Math 113 Spring 2024
My Complex Analysis Course
Last updated May 24, 2024 12:34:22 EDT

General Information HW Assignments Canvas Page


Announcements:


Week of March 25 to 29
Assignments Made on:
Monday:
  • Homework 0: Before class on Monday, you should download a copy of the text from the "Files" tab on our canvas page and read/skim sections 1.1 thru 1.4. Also download homework-0 from the gradescope page and answer the survey questions there. Then upload a PDF file with your answers in gradescope. Part of the point of homework-0 is to make sure everyone is comforable with gradescope and the process of uploading solutions and linking the problems to their respective pages. Also, it's fun to tell your friends that there was an assignment due the first day of class. (Note that you can actually turn in homework-0 any time this week.)
  • Solutions: I have uploaded solutions to (most of) the exercises in Sections 1.1 to 1.3 under "Files" on our canvas page. Please let me know if you have trouble accessing them. These are also fair game for the Typo competition.
  • Do: Ex 1.4.1, Ex 1.4.7(a), Ex 1.4.8, Ex 1.4.12.
  • Study: (Not to be turned in.) If you ever might teach this stuff, you should have a look at most of the exercises in Section 1.3 and 1.4. Especially Ex 1.4.3-5.
Wednesday:
  • Read Section 1.5.
  • Do: Ex 1.5.2 and Ex 1.5.8.
  • Study: Ex 1.5.9 and Ex 1.5.10.
Friday:
  • Read Sections 2.1 and 2.2. Since I've fallen a bit behind, we probably won't finish section 2.2 today. But you shuold be able to work on everything by Ex 2.2.3 and Ex 2.2.5. (Ex 2.2.5 is really untanglind definitions, but it will probably be tricky for most.) Please turn in the first three assignments by Tuesday evening. This will give you time to stop by office hours if you need some help or hints.
  • Do: Ex 2.1.8, Ex 2.2.3, Ex 2.2.5
  • Study: Ex 2.1.4, Ex 2.1.6, Ex 2.2.4, Ex 2.2.7 (This is tricky, but gives another version of Thm 2.6 without the unmotivated step of the proof).


Week of April 1 to April 5
Assignments Made on:
Monday:
  • Study: Catch up Day.
  • Do: No new assignment. Note that Homework 1 is due by tomorrow.
Wednesday:
  • Study:Read Sections 3.1 and 3.2 through Definition 3.15.
  • Do: EP-1: Suppose that $D$ is a domain and that $0\notin D$. Show that there is a branch of $\log z$ in $D$ if and only if $1/z$ has an antiderivative on $D$. Hint: if $f'(z)=1/z$ for all $z\in D$, consider the function $h(z)=z/\exp(f(z))$ and note that $\exp(z)=a$ has infinitely many solutions if $a\not=0$.
Friday:
  • Study: Read through section 3.3.
  • Do: Ex 3.2.2 and Ex 3.2.5
  • Study: You really should be aware of Ex 3.2.6, Ex 3.2.7, and Ex 3.2.8.


Week of April 8 to 12
Assignments Made on:
Monday:
  • Study: NO CLASS.
  • Do: Use Eye Protection.
Tuesday (x-hour):
  • Study: Read Section 4.1
  • Do: Ex 4.3.1
Wednesday:
  • Study: Read Section 4.2 and 4.3
  • Do:
    1. Ex 4.2.1, Ex 4.3.2, Ex 4.3.4, and Ex 4.3.8. The challenge of Ex 4.3.4 is the write up and elegant solution without hand waving.
    2. Complete the proof of Cauchys Theorem for Convex Domains by showing that if $f$ is continuous at $z_0$, then $$ \lim_{z\to z_0}\frac1{z-z_0}\int_{[z_0,z]} \bigl(f(w)-f(z_0)\bigr) \,dw=0.$$
Friday:
  • Study: Read Section 5.1. I should not have assigned Ex 4.3.2 until we have covered Moreras Theorems in lecture. (I suppose this is a hint as to how I would solve it.)
  • Do: Ex 5.1.2 and Ex 5.1.3.
  • Study: You should know how to do Ex 5.1.1. Notice that Ex 5.1.4 is supposed to be trivial since the identities hold in the real case and you apply Corollary 5.3. You should at least think about Ex 5.1.5 long enough to realize that it is not an immediate consequence of Theorem 5.2.


Week of April 15 to 19
Assignments Made on:
Monday:
  • Study: Read section 5.2. Last week's homework is due tomorrow.
  • Do: Ex 5.2.4, Ex 5.2.10, and Ex 5.2.11
  • Be Aware: Ex 5.2.3, Ex 5.2.5, Ex 5.2.7, Ex 5.2.12.
Wednesday:
  • Study: Read sections 5.3 and 5.4
  • Do: Ex 5.3.6, Ex 5.3.9, Ex 5.3.12, Ex 5.4.1, and 5.4.2. Especially in Ex 5.3.6, you can use the result of Ex 5.3.5 without proof.
  • Be Aware: You should be comfortable with Ex 5.3.5. If fact you should consider it assigned but ungraded. If you like Ex 5.3.12, have a look at Ex 5.3.13.
Friday:
  • Study: Read section 5.5 up to the Open Mapping Theorem.
  • Do: Try Ex 5.4.5. This is sort of a challenge problem and no one should get stressed if they can't do it.


Week of April 22 to 26
Assignments Made on:
Monday:
  • Study: We'll finish most of section 5.5 today. Remember, there is no class this Wednesday. We will make in up in a future x-hour.
  • Do: Ex 5.5.2 (assume $f$ is not constant), Ex 5.5.3, and Ex 5.5.4. (You can use Ex 2.2.4 without proof if you read the solution.)
Wednesday:
  • Study: No Class today.
  • Do:
Friday:
  • Study: Finish section 5.5 and start section 6.1
  • Do: Ex 5.5.6


3
Week of April 29 to May 3
Assignments Made on:
Monday:
  • Study: Read section 6.1
  • Do:
    1. In Homework 4, and number of folks wanted to use L'Hospital's Rule. We haven't proved any such result, nor do the proofs from calculus work in our setting of analytic functions. So I created a few problems, four actually, to see what we can recover. The assignment can be found under the Files -> Homework Solutions tab on our canvas page. The file is lhospital.pdf.
    2. Verify the claim made in lecture during the proof of the general Cauchy Theorem: suppose $D$ is a domain and that $g$ is continous on $D\times D$. If $z_n \to z$ in $D$ and if $K\subset D$ is compact, show that $f_n(\cdot) =g(z_n ,\cdot)$ converges uniformly on $K$ to $f(\cdot)=g(z,\cdot)$. (Hint: there are lots of ways to approach this problem. I used the fact that a continuous function on a compact metric space is necessarily uniformly continuous -- a reference is Theorem 4.19 in Rudin's Principles of Mathematical Analysis or Theorem 1.33 of Foland's Advanced Calculus.)
Wednesday:
  • Study: Read section 6.2.
  • Do: When we finish section 6.2: Ex 6.2.2, Ex 6.2.4 (use Ex 6.2.3), Ex 6.2.6 (consider Corollary 6.17).
Friday:
  • Study: Read section 7.1
  • Do: Ex 7.1.3


Week of May 6 to 10
Assignments Made on:
Monday:
  • Study: Read section 7.2. Recall that we are meeting in our x-hour (aka tomorrow) this week.
  • Do: Ex 7.2.2
Tuesday (x-hour):
  • Study: Start section 7.3. We will finish up (probably) on Wednesday.
  • Do: No new assignment
Wednesday:
  • Study: Finish section 7.3 and start section 7.4
  • Do: Suppose that $D$ is an open connected subset of $\mathbf C$.
    1. If $p\in D$, show that $D\setminus\{p\}$ is connected. (You can assume that any deleted neighborhood, $D_{\epsilon}'(p)$ is path connected and therefore connected.)
    2. Show that if $D$ is a neighborhood of $\infty$, then $D\cup \{\infty\}$ is connected in $\mathbf C^+$.
Friday:
  • Study: Finish up section 7.4. If there is time, we will skip ahead to Chapter 10.
  • Do: Ex 7.4.2, Ex 7.4.3, and as another challenge problem try Ex 7.4.4.


Week of May 13 to 17
Assignments Made on:
Monday:
  • Study: Read section 10.1 although we probably won't finish it today.
  • Do:
    1. Let $D=D_1(0)$ be the open unit disk and $\mathbf T=\{\,z\in \mathbf C:|z|=1\,\}$ its boundary so that $\overline D=D\cup \mathbf T$. Then the disk algebra, $A(\mathbf T)$, is the set of $f\in C(\mathbf T)$ that have a holomorphic extension to $D$. That is, $f\in A(\mathbf T)$ if there is a $g\in C(\overline{D})$ such that $g |_{\mathbf T}=f$ and $g|_D\in H(D)$. It is clear that $A(\mathbf T)$ is a subalgebra of $C(\mathbf T)$. Show that it is closed in $C(\mathbf T)$ so that it is a Banach subalgebra. (I suggest using Corollary 5.41 in the text.)
    2. Conclude that $C(\mathbf T;\{\infty\})\subset A(\mathbf T)$.
  • Hint for Ex 7.4.4: I used the following observation: show that if $a\notin f(\gamma^*)$, then $\operatorname{Ind}_{f(\gamma)}(a)=N_{g_a}$ where $N_{g_a}$ is the number of zeros of $g_a(z)=f(z)-a$ inside of $\gamma$.
Wednesday:
  • Study: No office hours today and Friday. Finish reading chapter 10. Read the beginning of section 11.1 through Proposition 11.1.
  • Do: Carefully study the proof of Proposition 11.1. No new written assignment, but think about all the exercises at the end of section 11.1.
Friday:
  • Study: No class or office hours today.
  • Do: No new assignment.


Week of May 20 to 24
Assignments Made on:
Monday:
  • Study: Read section 11.1 and 11.2
  • Do: Work Ex 11.2.1 and Ex 11.2.2. (Nearly full credit for showing that $d_\sigma$ is a pseudo-metric -- that is, showing that $d_\sigma$ satisfies all the requirements except for showing that $d_\sigma(p,q)=0$ implies $p=q$. Showing this can be considered another challenge assignment.)
Tuesday (x-hour):
  • Study: Read sections 11.3 and 11.4.
  • Do: Ex 11.5.2. (You don't need to read section 11.5 to do this problem.)
Wednesday:
  • Study: Start 11.5.
  • Do: Suppose that $(f_n) \subset H(D)$ converges normally to $f$ on a domain $D$. Suppose that $z_n\to z_0$ in $D$. Show that $f_n(z_n)\to f(z_0)$.
Friday:
  • Study: Finish section 11.5. We'll finish up the course with section 11.6 on Wednesday. (Remember, Monday is a holiday.)
  • Do: No more homework this term!


Dana P. Williams
Last updated May 24, 2024 12:34:22 EDT