Here is the distribution of scores on the Final Exam, in the following form: (Points,#Undergrads,#Grads). The medians for both were in the 180s.
(> 199, 1, 1)
(190's, 4, 4)
(180's, 4, 2)
(170's, 1, 1)
(160's, 1, 1)
(150's, 3, 1)
(< 150, 3, 0)
Three enrolled students did not take the Final.
Here is the distribution of grades in the Course, in the same format as above. Note: I do not give A+ to graduate students in undergrad courses.
(A+, 1, 0)
(A , 4, 5)
(A-, 5, 2)
(B+, 1, 1)
(B , 2, 1)
(B-, 2, 0)
(C+, 1, 0)
(C , 1, 1)
(Ab, 3, 0)
F 1/21 -- More on complex numbers, their standard and polar forms. First look at the argument function and some basic inequalities, and the solution of equations like z^n = w.
M 1/24 -- Topology and a first look at mappings and square roots. First homework distributed: Homework One, due M 1/31.
W 1/26 -- More on mappings and the standard material on limits and continuity in 1.4. Much of this was skipped in class; ask if you have any questions! Phone/email list sign-up, to be distributed on F 1/28. (Supplemental list will be passed out later.) Start reading the HW and bring questions.
F 1/28 -- First definitions of the complex extensions of the exponential and logarithmic functions and of the trigonometric functions. Phone/email list distributed, with a supplemental list to go out next week. Questions on the homework discussed up top.
Comments from the first homework: A few remarks on the problems. For
#4, keep in mind that it's 1.1, not
1.1.1. For #6, the number z = 2^{11} + 2^{11} i is given in standard
form. Its polar form is 2^{23/2}(cos(pi/4) + i sin(pi/4)), or
2^{23/2}e^{i*(pi/4)}. (Here and in #10, "*" just means multiplication.)
In #10, the region 0\le x \le y could also be described as those z
for which pi/4 \le Arg z \le pi/2, plus the origin, and the
right-hand plane could be describes as those z for which x \ge
0. (I'm using here the tex notation that "\le" means "less than or equal
to" and "\ge" means "greater than or equal to".)
A student writes: "I was wondering for problems 6 and 7 where we have to
decide whether a set is open, closed, connected, or a domain,
do you want us to give answers like they do in the book with
formal proofs showing explicitly that, for example, the set
{z=x+iy: x < 0} has no boundary points and is therefore
entirely composed of interior points and must therefore be an
open set, or is it enough just to say that it has no boundary
points so it must be entirely composed of interior points and
must therefore be an open set?
My reply: I will *accept* "informal" answers to problems 6 and 7 if they show
some thought, especially if they are accompanied by an accurate
picture. It is not terrible for a mathematics student, once in an
educational career, to prove such a thing in detail, but will not be
the focus of the class or the homework.
A student writes, in effect, that problem 11 makes no sense. My reply
is that, when a problem gives an "analytic
geometric" interpretation of a question in complex variables, it is
looking for the set of (x,y) so that z = x + iy satisfy the given
conditions.
A student writes: "Could you please
tell us again why we need to cut the negative real axis on the
complex plane when defining the domain of the Arg(z)?"
My reply: I will be going over this again (it is traditionally one of the
tricky points of the course!) The reason for the cut is that it is
impossible to define the argument function on C minus {0} in a continuous
way. Let's say that the argument of 1 is 0. As you go clockwise on the
unit circle, the argument will keep increasing until you approach 1 from
the fourth quadrant, and the argument is 2Pi. The only way to deal with
this is to decide that the argument will not be continuous on all of C
minus {0}. You make a cut, semiarbitrarily on the negative real axis, and
then you can define the argument continuously everywhere but on the cut.
You can actually make the cut in many different ways, but this is the
simplest.
M 1/31 -- Catch-up Monday. Lots of small remarks about material covered in the course so far, and some discussion of the homework. First homework solutions distributed (not linked, come to class!) Second homework distributed: Homework Two, due M 2/7.
W 2/2 -- We finished our discussion of section 1.5 and began 1.6. I handed out an (unlinked) Mathematical Cultural Notes 1, on the hyperbolic cosine and sine.
F 2/4 -- More on complex integration. Homework 1 returned. Two handouts: one a supplemental set of comments on HW1, the other a combination supplemental email and phone list and information on dropping the class late. (This is not a hint.)
A student writes: "For problem 6. what do you mean by polynomials in z? like polynomials of exp(x)?" My reply: as an oblique hint, observe that e^[Log z] = z for all non-zero z. If you want to break this up into real and imaginary parts, it could get tedious, but keep in mind that z = x + i y = |z|(cos[Arg z] + i Sin[Arg[z]), hence x = |z|(cos[Arg z]), etc. A student asked after class about the terminology "epsilon-sub-zero(x_0,y_0)". By this I mean a function epsilon-sub-zero, evaluated at the point (x_0,y_0). After writing the solutions, I'd like to point out two corrections to the text. First, the solution to 1.5#13 is wrong. You can't take 4 times the log. You have to work out the inside first, before taking the log. Second, in 1.5#28, replace the last symbol with {w: 0 < |w| < 1}; w=0 is not in the image of the map.
M 2/7 -- Second homework solutions distributed (not linked, come to class!) Third homework distributed: Homework Three, due M 2/14. With a suspiciously small number of questions, your instructor completed chapter one and started chapter two. Topics: line integrals and their combinations, Green's Theorem and how the Cauchy-Riemann equations show up, computing derivatives and how the Cauchy-Riemann equations show up. More on Wednesday. Show up!
W 2/9 -- Second Mathematical Cultural Notes, not linked, on why {u = f(t), v = g(t)} for polynomials f and g implies that there is a non-zero polynomial F so that F(u,v) = 0. We proved the key theorem that, under mild and always achieved conditions, C-R is equivalent to analyticity in a domain. More implications of being analytic.
F 2/11 -- A small, but brave and hardy troupe of students came to class, got homework two back (together with additional comments) and heard about homework three (summarized at the top of the page). We finished 2.1, skipped 2.1.1 and started in on 2.2. It's a power series world from now on!
In the homework, I am not asking for elaborate calculations, just a recognition that some steps are needed. In #7, write w = r e^{it} and then write out the n-th roots (induction is obliquely used in establishing a formula that will solve the problem, but you do not directly go from a sum of (n-1)-st roots to a sum of n-th roots. In #8, you can use computations for real polynomials without detailed comment. In the definition of C_2 in #9, that should be "z = i", not "z+i". In #13, please flip the symbol around for the Laplacian.
M 2/14 -- Makers of persian rugs would intentionally sew mistakes into them to remind the world of human fallibility. So it was with Monday's class, forced to move to a much smaller room by plumbers and taught by a guy coming down with a bigger cold than he thought. All mistakes will be rectified in the coming days. Homework 4, corrected was distributed: Homework Four, due M 2/21. We talked about harmonic conjugates and power series.
W 2/16 -- A brief class, more with descriptions of what's covered than with coverage. I was at home in bed and falling asleep before the 11:00 bell stopped ringing. More on power series
F 2/18 -- Recovery. Power series examples and the implications. Fortunately, not too many mistakes on the homework, only that in #13, 3 sin z should be 3 cos z.
M 2/21 -- Lots of handouts: Homework 3 returned with the usual additional comment sheet, Homework 4 solutions, and Homework Five, due 2/28. We decided on the date for the first exam, M 3/7, since there is no other mutually agreeable time. More discussion on the exam as the weeks progress. Mathematically, we reach Cauchy's Theorem, one of the most important parts of the course.
W 2/23 -- Two applications of contour integration: the average value of 1/(r + cos[t]) on [0,2pi] and the integral from 0 to infinity of 1/(z^n+1). Plus the magnificent elegance of the Cauchy-Goursat Theorem.
F 2/25 -- The wonderful representation of analytic functions by power series, Liouville's Theorem and its friends. Maybe the most beautiful mathematics of the semester so far. You should have been there.
M 2/28 -- More on section 2.4: Liouville's Theorem again, the order of zeros (which will be very important on W), analytic logarithms. Morera botched and punted until W. Homework 5 solutions distributed. HW 6 delayed until Tuesday noon: Homework Six, due 3/11. Please note that the first test will only cover through the first five homeworks. There will be some review on both W and F.
W 3/1 -- Some review. Morera's Theorem proved. The beginings of the discussion of isolated singularities.
F 3/3 -- More review, more isolated singularities. This is material that will not be on the first test.
Test 1 Corner
The test will be on Monday at 10 am in the usual place. It will cover the material from the first 5 homeworks, which means through section 2.3. As a general guide, if there is a topic in the book which I did not mention in class and did not give homework on, then it is unlikely to appear on the exam. I am leaving a folder containing the semester's handouts to date outside my office door (327 Altgeld). M 3/7 -- First test.
W 3/9 -- First test returned and discussion. Moving on to residues and a little bit more about the behavior of analytic functions near removable singularities.
Remember, HW 6 is due on Friday, March 11. What I mean in problem 9 is that, for example, if f is a polynomial of degree less than or equal to 2, you can always find constants a, b, c so that f(z)/(1-z)^3 = a/(1-z) + b/(1-z)^2 + c/(1-z)^3. This is most easily done by multiplying through by (1-z)^3. A student asks: For #9, I have a sum of 2 series. I can combine them, but it's not very pretty. What form would you prefer them in? My reply: I would prefer a single series. Make sure that the summations are over the same sets of integers. A student asks: For #10, I believe we've had 1 or 2 of these on previous hws. Can we do them the same way, using Cauchy's formula, or were you looking for use of residues or something? My reply: Any correct method. I was thinking residues.
F 3/11 -- Discussion of HW 6; Homework Seven, distributed. Please note due-date of F 4/1. Prof J. Tyson will substitute for me on M 3/14 and W 3/16. Then it's spring break, and we'll all be back on M 3/28. Be careful out there!