Math 43
Functions of a Complex Variable
Last updated July 18, 2017 09:28:22 EDT

Announcements:

• THIS IS A PAST INSTANCE OF THIS COURSE. It was given in the spring of 2014.
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• Generally speaking, this page, rather than the canvas page, will contain most of the information you need for the course.

### Homework Assigments

Week of March 24 to March 28
 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 $xy>0$. 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: 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 March 31 to April 4, 2014
(Due Monday, April 7th)
 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. Wednesday: Study: Read sections 2.3 and 2.4 Do: Section 2.3: (1), (3), 4a, (8, 11efg), 12, (13, 14), 16. Section 2.4: 1c, (2), 3, (4 mentioned in lecture), 5, (6), 8, 12, 14. 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. BUT MY FIRST SUGGESTION WAS TOO SIMPLE!! I thank Kyutae Han for pointing out my sillyness. 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 $D$. So I defined $$v(a,b)=\int_{y_0}^b u_x(a,t)\,dt -\int_{x_0}^a u_y(s,y_0)\,ds.$$ 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.)

Week of April 7 to 11, 2014
(Due Monday, April 14)
 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. 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. Here is a disscussion of problems 20 and 21 in section 2.5. These problems illustrate some key points. Friday: Study: Read section 3.5. (We are skipping section 3.4.) Do: Section 3.5: 1ae, (3, 4,) 5, 11, 12, 15a, 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 in Thursday, April 17th. 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.

Week of April 14 to 18, 2014
(Due WEDNESDAY, April 23)
 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. Wednesday: Study: Study for the exam. Remember that this week's homework is due Wednesday, April 23rd. 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. 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" $$\int_\Gamma \mathbf{F}\cdot d\mathbf{r}=\int_\Gamma P\, dx + Q\,dy,$$ 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.

Week of April 14 to 18, 2014
(Due Monday, April 28th)
 Monday: Study: Last weeks assignments are due Wednesday. Today and the rest of this weeks assignments will be due Monday the 28th. 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 Midterm: The prelim will be returned in class Monday. If you miss class, you can pick it up in my office. Some information about he exam follows. Here is a hist-o-gram of the scores on the exam. Note that the quartiles were Q1 = 63, Q2 (the median) = 56, and Q3 = 44. Here are some brief solutions. Grades: If you are concerned about your grade or progress in the course, it is probably best to talk to me personally. A crude idea of you current standing can be obtained as follows. Add your raw exam score to your scores on the first three homeworks. This gives you a number between $75+36=111$ and zero. If your number is above 91, then you probably have some sort of "A". If it is above 77, then some sort of B. If it is above 50, then some sort of C. If your score is below that, then perhaps we should talk soon. 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 18th) 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. 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? Friday: Study: Read section 4.6 Do: Section 4.6: 4, 5, 7, and 11. 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.

Week of April 28 to May 2, 2014
(Due Monday, May 5th)
 Monday: Study: Read section 5.1. Review power series as necessary. Do: Section 4.6: 13, 14, and 15. Section 5.1: (3, 4), 5, 6, and 10. Wednesday: Study: Read section 5.2 Do: Section 5.1: 16, 18, 20 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, 8, and 15. (For 15, see Example 2 in the reading.)

Week of May 5 to 9, 2014
(Due WEDNESDAY, May 16th)
 Monday: Study: Because of the exam on Thursday, this week's assignments will be due Wednesday, May 16th. The exam will cover through section 5.3 in the text. We will not cover section 5.4 at all. Today, you should read section 5.5. Do: In section 5.5: 1, 6, 7ab, 9 and 13. 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. Possible approach to problem 10: There is a better way to do problem 10 than I suggested in office hours. Note that the function $g$ must satisfy $$g(z)=\frac{f(z)}{(z-z_1)^{m_1}\cdots (z-z_n)^{m_n}}.$$ In particular, $g$ must have isolated singularities at the points $z_1,\dots, z_n$. It will suffice to see that these are all removable singularities.

Week of May 12 to 16, 2014
(Due Wednesday, May 21, 2014)
 Monday: Study: Today's assignment is due WEDNESDAY, May 21st. (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 \#10 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 $a_n \le M n^k$. Midterm: The midterm is graded and will be returned in class Monday morning. If you miss class, you can get your exam during office hours. On the whole, the class did extraordinarily well on the exam -- which I felt contained some challenging questions. The median was 110/125 with first quartile 116/125 and thrid quartile 96/125. Here is a hist-o-gram of the scores. A very crude estimate of you grade in the course to date can be computed as follows. Add your midterm score (out of 75), the sum of your first 5 homework scores (out of 60) and the midterm grade (out of 125). This gives you a number between 0 and 260. A score of 230 or above is probably some sort of A. Above 194, some sort of B and above 160, some sort of C. I'll send some solutions along via a canvas announcement for security reasons. Wednesday: Study: Finish section 6.1 Do: 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 -- that can be proved using Riemann's Theorem from lecture.), 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. 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.

Week of May 19 to 23, 2014
(Homework no longer collected!!)
 Monday: Study: Read section 6.3 Do: Section 6.3: 3, 5, 7, 9, 11 and 13. Wednesday: 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, May 12) 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 $0$. Friday: Study: Skim section 6.5. Concentrate on pages 337 to 341 -- especially Lemma 4 which will will prove in class. We are not covering section 6.6 (yay!). We're taking a different approact to the material in 6.7 and you should rely on lecture (but mostly your own homework) for that material. Do: Section 6.5: 5, 9 and 12. Prove the Walking the Dog Lemma: (PROVED IN LECTURE)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 $N_f=\operatorname{Ind}_{f(\Gamma)}(0)$.)
 Monday: Study: Memorial Day Do: Wednesday: Study: All good things .... No more homework of any kind. Please monitor this space for updates (such as typo corrections or clarifications) to the final. Final Exam: Instructions: The exam is due in my office between 10am and 11am on Monday, June $2^{\rm nd}$. Hand the exam to me personally. Do not leave it under my door. Your solutions are to be fully justified and written neatly on one side only of $8\frac12''\times 11''$ paper with smooth edges and stapled in the upper left-hand corner. You are to work all problems. The number of each problem should be clearly labeled, and unless you are using LaTeX, each problem should start on a separate page. Print your name clearly on the first page of your exam. Getting the exam: Sometime Wednesday, you will get a canvas announcement with a link to a PDF of the final. Once you download the exam, you are bound by the usual honor code caveats: you can use our textbook, class notes, the firs two exam solutions and graded homework. But no other sources are allowed excepting that you can ask me for clarification. Prior to downloading the exam, you can ask general questions in office hours. Corrections and Clarifications: As of 31 May 2014 at 5:00pm. In problem 2, $n$ is an integer greater than $1$. In problem 4(c), $n$ is an integer greater than $1$. In problem 6a, you can assume $f$ is analytic in the punctured disk $B_r'(z_0)$.