Read section 5.5
- Do: No assignment today. Study for exam.
|
Friday:
- Study: Read section 5.6
- Do:
- Section 5.5: 6, 7ab, 9, 13.
- Section 5.6: 10, 17 and 18.
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Monday:
- Study: Monday, the 25th is a holiday, so today's
assignment is due WEDNESDAY, May 27th. (That will be the last
assignment that will be turned in and graded.) Read section 5.6
and start section 6.1.
- Do:
- Section 5.6: (1), 4, (5), 6, 12 and 15.
- Let $\{a_n\}_{n=0}^\infty$ be the Fibonacci sequence as in
problem #11 on the exam. Then $$ f(z)=\sum_{n=0}^\infty a_n z^n =
\frac{-1}{z^2+z-1}=\frac{-1}{(z-\alpha)(z-\beta)} $$ where
$\alpha=\frac{\sqrt 5-1}2$ and $\beta=\frac{-\sqrt 5 -1}2$. Use a
partial fraction decompositon and geometric series to show that $$
a_{n-1}= \frac{(\frac{1+\sqrt 5}2)^n - (\frac{1-\sqrt 5)}2)^n }{\sqrt
5} $$ for $n\ge0$.
- Show that the Fibonacci numbers grow faster than any power of $n$
in the following sense. Use the comparison test and what you know
about the radius of convergence of $\sum_{n=0}^\infty a_n z^n$ to show
that given $M>0$ and positive integer $k$, there is no $J$ such
that $n\ge J$ implies $a_n \le M n^k$.
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Wednesday:
- Study: Finish section 6.1
- Do:
- Let $$f(z)=\sum_{j=1}^\infty \frac{b_j}{z^j} \quad\text{
for $|z|>r$. } $$ We want to see 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.
- Section 6.1: (1beh, 3beh), 4 (here and elsewhere, you can
assume that the Laurent series for $f'$ can be obtained from that
of $f$ by term-by-term differentiation as we showed in the problem
above.), 5 and 6.
- Suppose that $f$ is analytic in a simply connected domain $D$
and that $f$ has finitely many distinct zeros $z_1,\dots,z_n$ in $D$
with orders $m_1,\dots,m_n$. Use problem 10 in section 5.6 to show
that $$ \frac{f'(z)}{f(z)}=\frac{m_1}{z-z_1}+\cdots
\frac{m_n}{z-z_n}+\frac{g'(z)}{g(z)} $$ where $g$ is analytic and
nonzero in $D$. Conclude that if $\Gamma$ is a positively oriented
simple closed contour in $D$ that contains all the $z_i$ in its
interior, then $$ \frac1{2\pi i}\int_\Gamma \frac{f'(z)}{f(z)}\,dz=
m_1+\cdots m_n. $$ Thus, in English, the contour integral counts the
number of zeros of $f$ inside $\Gamma$ up to multiplicity.
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Friday:
- Study: Read section 6.2
- Do:
- Section 6.2: (2), 3, 5, (7), and 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.
- Suppose $f$ has a pole of order k at $z_0$. What is
$\operatorname{Res}(\frac {f'}{f};z_0)$?
- Use the Residue Theorem to
restate the conclusion to the
written problem at the end of Wednesday's assignment 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
a simply connected domain $D$ except for possibly finitely many
poles. Suppose $f$ has finitely many zeros in $D$ and that
$\Gamma$ is a postively oriented simply closed contour in $D$
containing all the zeros and poles of $f$ in its interior. Show
that $$ \frac1{2\pi i}\int_\Gamma \frac{f'(z)}{f(z)}\,dz= M-P, $$
where $M$ is the number of zeros of $f$ counted up to multiplicty
and $P$ is the number of poles of $f$ counted up to
multiplicity.
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Monday:
- Study: No Class
- Do: Take a moment of contemplation for those who
sacrificed for your country.
|
Wednesday:
- Study: Read section 6.3
- Do:
- Section 6.3: 3, 5, 7, 9, 11 and 13.
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Friday:
- Study: Read section 6.4
- Do:
- Section 6.4: 2 and 3.
- Show that if $a>0$ and $b>0$, then
$$
\int_0^\infty \frac{\cos(ax)}{x^4+b^4}\,dx= \frac{\pi}{2b^3}e^\frac{-ab}{\sqrt 2} \sin\bigl( \frac{ab}{\sqrt 2}+\frac \pi4\bigr).
$$
- Show that if $a>0$ and $b>0$, then
$$
\int_0^\infty \frac{x^3\sin(ax)}{(x^2+b^2)^2}\,dx = \frac\pi4(2-ab)e^{-ab}.
$$
- Fun with the index: Let $\Gamma$ be a (not
necessarily simple) closed contour with $a\notin \Gamma$. Then we
define the index of $a$ with respect to $\Gamma$ to be $$
\operatorname{Ind}_\Gamma(a):= \frac1{2\pi i}\int_\Gamma
\frac1{z-a}\,dz. $$ If you draw a few pictures and think about the
Deformation Invariance Theorem, you should guess that
$\operatorname{Ind}_\Gamma(a)$ counts the number of times $\Gamma$
wraps around $a$ in the counterclockwise direction. (Thus, clockwise
encirlements count as $-1$.) Let's at least prove that
$\operatorname{Ind}_\Gamma(a)$ is an integer in the case that $\Gamma$
has a smooth parameterization $z:[0,1]\to \mathbf C$ so that $$
\operatorname{Ind}_\Gamma(a) =\frac1{2\pi i}\int_0^1
\frac{z'(t)}{z(t)-a}\,dt. $$ Define $$ \phi(s)=\exp\Bigl(\int_0^s
\frac{z'(t)}{z(t)-a}\,dt\bigr). $$
- Observe that it will suffice to see that $\phi(1)=1$.
- Let $\psi(t)=\displaystyle{\frac{\phi(t)}{z(t)-a}}$. Show that $\psi$ is
contstant and conclude that $\phi(t) =
\displaystyle{\frac{z(t)-a}{z(0)-a}}$.
- Since $\Gamma$ is closed, conlude that $\phi(1)=1$ as
required.
- Even more fun with the index: Recall from homework
(Wednesday) that if $f$ is analytic on and inside a simple
closed contour $\Gamma$, then if $f$ is nonzero on $\Gamma$, $$ N_f :=
\frac1{2\pi i}\int_\Gamma \frac{f'(z)}{f(z)}\,dz $$ is the number of
zeros of $f$ inside $\Gamma$ counted up to multiplicity. Let $f(\Gamma)$
be the closed contour which is the image of $\Gamma$ by $f$; thus if
$\Gamma$ is parameterized by
$z:[0,1]\to \mathbf C$, then $f(\Gamma)$ is parameterized by $t\mapsto
f(z(t))$ for $t\in [0,1]$. Note that $0\notin f(\Gamma)$. Show that
$N_f=\operatorname{Ind}_{f(\Gamma)}(0)$. In English, the number of zeros
of $f$ inside $\Gamma$ is equal to the number of times $f(\Gamma)$ wraps
around $0$.
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Monday:
- Study: We will call it quits with section 6.4 in the text.
- Do:
- Prove the Walking the Dog Lemma:Let
$\Gamma_0$ and $\Gamma_1$ be closed contours parametrized by
$z_k:[0,1]\to\mathbf C$ for $k=0,1$, respectively. Let $a\in\mathbf
C$ and suppose that $$ |z_1(t)-z_0(t)|<|a-z_0(t)|\quad\text{for $t\in
[0,1]$}. $$
- Note that $a\notin \Gamma_k$ for $k=0,1$.
- Parameterize $\Gamma$ by $z:[0,1]\to \mathbf C$ where
$z(t)=\frac{z_1(t)-a}{z_0(t)-a}$. Observe that $\Gamma\subset D=B_1(1)$ and conclude that $\operatorname{Ind}_\Gamma(0)=0$.
- Conclude that
$\operatorname{Ind}_{\Gamma_0}(a)=\operatorname{Ind_{\Gamma_1}}(a)$.
In other words, $\Gamma_0$ and $\Gamma_1$ wrap around $a$ exactly the
same number of times.
- Prove Rouche's Theorem: Suppose that $f$ and $g$ are
analytic on and inside a simple closed contour $\Gamma$, and that for
$z\in \Gamma$, $|f(z)-g(z)|<|f(z)|$. (Notice that this implies
neither $f$ nor $g$ has zeros on $\Gamma$.) Show that $N_f=N_g$, where
$N_f$ is the number of zeros of $f$ inside $\Gamma$ counted up to
multiplicity. (Use the Walking the Dog Lemma and the observation
$N_f=\operatorname{Ind}_{f(\Gamma)}(0)$.)
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