Math 43 Spring 2023
Functions of a Complex Variable
Last updated May 19, 2023 16:04:31 EDT

Announcements:

• Copies of lecture slides will be available prior to lecture from our canvas page. I suggest annotating these during lecture as a form of note taking.

### Homework Assigments

Week of March 30 to April 3
(Solutions available on Monday)
 Monday: Read:Sections 1.1 and 1.2. 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: Files:The slides from Monday's lecture can be found under the "Files" tab on our canvas page. I also posted a solution to the ellipse example under the "Files" tab in the Solutions folder. 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. Hint: In section 1.3 number 11, you can use the fact that $(1+i)(5-i)^4=956-4i$. 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: Read: 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. It is possible that we won't finish section 1.6 until Monday. Do: In section 1.5, work: 10, 11, 14, 16. In section 1.6: 1, 10, 18 and 20. Suggested Only:In section 1.5: 5acf, 12 and 13. In section 1.6: 2-8 and 19.

Week of April 3 to April 7
Due Monday, April 10
 Monday: Study: Sections 2.1 and 2.2. Remember that we are meeting in our x-hour on Tuesday. Do: 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 authors 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.) Tuesday (x-hour): Study: Read Sections 2.3 and 2.4. We will not finish section 2.4 today. Please remember that there will be no lecture on Wednesday. Do: In Section 2.3: 4a, 12, 16. In Section 2.4: 3, 5. Friday: Study: Read Section 2.5 and 3.1. We will only cover a small part of section 3.1 in lecture, so you skip the parts we don not talk about. Do: In Section 2.4: 8, 12, 14. 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 will 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 are 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 we will agree to make this problem less messy by making the above assumptions.)

Week of April 10 to 14
Due Monday, April 17
 Monday: Study: For today, you should read Sections 3.1 and 3.2. We covered only part of Section 3.1 in lecture but the reading may help firming up that material. We may not finish all of 3.2 in lecture on Monday. 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 not apply L'Hopital as we haven't proved it in the complex case.) Wednesday: Study: Read Section 3.3. Some care is required as students often find the complex logarithm confusing the first time they see it. Do: In Section 3.3: 3, 4, 9, 14. Friday: Study: Read Section 3.5. (We are not covering Section 3.4 although a quick skim of the material could still be helpful.) We may start of Section 4.1 if there is time. The preliminary exam on Tuesday will cover through Section 3.5. Do: In Section 3.5: 1ae, 5, 11, 12, and 19.

Week of April 17 to 21
Due Monday, April 24
 Monday: Study: Read Section 4.1 and get started on 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. Do: In Section 4.1: 3, 4, 8. Tuesday (x-hour): Preliminary Exam: There will be a 50 minute "in class" portion in our x-hour. The "take-home" portion will be available on gradescope must be uploaded prior to the start of class on Wednesday. Wednesday: Study: Finish Section 4.2 and read the first part of section 4.3. Do: In Section 4.2: 5, 6a and 14. (EP-1) 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 $$\int_\Gamma f(z)\,dz=\int_\Gamma P\,dx+Q\,dy +i \Bigl(\int_\Gamma R\,dx + T\,dy\Bigr).$$ Friday: Study: Finish Section 4.3. Do: In Section 4.3: 2, 3, 5.

Week of April 24 to 28
Due Monday, May 1st
 Monday: Study: Read Section 4.4a. We are not covering Section 4.4b. Do: In Section 4.4: 2, 15, 18, 19. Wednesday: Study: Read Section 4.5 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: Study: Finish or re-read Section 4.5. Do: In Section 4.5: 6, 8, 13, 15.

Week of May 1 to May 5
Due Monday, May 8
 Monday: Study: Read Section 4.6. Do: In Section 4.6: 4, 5, 7, 11, 13, 14. Suppose that $f$ is entire and that $|f(z)|\ge 1$ for all $z$. Prove that $f$ is constant. Chapter 5: Today and Wednesday, we start working with power series. A review of series as well as both Talyor and MacLaurin series would not go amiss. Wednesday: Study: Read Section 5.1 Do: In Section 5.1: 5, 6, 16, 18, 20. Friday: Study: Read Section 5.2. Do: In Section 5.2: 4, 10, 11bc, 13.

Week of May 8 to 12
Due Monday, May 15
 Monday: Study: Read Section 5.3. Do: In Section 5.3: 1, 6, 8. Prove the following result from lecture: (EP-1)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 $\overline{B_r(0)}$. Study for the Midterm tomorrow Wednesday: Study: Read Section 5.5. We are not formally covering Section 5.4. Do: In Section 5.5: 1ac, 6, 7ab, 9, 13. Friday: Study: Start Section 5.6. We probably will not cover all of Section 5.6 today. Do: (EP-2) 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 15 to 19
Due Monday, May 22nd
 Monday: Study: Finish Section 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. For problem 10, what kind of singularities does $g(z)=(z-z_1)^{-m_1}\cdots(z-z_n)^{-m_n}f(z)$ have? Wednesday: Study: Read Section 6.1. Do: In Section 6.1: 4, 5, 6. Friday: Study: Read Section 6.2 and start Section 6.3. Do: In Section 6.2: 3, 5, 9. (For problem 9, the binomial theorem might be helpful.) I am only assigning a few of these as the answers are provided. Use your own judgement about how much practice you need. I suggest you DO NOT try problem 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.
 Monday: Study: Read Section 6.3 and start Section 6.4. Note that we are no longer collecting homework. But this material will be on the final and I will distrubute some solutions in due course. 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. Wednesday: Study: Read Section 6.4 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. Friday: Study: Read Section 6.7. Our approach is more streamlined than that of the text. You are only responsible for what we do in lecture. Also be aware that Monday, May 29th, is a holiday. Our last class will be Wednesday, May 31st. Do: In Section 6.7: 2, 4, 6. Note we already did problem 3.