Math 43
Functions of a Complex Variable
Last updated June 01, 2020 09:13:08 EDT

### Homework Assigments

Week of March 30 to April 3
(Due Monday, April 6)
 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 $xy>0$. 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. 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.

Week of April 6 to 10
Due Monday, April 13
 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. 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. 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!

Week of April 13 to April 17
Due Monday, April 20
 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. 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. 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.

Week of April 20 to 24
Due Monday, April 27
 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 $$\int_\Gamma f(z)\,dz=\int_\Gamma P\,dx+Q\,dy +i \Bigl(\int_\Gamma R\,dx + T\,dy\Bigr).$$ 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. 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.

Week April 27 to May 1
Due Monday, May 4
 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. 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. 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.

Week of May 4 to May 8
Due Monday, May 11
 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. 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. 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.

Week May 11 to 15
Due Monday, May 18
 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 $\overline{B_r(0)}$. 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. 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.

Week of May 18 to May 22
Due Monday, May 27
 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.) 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. 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.

Week of May 25 to 29
Due Monday, May 18
 Monday: : Memorial Day Holiday Do: Chill 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. 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. 