Numerical grades will be retained for use in computing the final letter grade in the course. Here are approximate letter grade assignments for this exam:
Problem 1 (9 points)
3 points for the graph, and 2 points for each of the roots, a total of 6 points. 3 points partial credit if some correct algebraic work is shown. 3 of 6 points for the values of the roots are earned if the values of sine and cosine are not simplified.
Problem 2 (14 points)
a) 7 points: 3 points for the graph of U, which should be the graph of a region; 1 of those points is for locating Q. 2 points for some correct algebraic desciption of of U. 2 points for a correct rectangular description of Q.
b) 7 points: 3 points for the graph of V, which should be the graph of a region; 1 of those points is for locating R. 2 points for some correct algebraic desciption of of V. 2 points for a correct rectangular description of R.
Problem 3 (12 points)
a) 7 points: 5 points for correct computation of (1+i)i, and the answer should include infinitely many possible values (the answer must include a specification of any variable); 2 points for any correct computation of (1+i)2.
b) 5 points: I gave 2 points for even mentioning the capital A variant of Arg. I also tried to be sensitive to students who might want to provide examples for a definition of Arg where the values are in [0,2Pi), not, as our textbook would have it, in (-Pi,Pi]. I did want "an explicit pair of complex numbers" so a discussion without giving a specific example did not receive full credit.
Problem 4 (13 points)
a) 9 points: 3 points for each piece of the computation. 1 point for the answer, and 2 points for the "mechanics" of the computation.
b) 4 points: 2 for recognition that the problem can be done by antidifferentiation, and 2 points for completing the computation.
Problem 5 (16 points)
a) 8 points: 2 points for using the Cauchy-Riemann equations. 3 points for verifying each equation. Recognition of harmonic is 2 points of one 3 point score and use of equality of mixed partials is 2 points of the other 3 point score.
b) 8 points: Verification that the function is harmonic is 4 points, and creating a harmonic conjugate is the other 4 points.
Problem 6 (10 points)
a) (8 points) The simplest definitions for this problem (4 points) used the complex exponential function, and then the "proof" (4 points) is rather simple. I tried to give appropriate credit to students who used other definitions.
b) (4 points) A specific correct z earns 1 point. Verification gets the other 3. Probably some use of the triangle inequality or reverse triangle inequality is needed.
Problem 7 (12 points)
1 point for citing the ML bound and 1 point for a correct value of the length.
4 points for treating the top of the fraction correctly. This means using the triangle inequality, and finding correct bounds for both the exponential function and the monomial. The delicate part here is certainly the bound for the exponential.
4 points for treating the bottom of the fraction correctly. So here an estimate using the "reverse triangle inequality" needs to be given, with a correct selection of BIG and small when R is large. 1 point is decucted for saying nothing about how large R should be to avoid dividing by 0, as requested. Any acceptable value of R earns this point.
1 point for putting together the estimates and 1 point for deducing the required result (with a limit).
I deducted 2 points for asserting that complex numbers can directly participate in inequalities. C is not an ordered field.
Problem 8 (12 points)
The radius of convergence and supporting computation was worth 8 points. 1 point was reserved for the value of the radius of convergence, which could have been obtained in various ways. I looked for supporting evidence using, for example, the Ratio Test for the other 7 points. I also looked for the word/symbol, "limit/lim" appropriately.
1 point was reserved for the requested value of the series sum, and 3 points for supporting material.
The second exam
Numerical grades will be retained for use in computing the final letter grade in the course. This exam was challenging. There are comments on the student problem solutions are included below. Here are approximate letter grade assignments for this exam:
Problem 1 (18 points)
a) (9 points) 5 points for work at z=0 and 4 points for work at the other singularities.
b) (9 points) Use of the Residue Theorem on the "big loop" is worth 6 points; use of the Residue Theorem on the "small loop" is worth 2 points; putting the results together for the answer is worth 1 point.
Comment The instructor would have lost a point in this problem.
Problem 2 (18 points)
There was little difficulty grading this for students who did the problem correctly. For students who made little progress, I graded this more "holistically" than I ordinarily like. I looked for a change to a complex line integral, and then use somewhere of the power series for the exponential function. Those were essential steps, good for about a quarter to a third of the credit. Invalid algebra or arithmetic was bad, as were logical assumptions connecting w and θ. I tried to see if people were making headway towards solving the problem and used that to assign a grade.
Comment Students did better on this problem than I expected. Many students gave more efficient solutions than what I wrote initially! One student cleverly "reverse engineered" a solution using properties of Bessel functions. In fact, the Bessel function properties are frequently (usually?) proved using similar Residue Theorem arguments!
Problem 3 (14 points)
Use of a power series for f is worth 6 points. Substitution and clarification gets 7 more points. The value of g at a earns the last 1 point.
Alternate approaches using L'Hôpital's Rule are also valid and I tried to assign appropriate partial credit.
Comment Students did this problem well. It follows directly from the power series/Taylor series or using a L'Hôpital's Rule argument.
Problem 4 (12 points)
Use of the Cauchy Integral Formula for the first derivative earns 6 points. The other 6 points are earned for successful estimation by ML.
Comment This is almost a characteristic complex variables problem, with no analog in real calculus. Again, students did well (look at the median).
Problem 5 (12 points)
A direct use of the Cauchy estimates easily shows that f is constant. Also an approach similar to our Liouville's Theorem proof can be used, but then more evidence supporting an estimate centered at points other than 0 must be given. I subtracted 2 points if such detail is not given. I will give 3 points for identifying the candidate for the unique function.
Comment I was unhappily surprised by the relatively poor student performance. Again, various estimation techniques work. I had thought this would be relatively straightforward.
Problem 6 (16 points)
a) (8 points) I will try to assign points depending on the progress towards a solution. A valid series centered at i in the annular region for 1/z alone will earn 4 points.
b) (8 points) Again, I will try to assign points depending on the progress towards a solution, and 1 point will be reserved for an explicit radius of convergence. The exceptional constant coefficient earns 1 point.
Comment This is one of the two problems (together with the previous one) which I would have no hesitation giving on a time-limited exam in an undergraduate complex analysis course. I also worked out a number of examples similar to these problems in class, and certainly the textbook has a number of worked-out examples. So I was rather surprised at the low scores on this problem. Perhaps people just got tired in the last three problems?
Problem 7 (10 points)
Getting an appropriate formula for an analytic f(z) is worth 4 points (just writing zz without further verification ignores the difficulty!), and the correct value of f(i) is worth 1 point. Explaining why this must be the only valid formula is worth the other 5 points. I'll give 3 of these 5 points for just asserting that they should be the same because they agree on the positive real axis but I will need some further "theoretical" statement to earn the last 2 points.
Comment Here there actually seemed to be serious misunderstanding about what was wanted. I did give some (minimal!) credit for a straightforward "value" of ii, but the real difficulty of the problem is what that particular complex number (actually a real number!) is the correct value. And few students seemed to see that the connection between xx and e-π/2 needed to explained. That is most of what I was looking for.
Comment on the totals Anyone who got a B or B+ or A on this exam (there were 14 of you!) should be proud. This was a rather stiff exam for an undergraduate course, even if it was a takehome exam. The final exam will be simpler.
The final exam and the essays
Let me address grading the essays first. I did read them, and reread
them, and grade them, and contrast the grading, and try to make a fair
assessment. I read them for both math content and exposition. Students
who responded straightforwardly to the
the essay assignment generally wrote essays which were direct,
easy to read and understand, and easy to grade. Almost invariably
these students earned 50 out of 50 points. Other students decided (I
don't know why!) to make more of an adventure out of the assignment. I
found those essays more problematic, and when I read them, I
frequently needed more information than was supplied (and that's a
defect!) or found the language difficult to follow. Those grades
ranged from 30 to 50. The essays are all available for pickup whenever
you wish, with comments and large red question marks as appropriate.
I mentioned the following to several students. I did the exam, all of the problems, completely written out with details, about a week before you all did. I do this to estimate the exam difficulty and suitability. Please realize that I'm supposed to know this stuff well, and that I can write solutions quickly. The following is not a boast but more of a sad confession. I wrote complete answers to 9 of the 10 problems with little strain in about 20 to 25 minutes. Then I spent about an hour working on the darn complex evaluation of the improper integral (problem #3). Indeed, for over half that time I kept getting a pure imaginary answer for the value of the integral. I made numerous errors. I was trying to work so fast that diligence was completely forgotten. Finally I got the answer given by Maple. This was both irritating and very embarrassing. Unsurprisingly, I graded the third problem last because of my distaste! Here are approximate letter grade assignments for the final exam:
Problem (20 points)
4 points for the idea, including explicitly stating the unique candidate for f(z). 12 points for some explanation (using "If ... then ..." and sentences) so I could read and have confidence that the writer understood what's going on.
Problem 2 (20 points)
This was to be the cute problem. I wanted people to cite the Cauchy Integral Formula for a) (7 points), the Cauchy Integral Formula for derivatives for b) (7 points), and Cauchy's Theorem for c) (6 points). A few students in fact did this. But many people just threw the Residue Theorem at all three of these integrals, and, of course, that works. This powerful result indeed does include all of the situations described by the previous three theorems.
Problem 3 (20 points)
3 points for the residue computation, 8 points for the contours and integral estimations, and then 6 points for putting everything together (so the answer comes out correct and is a real number!). There are several different strategies for the solution, but unless you are careful about the formulas for the analytic function z1/3 and stay away from the appropriate "branch cut", successful completion of the problem is impossible.
Problem 4 (20 points)
Part a) was worth 8 points. I wanted people to declare "geometric series" and look at the appropriate ratio: not much of a solution, but that's what is needed. Part b) was worth 12 points. I wanted the real part of both sides, and some care must be used. The formula involved can be verified with standard trig identities but this is a lot of work. The formula is useful when studying Fourier series.
Problem 5 (20 points)
The singularity at 0 is special, and earned 6 points for its consideration. There are infinitely many other singularities, and I wanted the student's work to apply to all of them. I gave 6 points for the order of the pole (and it is not enough to only consider the denominator in analyzing the order of the pole: you'd better also look at what's on top [z/z2!]). I gave 8 points for a calculation of the residue. It was not enough to just write an answer -- I wanted some explanation of what you were doing.
Problem 6 (20 points)
This is the stupid problem of the exam. I can only confess I copied it from some textbook whose author was not very creative (but the lack of creativity and care rebounds to me, doesn't it?). Explicit roots of this cubic polynomial can be given without much difficulty. Then the problem is rather simple. A more complicated polynomial would have been nice. Oh well. I did get to penalize people who wrote inequalities the wrong way, or who had complex numbers directly (without moduli) participating in inequalities.
Problem 7 (20 points)
10 points for each part. This was not a novelty, since part a) and even part b) (slightly simplified) were on an earlier exam. For part a), I gave 4 points for citing the Cauchy-Riemann equations, and 3 points each for verifying each one of them. In b), I gave 4 points for applying the Laplacian to the indicated function, 3 points for determining the value of A, and 3 more points for finding all harmonic conjugates. I was nasty: people had to compute derivatives of polynomials and trig and hyperbolic functions.
Problem 8 (20 points)
Maybe this was a slightly imaginative problem, asking for students to explain a calculus result which might be difficult to forecast using only calculus. I looked for radius of convergence, distance to singularities, etc. in the explanation.
Problem 9 (20 points)
This is a rather routine problem about linear fractional transformations, but we covered the material in the last few days of the semester. The plurality of the points, 9, went for discovering a formula for T(z). Please note that a linear fractional transformation has infinitely many descriptions because (αaz+αb)/(αcz+αd) is the same as (az+b)/cz+d). Indeed, abstractly the group of these transformations is called the projective general linear group, partly because of this. I starting writing a table of the valid formulas discovered by students. I am sure there were at least 7 or 8. 1 point for b), 2 points for the image in c) and 1 point for the brief description, 2 points for the image in d) and 1 point for the brief description, 1 point for e), and 2 points for the image in f) and 1 point for the brief description. A few of the graphs were bizarre, but many were correct.
Problem 10 (20 points)
I had hoped that this would be an easy and elementary problem. 10 points could be earned for each part. In a), I gave 2 points for the correct modulus and 2 points if the points in the picture formed a square centered around the origin. Then full credit was earned with the rectangular form of the roots. In b), the picture could earn 3 points, and the general formula, in rectangular form, would earn the balance of 7 points.
I computed a number for each student using the information I had. I then attempted to assign letters using this set of numbers and a scale with percentages similar to the final exam scale. There were few differences from the exam grades. I entered the course grades into the Registrar's computer system on Monday, May 12. I hope students will be able to see them soon.
If I ever happen to have students in this course in some other course, I will be sure to request or even require that assertions in all written work be accompanied by clear and convincing supporting reasons.
Indeed: reasons and logic, written well using complete English sentences.
If you have questions ...
Rutgers requests that I retain the final exams. Students may ask to look at their exams and check the exam grading. These students should send me e-mail so that a mutually satisfactory meeting time can be arranged. Students may also ask how their course grades were determined using the process I described. Probably e-mail will be sufficient to handle most such inquiries.
Maintained by email@example.com and last modified 5/12/2008.