Assignments Page

Math 43
Functions of a Complex Variable
Last updated May 29, 2019 13:48:26 EDT

Homework Assigments

Week of March 25 to 29
(Due Monday, April 1st)
Assignments Made on:

Monday:

lecture: fields.
Study: Sections 1.1 and 1.2
Do:
1. If you haven't already, complete "Homework Zero" on the Canvas Website.
2. In section 1.1 work problems: 4, 8, 12, 21 and 30.
Suggested Only:
1. In section 1.1 look at: 15, 19, 22, 24 and 28.
2. 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 $xy>0$.

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.

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.

Friday:

Lecture: Last time.
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.

Week of April 1 to 5
(Due Monday, April 8th)
Assignments Made on:

Monday:

Study: Read sections 2.1 and 2.2
Do:
- Section 2.1 Work: 10, 12, 13.
- Section 2.1 Suggested: 1ace, 3d, 5, 6ab, 7, 8, 9.
- Section 2.2 Work: 5, 11de, 15, 22, 25bde.
- Section 2.2 Suggested: 2, 4, 6, 12, 18.
- 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, \#11d. The answer in the back of the book is incorrect.

Wednesday:

Lecture: Here are careful statements of our Cauchy-Riemann theorems.
Study: Read sections 2.3 and 2.4
Do:
- Section 2.3 Work: 4a, 12, 16.
- Section 2.3 Suggested: 1, 3, 8, 11efg, 13, 14.
- Section 2.4 Work: 3, 5, 8, 12, 14.
- Section 2.4 Suggested: 1, 2, 4 (mentioned in lecture), 6.

Friday:

Last Time: Some review.
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 Work: 5, 6, 8, 18, 20* and 21*.
- Section 2.5 Suggested: 1b, 2, 3cd, 10.
- 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 $z\mapsto \ln(|z|)+i\operatorname{Arg}(z)$ 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 $Arg (x+ i y)$ 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.)

Week of April 8 to 12
(Due Monday, April 15)
Assignments Made on:

Monday:

Lecture: Here are our remarks on partial fraction decompositions.
X-hour: We will meet in our x-hour (Thursday) this week.
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), 7, 10, and 15ac.
Suggested only: Section 3.1: 4 and 12.

Wednesday and Thursday:

Lecture: Some reviewfrom Monday's lecture and Wednesday's lecture.
Study: Read section 3.3. We will meet both Wednesday and in our $x$-hour this week. You can work the material for Section 3.2 after Wednesday's lecture and finish Section 3.3 after Thursday's lecture.
Do:
- Work Section 3.2: 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 $z$.)
- Suggested only Section 3.2: 5de, 8, 9, 11.
- Work Section 3.3: 3, 4, 9 and 14.
- Suggested only Section 3.3: 5 and 6.

Friday:

Study: Read section 3.5. (We are skipping section 3.4.)
Do:
- Section 3.5: 1ae, 5, 11, 12, and 19.
- Suggested only Section 3.5: 3, 4, and 15a.
- Please also work this problem: Is there an analytic branch of $\log z$ defined in the annulus $D=\{\,z\in\mathbf{C}:1<|z|<2\,\}$?
- Recall that our preliminary midterm is Friday, April 19th. It will cover up to and including Section 3.5 which I hope to complete Friday or Monday.
- Be aware that it is not likely that this assignment will be returned prior to the exam.

Week of April 15 to 19
(Due WEDNESDAY, April 24)
Assignments Made on:

Monday:

Schedule: At the moment, we aren't planing to meet in our x-hour this week. But that will depend on how far we get Monday and Wednesday. Stay tuned.
Lecture: From last time.
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.

Wednesday:

Lecture: Contour Integrals.
Study:Review sections 4.1 and 4.2. Read section 4.3. Remember that this week's homework is due Wednesday, April 24th.
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.

The Exam: The exam will cover through and including section 3.5 in the text. (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.

Friday:

In Class Portion of Preliminary Exam
Do: The take-home portion of the exam is due Monday. This week's homework is due Wednesday

Week of April 22 to 26
(Due Monday, April 29th)
Assignments Made on:

Monday:

Lecture: Review of Wednesday's lecture. A look at deformations.
Study: Last weeks assignments are due Wednesday. Today's and the rest of this weeks assignments will be due Monday the 29th of April. 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: 2, 15, 18, and 19.
Suggested Section 4.4: 1, 3, 5, 9, and 11.

Wednesday:

Lecture: Last time.
Study: Read section 4.5. Note that we are unlikely to finish all of section 4.5 today.
Do:
- Work Section 4.5: 2, 6, 8, 13, 15 and 16.
- Suggested only Section 4.5: 1, 3, and 10.
- 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.

Friday:

Lecture: more Cauchy and even Riemann.
Running Behind: I've fallen behind this week, so there will be no additional assignment today.
Study: Read section 4.6
Next week: We start working with power series on Monday. Reviewing power series as well as MacLaurin and Taylor series would not go amiss.

Week of April 29 to May 3

(Due Monday, May 6th)

Assignments Made on:

Monday:

Lecture: Last time.
Riemann's Theorem: Here is a short proof of Riemann's Theorem.
Study: Read section 4.6

Do:

Section 4.6: 4, 5, 7, 11, 13, 14 and 15.

Also:Suppose $f$ is an entire function such that $|f(z)|\ge1$ for all $z$. Show that $f$ is constant.

Wednesday:

Last time.
Study: Read section 5.1. Review power series as necessary.

Do:

Section 5.1: 5, 6, and 10.

Suggested only Section 5.1; 3 and 4.

Friday:

Study: Review power series as necessary.

Do:
Section 5.1: 16, 18, 20 and 21.

Section 5.2: 4, and 10 .

Week of April 29 to May 3
(Due Monday, May 6th)
Assignments Made on:

Week of May 6 to 10

(Due Wednesday, May 15)

Assignments Made on:

Monday:

Exam: The exam will cover through and including section 5.3. That will include at least half of today's lecture and all of today's assignment. Of course, due the exam on Friday, this week's assignments will be due WEDNESDAY (May 15).
Lecture: Last time.
Study: Read section 5.3.

Do:

Section 5.2: 11bc, 13.

Section 5.3: 1, 6, and 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 $\overline{B_r(0)}$.

Wednesday:

Lecture: last time.
Study: We are NOT covering section 5.4 in the text. Read section 5.5.

Do: In section 5.5: 1ac, 6, 7ab, 9, 13.

Friday:

: The in-class portion of the midterm will be given in lecture. The take-home portion is due at the beginning of lecture on Monday. Turn your exam in in class. DO NOT turn it in to the homework boxes.

Week of May 6 to 10
(Due Wednesday, May 15)
Assignments Made on:

Week of May 13 to 17

(Due Monday, May 20)

Assignments Made on:

Monday:

Study: Read Section 5.6

Do: Section 5.6: 10, 17 and 18.

Wednesday:

Last time: Zeros and isoloated singularities.
Study: Read section 5.6. We will not meet Friday this week, but we will meet in our x-hour on Thursday.

Do: (EP-1) 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.

Do: Section 5.6: 4, 6, 12, and 15.

Suggested only: Section 5.6: 1 and 5.

Thursday (X-hour):

Study: Read Section 6.1

Do: Section 6.1: 4, 5, and 6.

Suggested only: 1beh and 3beh.

Friday:

Study: NO CLASS FRIDAY

Do:

Week of May 13 to 17
(Due Monday, May 20)
Assignments Made on:

Week of May 20 to 24

(NOT TO BE TURNED IN)

Assignments Made on:

Monday:

Lecture: Residues.
Study: Read Section 6.2

Do: In section 6.2: 3, 5, 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. For example, you might also want to consider, at least, 2 and 7.
(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.

(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. $f$.

Comment: This is not to be turned in, I just thought you might be interested. I meantioned in lecture that these sorts of definite trigometric integrals 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 $$d\theta=\frac{2dz}{z^2+1}dx.$$

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.

Wednesday:

Lecture: Last time.
Study: Sections 6.3 and 6.4. Although you no longer need to turn in your homework, this material will definitely be on the final exam -- and practice is crucial.

Do: Section 6.3: 1, 2 and any others where the answer is provided. But have a good look at 9, 11, and 13. (13 is a little harder.) The material developed in problems 14 -- 19 will NOT be on the exam.

Do: Section 6.4: 2, 3, and 7. Note that we only delt with imporoper integrals of real-valued functions in class. Hence the techniques in the text are not required and you are NOT responsible for problems such as 4, 9, 10, and 12. You are welcome to use the techniques in the text, but only if you explain and justify what you are doing.

Friday:

Lecture: Last time.
Study: We will cover only enough of Section 6.5 to honor the memory of the epic battle between Henry Helson and Jeff Fox back in the day. You can try problem 10 if you like.

Do: The remainder of the course is based on Section 6.7, although we are not following the book closely. I will post some homework here once we've covered enough.

Final Exam: The final exam is Friday, May 31, from 3pm to 6pm in our regular class room: 006 Kemeny. The final will be all "in class".

Week of May 20 to 24
(NOT TO BE TURNED IN)
Assignments Made on:

Week of May 27 to May 31

(FINAL EXAM IS FRIDAY)

Assignments Made on:

Monday:

Study: NO CLASS

Do:

Wednesday:

Lecture: Last time.
Study: Skim Section 6.7. You are only responsible for what we cover in class

Do: Section 6.7: 2, 3, 4, 5, and 6.

Friday:

Study:

Do: Our Final Exam is from 3:00 pm to 6:00 pm in 006 Kemeny Hall.

Week of May 27 to May 31
(FINAL EXAM IS FRIDAY)
Assignments Made on:

Dana P. Williams
Last updated May 29, 2019 13:48:26 EDT