Monday:
- Watch the welcome video as well as both parts of lecture
1. You can access these either via our canvas page or via
the Lectures link on this page. Lecture 1 is meant to
supplement Sections 1.1 and 1.2 and you should review those
as well.
This will be a busy week.
Normally, we meet in our x-hour this week in order to jump
start the course and get the fundamentals under our belt. In
the current remote format, we cover this materal in the usual
three MWF bits.
- 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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Wednesday:
- Lecture: Watch all three parts of lecture 2. This
material is supplemented by sections 1.3, 1.4 and 1.5 and you
should review these as well.
- 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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Friday:
- Lecture: Watch all three parts of lecture 3. This
material is supplemented by 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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Monday:
- Summary: Each class period will start with a short
summary of the lecture for that day. I will keep the meeting
open for the remainder of the class period to field questions.
Feel free to treat this time just like office hours. In
partiular, you should feel free to drop in and out of the
meetin if you desire.
- Homework solutions: Here
are selected solutions for the
first week's homework.
- Lecture: Watch both parts of Lecture 4. This material
is supplemented by sections 2.1 and 2.2 in the text.
- Do:
- Download the practice
document, answer the questions, scan your solutions to a PDF
document, and then upload your completed version via the canvas
page by 10:10 am on Wednesday (April 8th).
- 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 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 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:
- Lecture: Watch both parts of Lecture 5. This material
is supplemented by sections 2.3 and 2.4 in the text.
- Do:
- In Section 2.3: 4a, 12, 16.
- In Section 2.4: 3, 5, 8, 12, 14.
- 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:
- Lecture: Watch both parts of Lecture 6. This material
is supplemented by secions 2.5 and parts of section 3.1. We will
cover only a small part of section 3.1 in the lecture, so you
should have a closed look at the entire section on your own.
- Do:
- 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'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 $z\mapsto
\ln(|z|)+i\operatorname{Arg}(z)$ 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'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.)
- Suggested: In Section 2.5: 1b, 2, 3cd, 10. Yikes!
Problem 10 is ridiculously complicated. Don't try it!
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Monday:
- Homework Solutions: Here are
some selected solutions
to the homework for week 2.
- Lecture: Watch both parts of lecture 7. This material
is supplemented by sections 3.1 and 3.2 in the text.
- 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't 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:
- Lecture: Watch all three parts of lecture 8. This material
is supplemented by section 3.3 in the text and introduces the
complex logarithm. Students generally find the logarithm a bit
mysterious, so careful study is called for.
- Do: In Section 3.3: 3, 4, 9, 14.
- Suggested: In Section 3.3: 5, 6.
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Friday:
- Lecture: Watch both parts of lecture 9. This material
is supplemented by sections 3.5 and first part of section 4.1 in
the text. The preliminary exam will cover through and including
section 3.5. The exam will be available about noon on Wednesday,
April 22nd and must be uploaded to our canvas page by 10:10 am on
Friday. Sorry, all times are EDT.
- Do: In Section 3.5: 1ae, 5, 11, 12, and 19.
- Suggested: In Section 3.5: 3, 4, and 15a.
Suggested only Section 3.5: 3, 4, and 15a.
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Monday:
- Homework Solutions Here
are selected solutions for
last week's homework.
- Lecture: Watch both parts of lecture 10. This material is
supplemented by section 4.1 and part of 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.
- Reminder:Recall that the preliminary exam will be available
Wednesday and will be due before class on Friday. Since several
people requested it, here is a copy of last
year's preliminary
exam. Note that this exam was partially in class and partially
take-home.
- Do:
- In Section 4.1: 3, 4, 8.
- 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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Wednesday:
- Lecture: Watch both parts of lecture 11. This material
is supplemented by section 4.2 and the first part of section 4.3.
We will finish up section 4.3 Friday, but the lecture will be
recorded live during out class period to allow folks to work on
the exam all day Thursday.
- Do: In Section 4.2: 5, 6a and 14. (While there is probably no
harm is waiting to work these problems until after you turn in
your exam, it might be useful to try to keep up if you have
time.)
- Preliminary Exam: Here is
the Preliminary Exam.
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Friday:
- Lecture: Because of the exam, lecture 12 will be
recorded live during our class period on Friday (April 24th).
This material is supplemented by section 4.3 in the text.
- 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:
- Preliminary Exam: Your graded preliminary exams will be
returned via canvas. Here is a link to
a hist-o-gram of the classes
scores. Additional information about the exam including
solutions will be emailed via canvas.
- Homework Solutions: Here
are selections solutions to
last week's homework.
- Mathematica Demo: Here is
the deformation demo from
lecture. You will need Mathematica to run it.
- Lecture: Watch both parts of lecture 13. This material
is supplemented by section 4.4a in the text. We still have a bit
more of section 4.4a to do on Wednesay.
- Do: In Section 4.4: 2, 15, 18, 19.
- Suggested: In Section 4.4: 1, 3, 5, 9, 11.
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Wednesday:
- Lecture: Watch both parts of lecture 14. This material
is supplemented by sections 4.4 and part of section 4.5. We are
getting into the heart of the course now and it is important to
try to digest the statements of important theorems like the Cauchy
Integral Theorem and the Cauchy Integral Formula as well as their
implications.
- 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:
- No Class Meeting Monday: We will NOT have our regular
class meeting on Monday (May 4th). There will be a recoreded
lecture for day as usual. We just won't meet to review it. I
will hold an office hour Monday afternoon (1:30-2:30) in addition
to my regular TTh 1:30-2:30 office hours.
- Lecture: Watch both parts of lecture 15. This material
is supplemented by section 4.5 in the text.
- Do: In Section 4.5: 6, 8, 13, 15.
- Suggested:In Section 4.5: 1, 3, 10.
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Monday:
- Homework Solutions: Here
are selected solutions to
last week's homework.
- Lecture: Watch both parts of lecture 16. This material
is supplemented by section 4.6 in the text.
- Do:
- In Section 4.6: 4, 5, 7, 11, 13, 14, 15.
- Suppose that $f$ is entire and that $|f(z)|\ge 1$ for all
$z$. Prove that $f$ is constant.
- Chapter 5: On 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:
- Lecture: Watch both parts of lecture 17. This material
is supplemented by section 5.1 in the text.
- Do: In Section 5.1: 5, 6, 10, 16, 18, 20, 21.
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Friday:
- Last Year's Midterm: Here is
the midterm from last year. Of
course, we had a different format for exams in that class.
- Lecture: Watch both parts of lecture 18. This material
is supplemented by section 5.2 of the text.
- Do: In Section 5.2: 4, 10, 11bc, 13.
- Optional: This is just for fun and not really part of
Math 43. Define $f:\mathbf R\to\mathbf R$ by $$f(x)=\begin{cases}
e^{-\frac1{x^2}}&\text{if $x\not=0$, and}\\ 0&\text{if
$x=0$.}\end{cases}$$ As claimed in lecture, we want to show that
$f^{(n)}(0)$ exists and is equal to $0$ for all $n\ge1$. The
problem is that we can't find a formula for $f^{(n)}(x)$ when
$x\not=0$. Instead, I suggest
the following.
- If suffices to see that $\lim\limits_{x\to 0} \frac{f^{(n)}(x)}x=0$
for $n\ge0$.
- Use L'Hopital's Rule to show that
$\lim\limits_{x\to\infty}\frac{P(x)}{e^{x^2}}=0$ for any polynomial
(with real coefficients) $P$.
- Show that if $x\not=0$, then
$f^{(n)}(x)=P(\frac1x)e^{-\frac1{x^2}}$ for some polynomial
$P$. (You aren't supposed to be able to figure out what $P$ is,
just that there is one.)
- Use parts 2 and 3 to establish 1.
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Monday:
- Homework Solutions:Here
are selected solutions to
last week's assignments as well as todays.
- Lecture: Watch both parts of lecture 19. This material
is supplemented by section 5.3 of the text.
- Midterm: Our midterm will be made available after our
class meeting on Wednesday. You will need to upload your
solutions before the beginning of our class period on Friday
(10:10am EDT). The exam will cover thru and including lecture 19
and section 5.3 in the text.
- 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:
- Friday's Lecture: Friday's lecture will be recored live
during our regular class meeting due to the exam.
- Midterm: The midterm will be emailed out today after
our class meeting. You must upload your solutions before our
class meeting on Friday.
- Lecture: Watch both parts of lecture 20. This material
is supplemented by section 5.5 in the text. We are not formally
covering section 5.4.
- Do: In Section 5.5: 1ac, 6, 7ab, 9, 13.
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Friday:
- Lecture: Today's lecture will be recorded during our
class meeting. It will be supplemented by section 5.6. We will
not finish 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:
- Midterm: The midterm has been graded and returned
together with solutions. Here is
a hist-o-gram of the scores
on the midterm.
- Homework Solutions: Here
are selected solutions to last week's
homework. Recall that Monday's assignment was included with last
week's solutions.
- Lecture: Watch both parts of lecture 22. This material
is supplemented by section 5.6. We have finished 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, 17, 18. (You
should compare 17 and 18 to problem 8 on the midterm.)
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Wednesday:
- Lecture: Watch both parts of lecture 22. This material
is supplemented by section 6.1 in the text.
- Do:
- In Section 6.1: 4, 5, 6.
- In Section 5.6: 11. I was unable to use the hint in the
back of the book. I would be very interested if someone can
show me how to use it! Instead I suggest the following.
- Show that it suffices to see
that if $f$ has a pole at $z_0$, then $\operatorname{Re}f$ is
unbounded in every deleted
neighborhood of $z_0$.
- If, to the contrary, $\operatorname{Re}(f(z))\le M$
for all $z$, then consider $g(z)=\exp(f(z))$ and note that
$f'=g'/g$.
- I found #6 in Section 5.6 as well as #6 is section 6.1
useful.
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Friday:
- Lecture: Watch both parts of lecture 24. This material
is supplemented by section 6.2 and a little bit of section 6.3.
We will be following a slightly different path through section 6.3
and 6.4 that the text does. Be aware of the differences.
- 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. For example, you might also want
to consider, at least, 2 and 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:
- : Memorial Day Holiday
- Do: Chill
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Wednesday:
- Homework Solutions: Here
are selected solutions for
last week.
- Lecture: Watch all three parts of lecture 25. The
material is supplemented by parts of both section 6.3 and 6.4. We
are taking a slightly different approach than in the text.
- 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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Friday:
- Last Year's Final: Here
is last year's final exam.
Note that this was an "in class" exam.
- Mathematica Demos: The plus-one
demo and the
exponential demo.
- Lecture: Watch all three parts of lecture 26. This
material is supplemented by sections 6.3 and 6.4 in the text.
- 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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