Math 73/103
Measure Theory and Complex Analysis
Last updated July 18, 2017 09:28:13 EDT

Announcements:

• Solutions to the fifth assignment are available.
• The final is available from the assignments page and will be due my noon on Monday the 24th.
• Monitor the assignments page for comments and corrections for the exam.

### Homework Assigments

Math 73/103 2014
 Monday, September 25: Read: Review the Riemann integral: $\mathcal R\int_a^b f(x)\,dx$. Homework: Work problems 1, 2 and 3 from the First homework assignment. Due: Wednesday, September 24. (The assignment was last modified: 9:28 am, July 18, 2017). Update Wednesday, September 27: Read: We're more or less in the very beginning of Chapter 1 of Rudin. Homework: Now you're ready for problems 4, 5 and 6. Optional: Here's a little worksheet on Borel sets. Let me know if you want to talk about it. Assignment Two: Friday, September 26 Read: We've finished through page 20 in the text. Assignment One: Assignment One is due today. Assigment One: suggested solutions. Comments and corrections welcome. (Typo fixed 7:00 pm, December 31, 1969.) Borel Sets: suggested solutions. Comments and corrections strongly encouraged. Assignment two: Here is the first draft of the second assignment. (This version dated 9:28 am, July 18, 2017). Today's assigment:Start with problems 1 and 2. Monday, September 29: $\LaTeX$: Here is the source for the first homework assignment with solutions. You'll need my class file to $\LaTeX$ it. This is also the class file I use for undergraduate exams. Here is the source for the second homework assignment with the answers removed. Homework: No new assignment Wednesday, October 1st: Second Homework: You can now work on problems 1-4 and 7ab. Scores on First Homework: The first assignment has been returned to mailboxes (for those with mailboxes in Kemeny). The rest will be returned in class. The distribution of scores, out of 45, was: 39, 39, 36, 34, 33, 32, 32, 15. Please come see me if you have questions (after comparing with the suggested solutions). x-hour: We are meeting in our x-hour this week and next. Second Homework Due Wednesday: Second Homework: The second assignment is due Wednesday (October 8th). Class the Week of October 13 -- 17: Second Homework: Here are some solutions to the second assignment. Class Schedule: We meet tomorrow (the 9th) in our x-hour as well as "as usual" Friday. I am away next week (October 11 -- October 20). Carolyn Gordon will cover our lecture on the 13th and 15th. NO CLASS Friday the 17th or Monday the 20th. Third Assignment: Here is the third assignment. You should be able to start on all the problems after Friday's lecture (or whenever Carolyn or I get around to defining a premeasure on an algebra of sets). I will be due as soon as I can get the second assignment graded. Second Homework Returned: Second Homework: I've returned the second assignment to your mail boxes (or will return them in class Monday if you don't have a mailbox). Frankly, the work was generally sloppy. Please work on your exposition. It is what we do, so you need to take pride in producing elegant -- as well as correct -- solutions. Solutions: I made some "improvements" to the solutions I handed out earlier. Here are some updated solutions to the second assignment. Scores: Here are the scores on the second assignment (our of 40): 33, 27, 26, 26, 25, 24, 23, 15. I did not grade one problem and one problem was worth 10. Source: Here is the $\LaTeX$ source for the third assignment. I'm Back! Third Assignment: There was a bad typo on problem 4c. The $\bigcup G_k$ should have been $\bigcap G_k$. The file hw3.pdf has been updated. Due Date: Please turn in the third assignment on Friday. Complex Variables: Friday, October 24 Fourth Assignment: Here is the fourth assignment. It is largely independent of lecture. If you get stuck, you are welcome to look for hints/solutions in basic texts. Third Assignment: Here are some brief solutions for the third assignment as well as for the optional problem on Riemann integrals. Monday, October 27 Third Assignment returned: The third assignment will be returned by class time on Monday (the 27th). The scores were 34, 31, 30, 27, 25, 23 and 23. In the process of grading the third assignment, I made a number of "improvements" to the solutions: hw3-solns.pdf. Monday, November 3 Fifth Assignment : Here is the fifth and last assignment. (There is still a takehome final to come.) Solutions to fourth assignment: Here are some partial solutions to the fourth assignment. Last Days Fifth Assignment : The fifth last assignment is due on Monday, November 17th (aka the last day of class). There were some annoying typos on problem 3, and I've officially added problem 5 in the text. (There is still a takehome final to come.) The Riemann Sphere: The final will be available on Monday or Tuesday. It will be "take-home" and due by Monday the 24th at noon. Problems with Fifth Assignment 4 (Rouche's Theorem): I was way too cavilier with my statement of Rouche's Theorem in homework problem #4 on the fifth assignment. (Full disclosure: I borrowed it from the last time I taught Math 43.) First of all, saying "$f$ and $g$ analytic on and inside $\gamma$" means that $f$ and $g$ are analytic on a simply connected region $\Omega$ containing $\gamma$. You may simply assume that we proved in lecture that $$N_f := \frac1{2\pi i}\int_\gamma \frac{f'(z)}{f(z)}\,dz \tag{**}$$ is the number of zeros of $f$ inside of $\gamma$. Or if you want to be fussy -- since we only proved (**) when $\gamma$ was a circle -- you can establish (**) using the Residue Theorem. But even then you will need to assume $\operatorname{Ind}_\gamma(a)=1$ if $a$ lies inside of $\gamma$. The Final Here are some brief solutions for the fifth and last homework assignment. The final exam is available by clinking on the link. It is due Monday. Please monitor this page for corrections or comments. Here are some solutions for the final. Have a great break.