Homework Assigments

Week of May 7 to 11
(Due WEDNESDAY, May 16th)
Assignments Made on:

Monday: Lecture: Last time. Study: Because of the exam on Friday, this week's assignments will be due Wednesday, May 16th. 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. We are not meeting in our x-hour this week. Do: In section 5.5: 1ac, 6, 7ab, 9, 13.

Wednesday: Lecture: Last time. Study: Read section 5.6 Do: Section 5.6: 10, 17 and 18.

Friday: Exam: Take home portion is due Monday.

Week of May 14 to 18
(Due WEDNESDAY MAY 23.)
Assignments Made on:

Monday: Lecture: Last time. Study: Read section 5.6. Do: (EP-1) 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. (EP-2) Let $\{a_n\}_{n=0}^\infty$ be the Fibonacci sequence: $a_0=1$, $a_1=1$, and $a_n=a_{n-1}+a_{n-2}$ for $n\ge2$. On the practice exam (the 2015 midterm for this course), we showed that $$ f(z)=\sum_{n=0}^\infty a_n z^n = \frac{-1}{z^2+z-1}=\frac{-1}{(z-\alpha)(z-\beta)} $$ where $\alpha=\frac{\sqrt 5-1}2$ and $\beta=\frac{-\sqrt 5 -1}2$. Note that the raduis of convergence is $R=\alpha<1$. 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 $n\ge1$. (EP-3) 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: Lecture: Last time. Study: Finish section 5.6 Do: Section 5.6: (1), 4, (5), 6, 12 and 15.

Friday: Lecture: Last time. Study: Read 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 as we showed in the problem above.), 5 and 6. (EP-4) Suppose that $f$ is analytic on and inside a positively oriented simple closed contour $\Gamma$. 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-5) Suppose $f$ has a pole of order k at $z_0$. What is $\operatorname{Res}(\frac {f'}{f};z_0)$? (EP-6) Use the Residue Theorem to restate the conclusion to the written problem (EP-4) 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 $$ \frac1{2\pi i}\int_\Gamma \frac{f'(z)}{f(z)}\,dz= N_f-P_f, $$ where $N_f$ is the number of zeros of $f$ counted up to multiplicty and $P_f$ is the number of poles of $f$ counted up to multiplicity. NOTE: This problem was poorly worded. I hope it was clear that I meant that $N_f$ is the number of zeros inside $\Gamma$ (up to multiplicity), and that $P_f$ is the number of poles inside $\Gamma$ (up to multiplicity). Moreover, we need to assume that $\Gamma$ avoids the zeros and poles of $f$.

Week of May 21 to 25
(Homework No Longer Collected)
Assignments Made on:

Monday: Lecture: Last time. Study: Section 6.2 HOMEWORK: Only this assignment is due Wednesday along with all of last week's asignments. The rest of this week's assignments will not be collected. 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. 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: Read section 6.3 HOMEWORK: Recall that homework is no longer being collected. Do: Section 6.3: 3, 5, 7, 9, 11 and 13. 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 (EP-4) that if $f$ is analytic on and inside a positively oriented 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: Lecture: Last time. Final Exam: Our final exam will given on Saturday, June 2, from 8:00 to 11:00 in 007 Kemeny Hall. 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 $$ \int_0^\infty \frac{x^3\sin(ax)}{(x^2+b^2)^2}\,dx = \frac\pi4(2-ab)e^{-ab}. $$

Last Week
(The Final is June 2)
Assignments Made on:

Wednesday: Lecture: Last slides. We will finish up with Section 6.4 and today's lecture. The final is Saturday, June 2nd, from 8-11 in Kemeny 007. 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. 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 $N_f=\operatorname{Ind}_{f(\Gamma)}(0)$.) The Open Mapping Theorem: If $f$ is a nonconstant analytic function on a domain $D$, then $f(D)$ is open.

Thursday: Office Hours: 9-11 as usual.

Friday: Office Hours: 10-11 and 2-3 Final Exam: Saturday, 8-11 in 007 Kemeny Hall.

Dana P. Williams
Last updated May 30, 2018 10:46:07 EDT