Math 448 D13 Home Page

This is the home page for Math 448 D13. "Complex Variables". This class meets for the Spring 2017 semester at MWF 11:00-11:50 in 443 Altgeld. The mathematical links page is here. Suggest a new one!

The semester so far

Administrative comments

The class did comparably on the final as on the hour exams. There are some incompletes, so I don't want to talk about the questions here. (Will add more later this summer.) Course grade distribution: A+ (4), A (16), A- (3), B+ (3), B (1), B- (2). This was an excellent class. Thanks for your participation!

Please note that HW 10 has been amended; the new version is below, and will be emailed to the class and distributed in class.

F 4/28 and M 5/1 -- Additional discussion of conformal mappings and a few other topics which won't be tested, plus Test 2 returned.

W 4/26 -- Test 2, not online

M 4/24 -- Two handouts: Homework Ten retro (including a correct solution to #7) and Second Exam review terms (including an incorrect statement of Cauchy's Theorem, missing 2 pi I, and therefore causing me to lose two points.)

M 4/17, W 4/19 & F 4/21. Yeah, I've fallen behind. Finished all the testable material and will do bonus topics in between reviewing for the second exam and the Final. Pick up your 3 x 5 card if you didn't get it on F. Only handout: Homework Ten Solutions.

W 4/12 & F 4/14 -- We're now into 3.3, on Fractional Linear Transformations. Passed out on Friday: Homework Nine Solutions and Homework Ten, due 4/21/17. This is the last homework that will be covered on the second exam. I will do additional material in the course, however.

F 4/7 & M 4/10 -- We're zooming into 3.2. Homework Eight Solutions and Homework Nine, due 4/14/17.

1. On problems 1, HW8 (2.6#19), use the keyhole integral. You will need to know a couple of other integrals, one of which is 2.6#18, which I will do in class on Wednesday, and you can use that value, as well as the integral of 1/(1+x^2), without having to do it separately.

W 4/5 -- Did parts of problem 2.6#18, and moved on to section 3.1, including the argument principle and Rouche's Theorem. May well do a few more contour integrals later.

F 3/31, M 4/3 -- F was mostly going carefully through the solutions to HW7, and then basically, the books 2.6 HW 27 -> 31 with considerable detail. Also, the integral of x^(1/2)(log x)/(1+x^2) from 0 to infinity. There will eventually be handouts. Homework Seven Solutions and Homework Eight, due 4/7/17.

HW7 -- In #7, The sentence ``Suppose further that, for all z, |f(z)| <= 5 |g(z)|, this implies that f and g have the same zeros'' is misleading, but not incorrect. If g = 0, then f = 0. The goal of the problem is to establish the converse!

I have no clear idea of when the second test will be.

W 3/29 -- Test 1 was returned and I went carefully over examples 3, 7 and 8 from the text. Bring (or email) questions; they are very helpful to me in preparing the presentations.

M 3/27 -- Back in the saddle. Did some review and ended with discussing the handout on a couple of contour integrals. Don't worry about trying to figure out which method to use; the course will show you lots of approaches and I know that time on exams is limited.

F 3/17 -- Exam will be returned W 3/29. I did a few examples of integrals evaluated by the residue method (and will have a handout on M 3/27). Did a bit of an overview of the rest of the course. Homework 7 was distributed, due F 3/31: Homework Seven. Be ready to start moving once classes start again! We'll do a quick review of 2.5 and move on to 2.6 in the text.

First exam will be in class on W 3/15, covering through 2.4.

Next Homework will be due the Monday after Spring Break.

M 3/13 -- Review for the test on Wednesday. I will rewrite my list of review terms nicely and post it Tuesday by noon. But it will be more helpful in your own handwriting! Here is the List. The last line was cut off. It read: "NO: Proofs of Morera, Cauchy-Goursat, problems involving hyperbolic trig functions or the formulas for inverse trig functions."

Class on M 3/6 and W 3/8 will be by Prof. Harold Diamond. Bring questions. He will discuss both new material and supplemental material.

W 3/1, F 3/3 -- We've mostly finished 2.5, with an application of Liouville's Theorem to proving Partial Fractions in calculus. Homework Six Solutions were distributed and discussed. Did I miss anything?

HW6 Question about the Laurent Series for z/(sin^2(z)), about z_0 = 0
My reply: Think of this function as 1/z * (z / sin(z))^2 or (1/z) (z csc(z))^2. You know how to get sin(z) / z. Use Wed.'s class technique to get 1 / (sin(z) / z) = z / sin(z) = z csc (z), then square it. Since sin(z) has a zero at z=0, csc z has a pole at z=0 and you have to deal with that by putting in a negative power of z in z*csc^2(z), which is not analytic at z = 0, so you can't just take derivatives directly.
More generally, of a function has a pole of order 1, say at z = 0, then its Laurent series will look like f(z) = b1 / z + a0 + a1 z + a2 z^2 + ....

W 2/22, F 2/24, M 2/27 -- Sorry for the slowness. We have finished 2.4, have classified all isolated singularities and are ready to talk about the implications of residues. On these three days, the following were distributed: Homework Six, due F 3/3, Homework Five Solutions and Homework Four retrospective and some bonus notes. Please let me know if I have forgotten anything.

F 2/17 and M 2/20 -- Sections 2.3, 2.3.1 and 2.4. We proved Cauchy's Theorem and worked through the examples in 2.3. Class Monday ended with the proof of the first part of Theorem 1 in section 2.4. There will be notes. On Friday, distributed Homework Four Solutions and Homework Five, due F 2/24, though the last one was forgotten until after the bell, and some missed it. The first test will be during the week of March 13.

M 2/13 and W 2/15 -- We have finished section 2.2 and started section 2.3. Cauchy's Theorem! Distributed with Homework Three retrospective. Talked about exam, but I received new information and may change the date on Friday. I'll also explain why the restriction at one point to vertical/horizontal paths is a bit of a red herring.

HW 3 Comments
1. Please note that, as mentioned in class, Problem 7 is not extra credit and not worth 1/2 point. It's a regular problem worth 1 point.
2. In problem #5, the comment about analyticity is just there because if f is analytic in a domain that contains the curve, then it's easier to integrate f(z)dz. In this case, go to the definition.
3. In problem #6, it is desirable (and highly recommended) to use the quotient rule formula in the text!
4. By Section 1.4, if Re(f) and Im(f) are continuous at a point, then so is f.

W 2/8 and F 2/10 -- Catch up webpage. Distributed with Homework Two retrospective, Homework Three Solutions were distributed in the beginning, and Homework Four, due F 2/17. We finished the discussion of the Cauchy-Riemann equations and the distinction between being differentiable and being analytic. Whave talked about power series, which are essential to understanding analytic functions and covered the examples and properties found in section 2.2. We are in the middle of proving that we can differentiate power series term by term and will finish the proof on M 2.13

M 2/6 -- Handed out Bonus Notes on the CR => differentiable proof of 2/3, and talked about some more examples, including a careful differentiation of Log on C minus the non-positive real axis. We are nearly ready to go to section 2.2.

Some relevant, non-class links. One of the most distinguished graduates of the UIUC Mathematics Department is David Blackwell (Ph.D 1941), who was the first African-American scholar in the National Academy of Science. Here are links to the LAS page honoring David Blackwell in the Gallery of Excellence and his New York Times obituary. Also, I was requested to give class time for a presentation on healthy eating from ARC. I thought it was a worthwhile idea instead to put some links here: Cooking Classes and Nutrition Check up. Eat your vegetables!

F 2/3 -- Class began with a brief discussion of Homework 2. Homework Two Solutions were distributed in the beginning, and Homework Three, due Friday, Feb. 10, was distributed at the end. See comments above for correction to Problem 7. We talked more about differentiability and I showed that if u and v are differentiable functions from R^2 -> R (in the standard sense) and if f(z) = f(x+iy) = u(x,y) + iv(x,y) and if u and v satisfy the Cauchy-Riemann equations in a domain D, then f is analytic in D. There is an error in the book in discussing the differentiability of e^z, because division by zero is possible.

Homework 2 Comments
1. A student asks: is an epsilon/delta proof needed on #1.
My reply: If you use my hint, you will have f(z) as a polynomial for z \neq 1, and you can use the fact (p.37) that polynomials are continuous. I don't see a need for epsilon delta here.
2. A student asks about #5c Log z = - 1 + i \pi , which is impossible, because the book defines Arg(z) as lying in [-pi, pi).
My reply: An embarrassing mistake on my part. Please change the problem to Log z = - 1 - i \pi.
3. Hint for dealing with equations like cos(z) = a, sin(z) = a is included at the bottom of W 2/1 handout.
4. In #8, there shouldn't be "i" in the description of \zeta_7 in terms of cosine and sine, only in the exponential. It's a 7th root of unity.

W 2/1 -- HW 1 graded and returned, with retrospective. A proof of Green's Theorem was given, with implications for complex contour integration, and the definition of differentiability was presented. An unfortunate chalkographic error at the end that was pointed out by a couple of people, if f = u + iv and "_x" denotes partial differentiation by x, then "f_x = u_x + i v_x", not "f_y = u_x + i v_y", as I mistakenly wrote.

M 1/30 -- No handouts. More on integration, as in section 1.6. Lots of examples and a crucial estimate, that the absolute value of the integral of f(z)dz on a curve C is bounded by the product of a bound for |f(z)| and the length of the curve. This is a good time to review Green's Theorem, if you haven't thought about it for a while.

F 1/27 -- Class began with a brief discussion of Homework 1. Homework One Solutions were distributed in the beginning, and Homework Two, due Friday, Feb. 3, was distributed at the end. In the middle, I talked about the geometric effect of simple mappings on the complex plane, with an emphasis on the mappings az + b, z^n, e^z and Log z on sets line horizontal and vertical lines and circles |z| = r. Pictures were circulated, and will come to class on Monday as well. We started to talk about integration, which will be a crucial topic.

HW1 comments -- 1. A student asks for a hint on #7b, and I suggest induction.

W 1/25 -- No handouts. Introduction to arg(z), Arg(z), e^z (at first heuristically), log(z) and Log(z). Trig functions and, somewhat incorrectly, Hyperbolic trig functions. (This will be fixed on F 1/27, after we go over the homework. Forgot about the email list; will bring on F 1/27 as well.)

M 1/23 -- No handouts, but I passed around an email/phone study list. It will go around again on W and then be distributed in class on F. It will not go online for privacy reasons. We are pretty much through section 1.4, which is a bit faster than I thought. Please feel free to ask questions, either in class or by email.

F 1/20 -- Second day of class. We are being taped for use by NetMath. The material of the day was the geometry and topology of complex numbers. These might seem unmotivated, but we'll use them later. We are basically through section 1.3. Homework One, due Friday, Jan. 27, was distributed.

W 1/18 -- First day of class. The basic arithmetic of complex numbers in three ``modes'' as a + i b where a and b are real and i*i = -1, as ordered pairs (a,b), and as points in the plane. Why i is not "the square root of -1". DeMoivre's Theorem, and U(t) = cos(t) + i sin(t), but we know better. Hints of wondrous things to come. First assignment out on Friday. Handouts: Course Organization, How to Solve It guide, Class questionnaire

In advance of the semester, here are three links to lists of REU sites: one run by Prof. Steve Butler of Iowa State, and the others by the NSF and the AMS.