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 straigthforward "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.
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