Math 113 Spring 2022
My Complex Analysis Course
Last updated May 24, 2022 16:38:18 EDT

Announcements:

• The last homework problem is from Wednesday, May 18th. We'll sort out when the last assignment is due in class. I may make additional suggested exercies to look at, but no additional problems will be graded. ### Homework Assigments

 Monday March 28: Homework 0: Before class on Monday, you should down load a copy of the text from our canvas page and read/skim sections 1.1 thru 1.4. Also download homework-0 and answer the survey questions there. 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 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).

April 4 to April 8
 Monday: Study: Read Section 3.1. Do: No new assignment Jordan Papers: Here is the paper with the Jordan curve artwork and the paper asserting the nasty properties of Jordan curves (as opposed to Jordan paths). Wednesday: Study: Read Sections 3.1 and 3.2 Do: Ex 3.2.2. Study: Ex 3.2.1. If find Ex 3.2.4 interesting while Ex 3.2.5 relates our contour integrals to path integrals from physics (and multivariable calculus). Friday: Study: Read Section 4.1. Not many written assignments this week, but I hope EVERYONE is keeping up with the reading. (I have found more typos that everyone else!) I am really looking forward to feedback even if I do not necessarily take the advice. Do: Ex 4.3.1

Week April 11 to April 15
 Monday: Study: Read Section 4.2 and 4.3. Do: 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. Complete the proof of Cauchys Theorem for Convex Domains by showing that if $f$ is continuous at $z_0$, then $$\lim_{z\to z_0}\frac1{z-z_0}\int_{[z,z_0]} \bigl(f(w)-f(z_0)\bigr) \,dw=0.$$ Study: Ex 4.3.3 and Ex 4.3.6. Ex 4.3.6 is the "solution" to proving a form of Cauchy Theorem is many elementary texts. Wednesday: 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. Typo Competition: Here is the current typo contest (Last modified May 20, 2022) tally. Let us not let Dylan run away with it. Friday: Study: Read Section 5.2. Do: It follows from Theorem 5.2 that $f\in H(D)$ has a zero of order $m\in \mathbf N$ at $a\in D$ if there is a $g\in H(D)$ such that $g(a) \not=0$ and $f(z)=(z-a)^m g(z)$ for all $z\in D$. Show that $f$ has a zero of order $m\in \mathbf N$ if and only if $0=f(a)=f'(a)=\cdots =f^{(m-1)}(a)$ and $f^{(m)}(a)\not=0$. (This is probably how I should have defined zeros of order $m$ to begin with.) Ex 5.2.4, Ex 5.2.10, Ex 5.2.11.

Week of April 18 to April 22
 Monday: Typo Competition: Here is the current tally for the typo contest (Last modified May 20, 2022). I will keep updating this link even if it does not appear each week. I really appreciate the help. You should also feel free to make suggestions on the exposition if you like. Study: Read Sections 5.3 and 5.4. Do: Ex 5.3.6, Ex 5.3.9, 5.3.12, Ex 5.4.1, Ex 5.4.2, Ex 5.4.5. Study: I hope everyone is comfortable with Ex 5.3.5. If you were amused by Ex 5.3.12, have a look at Ex 5.3.13. Wednesday: Study: Read Section 5.5 through the Schwarz Lemma. We will finish 5.5 on Friday. Comments: As Steve pointed out, in Ex 5.3.6, it should say "of degree at most $n\ge1$". Furthermore, you can use Ex 5.3.5 here without proof. Also, Ex 5.4.2 was meant to be easy: just note that any compact subset of an open disk is contained in some closed subdisk. Friday: Study: Finish Section 5.5 and start on Section 6.1. Do: Ex 5.5.5 and Ex 5.5.6. (For Ex 5.5.6, you can use Ex 2.2.4.)

Week of April 25 to April 29
Due Monday, May 18
 Monday: Typo Competition: Here is the current tally for the typo contest (Last modified May 20, 2022). Homework 3: I put the third homework assignment up on gradescope. It covers the problems assigned up until Friday (the 22nd). It is due Friday. Study: NO CLASS TODAY Do: Wednesday: Study: Finish Section 6.1 and start Section 6.2 Do: No new assignment. Friday: Study: Finish Section 6.2 Do: Ex 6.2.2 and Ex 6.2.4.

Week of May 2 to May 6
Due Monday, May TBA
 Monday: Study: Read Section 7.1 Do: Ex 7.1.3. Note that, combined with Montel's Theorem, this gives another proof for Ex 5.4.5. Wednesday: Study: Read Section 7.2 Do: Friday: Study: Read Section 7.3 through Lemma 7.15. We will finish this section on Monday. Do:

Week of May 9 to May 13
Due Monday, May TBA
 Monday: Study: Finish Section 7.3 and read Section 7.4 Do: Ex 7.4.2, Ex 7.4.3, and given Ex 7.4.4 a try. From Here: For the moment, we will skip ahead to Chapter 10. After that, I am not sure what we should do. We could play with the residue calculus (Chapter 9) or see if we can prove some of the Picard theorems (Chapter 11). Wednesday: Study: Read Section 10.1 and start Section 10.2 Do: Friday: Study: Read Section 10.2 and 10.3 Do: No new assignments.

Week of May 16 to May 20.
Due Monday, TBA 