Math 43
Functions of a Complex Variable
Last updated May 28, 2015 13:05:25 EDT

Announcements:

• Generally speaking, this page, rather than the canvas page, will contain most of the information you need for the course.
• By popular demand, the final format will be Take-Home and due at 11am on Monday, June 8th.

### Homework Assigments

Week of March 30 to April 3
 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 April 6 to 11, 2014
(Due Monday, April 13th)
 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. Wednesday: Study: No lecture on Wednesday. We'll meet Thursday in our x-hour. Do: Catch up on your reading 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. 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.)

Week of April 13 to 17, 2015
(Due Monday, April 20)
 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. 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.

Week of April 20 to 24, 2015
(Due WEDNESDAY, April 29)
 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 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. 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. 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.

Week of April 20 to May 1, 2015
(Due Monday, May 4th)
 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. 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. 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, 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.

Week of May 4 -- 8, 2015
(Due Monday, May 11th)
 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 $D=\{\,z:|z|<1\}$. 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.

Week of May 11 to 15, 2015
(Due WEDNESDAY, May 20th)
 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 $|\sum c_n|\le\sum|c_n|$.) 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.

Week of May 18 to 22, 2015
(Due Wednesday, May 27, 2014)
 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 $a_n \le M n^k$. 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. 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.
 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. 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 $0$. 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 $N_f=\operatorname{Ind}_{f(\Gamma)}(0)$.)