The first exam

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:

Discussion of the grading

Grading guidelines

Minor errors (such as a missing factor in a final answer, sign error, etc.) were penalized minimally. Students whose errors materially simplify the problem were not be eligible for most of the problem's credit.
The grader will read only what is written and not attempt to guess or read the mind of the student.
The student should solve the problem given and should not invent another problem and request credit for working on that problem.

Problem 1 (10 points)
4 points for the graph (three points in correct positions at the vertices of an equilateral triangle), and 2 points for each of the roots, a total of 6 points. 3 points partial credit are earned if some correct algebraic work is shown. Only 3 of 6 points for the roots are earned if specific values of sine and cosine are not given.

Problem 2 (10 points)
3 points for correctly setting up a verification of what's proposed. And then I'll try to grade as much as possible towards a good conclusion.

Problem 3 (10 points)
Recasting the problem as "u=v" is worth 3 points. Then using the CR equations is worth another 2. The remaining 5 points is for solving the problem.

Problem 4 (10 points)
I think any attempts to solve this problem without using Cauchy's Theorem (or, equivalently, some form of Green's Theorem) are not likely to be successful. So 3 points for mentioning one of these forms of solution and 2 points for the answer. 5 points for some explanation of why the use of Green's Theorem or Cauchy's Theorem is relevant which must include the word "analytic" or "holomorphic" (2 points) and the phrase "piecewise smooth closed closed curve" (2 points). The answer alone is worth 2 points.

Problem 5 (10 points)
I believe only antidifferentiation is likely to "solve" this problem. 3 points for mentioning the technique, and then the remaining points are awarded, as much as possible, in the evolution (?) towards a solution. The answer is worth 2 points but only 1 point can be lost if a student insists on incorrectly "simplifying" the answer. I took off 2 points if the antiderivative is incorrect (I didn't check the arithmetic then).

Problem 6 (10 points)
The only technique at this stage in the course which will be successful is direct parameterization and computation. The beginning of that is worth 3 points, and the balance of the credit is obtained by carrying out the initial idea. Any incorrect "simplification" can lose only 1 point. The answer is worth 2 points. I'll give 2 points for noting that z⁴/4 has derivative z³ and the integral of that part around the closed curve is 0. People who don't "evaluate" e^8Πi (it is 1) lose 1 point.

Problem 7 (10 points)
Here the length of the integrand is worth 2 points. 6 points for the estimation of |f(z)|. Of this, 2 points for the top, and 4 points for the bottom of the fraction defining f(z). Finally, the estimate and the limiting argument can earn 2 points.

Problem 8 (10 points)
a) (6 points) Using some "test" is worth 2 points, and 3 points are for using the test correctly. Either the Ratio or the Root Test are not difficult to complete. 1 point for the answer.
b) (2 points) 1 point for answering the absolute convergence question, and 1 point for answering the divergence question.
c) (2 points) 1 point for answering the convergence question, and 1 point for giving a connection between absolute convergence and convergence.

Problem 9 (10 points)
a) (5 points) 2 points for finding the Laplacian of the function. 1 point for the answer, and 2 points for supporting evidence.
b) (2 points) 1 point for "Yes" and 1 point for some supporting reasoning.
c) (3 points) 1 point for "No" and 2 points for an example (1 for exhibiting a valid example, and 1 point for some verification).

Problem 10 (10 points)
a) (5 points) 1 point for the picture in general, and then 1 point for the vertices, 2 points for the edges (one straight edge and two curvy [parabolic!] edges), and 1 point for the inside.
b) (5 points) 1 point for the picture in general, and then 1 point for the vertices, 2 points for the edges (two straight edges and two circular arcs), and 1 point for the inside.

The second exam

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:

Discussion of the grading

This was a take home exam. Certainly the questions were serious and difficult, but I intended that students who thought about the questions would be able to solve them, helped by the longer time they had to think and consult references. I adjusted the letter grade "bins" as described above to take into account the difficulty. I would give this exam to students in an appropriate course at any college I know. Here are some things noticed as I read student answers. My comments may be helpful to you in the future.

• Students did not answer questions. One specific simple example is easy to cite. Part a) of problem 5 asks the following:

  If z is not equal to ±i explain briefly why there must be such a curve.

  A number of students did not answer this question. Other matters were discussed, but the question was not answered.
  Even when hints were given, as in problem 6:

  Show that there is such an entire function in this case.

  Many students just assumed there was an entire f with Re(f)=u and did not explain their assertion.
  Read the questions and answer the questions.
• Students made statements which were totally unsupported by citation or reasoning. For these I tended to write WHY? but this does not adequately represent my distress. This is an upper-level course, and students in it should be able to think and communicate. Assertions are not the same as verifications so writing the statement "The function is analytic" isn't magically convincing.
  Assertions should be supported by ideas carefully explained.
• On a take home exam, citations and references can be checked. A number of students wrote that Morera's Theorem verified path independence. It does not. It contains a sufficient condition to conclude analyticity. Other students confused "analytic" and "harmonic", even after this distinction was specifically mentioned. You can look things up, you can think ... One reason I gave this exam was to be able to evaluate student work in circumstances which were more authentic than a timed exam.
  Read your work; be sure it makes sense.

Grading guidelines

The grader will read only what is written and not attempt to guess or read the mind of the student.
The student should solve the problem given and should not invent another problem and request credit for working on that problem.

Problem 1 (16 points)
3 points for location of the singularities (some explanation should be offered).
4 points for the discussion of the singularity at 0.
9 points for the discussion of the non-zero singularities: 3 points for identifying the type of singularity (2 for some supporting reason), 2 for the order of the pole, and 4 of these points for the residue calculation.

Problem 2 (16 points)
2 points for the contour; 4 points for the residue computation; 3 points for the limit of the big semicircle; 3 points for the limit of small semicircle; 4 points for putting together the two intervals and the residue to get the result.

Problem 3 (16 points)
a) 4 points for getting the Laurent series for g(z).
b) 4 points for explaining why g(z) has a pole, and 4 points for explaining that only a finite number of the Laurent coefficients are non-zero.
c) 4 points for applying the previous information to show that f(z) is a non-constant polynomial.

Problem 4 (18 points)
a) 6 points, with 2 for starting some strategy which will be successful.
b) 6 points, with a Cauchy formula worth 3 of them.
c) 6 points, with a suggestion of a valid example worth 2, 2 more points for checking the limiting behavior of the function values, and the final 2 points for the derivative behavior.

Problem 5 (18 points)
a) 4 points. Some pictures and some supporting discussion are needed for full credit.
b) 6 points for discussion of A(1), and then 2 points for "transporting" this to a more general discussion.
c) 6 points: 2 points for a valid suggestion of a domain, and 4 points for supporting reasoning. (–2 if the domain is not maximal, and –2 if the domain is not open).

Problem 6 (16 points)
8 points for explaining why a harmonic conjugate exists, and the other 8 points for explaining why the resulting entire function is constant.

The final exam

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:

Discussion of the grading

Grading guidelines

Minor errors (such as a missing factor in a final answer, sign error, etc.) were penalized minimally. Students whose errors materially simplify the problem were not be eligible for most of the problem's credit.
The grader will read only what is written and not attempt to guess or read the mind of the student.
The student should solve the problem given and should not invent another problem and request credit for working on that problem.

Problem 1 (20 points)
6 points for the residue computation. 8 points for the contour and integral estimation, and then 6 points for putting everything together.

Problem 2 (20 points)
a) 6 points: 3 points for each equation (1 point for the answer and 2 for some reasoning).
b) 9 points: 3 points for an answer and 6 points for supporting evidence.
c) 5 points for all answers (2 points) with some supporting computation shown (3 points).

Problem 3 (20 points)
a) 5 points: the image of p is worth 1 point; each boundary curve is worth 2 points. One curve should be a straight line, and the other should be close to a circle with the correct center and radius.
b) 5 points: the image of p is worth 1 point; each boundary curve is worth 2 points. One curve should be a ray along the negative real axis, and the other should be a parabola with approximately the correct vertex and axis of symmetry.
c) 5 points: the image of p is worth 1 point; each boundary curve is worth 2 points. Each curve should be a circle with the correct center and radius.
d) 5 points: the image of p is worth 1 point; each boundary curve is worth 2 points. Each boundary curve should be a straight line with the correct slope (0!) and location.

Problem 4 (20 points)
An appropriate formula connecting f and f´´ is worth 5 points. Then using it is worth 10 points, and getting exactly the correct result is worth the last 5 points.

Problem 5 (20 points)
A use of Rouché's Thoerem would earn 8 points, and then completing the problem earns the other 12 points. Other verifications will be assessed appropriately (for example, a direct use of the Argument Principle could be tried).

Problem 6 (20 points)
a) 6 points: false (1 point); example (2 points); fixup with f´≠0 (3 points).
b) 7 points: false (1 point); example (3 points); fixup with interior of C inside U (3 points).
c) 7 points: false (1 point); example (3 points); fixup with convergence up to a boundary or singular point (3 points).

Problem 7 (20 points)
a) 8 points for using the Cauchy-Riemann equations.
b) 12 points: 4 points for verification of harmonicity, 4 points for a harmonic conjugate, and 4 points for writing a function of z.

Problem 8 (20 points)
a) (9 points) The integral of f(z)/z is worth 4 points and can be evaluated with the Cauchy integral formula. The integral of f(z)/(z-i)² is worth 5 points and 1 point of that is reserved for noticing that the curve winds (!) around i twice and then using this information to contribute to the answer.
b) (11 points) There are 3 singularities inside the circle (2 points). Finding the residue correctly at one singularity is worth 3 points, and then at the others, and the others together are worth 4 points. Assembling the answer is worth 2 points. Only noticing the pole at 0 is worth 1+3+1=5 points.

Problem 9 (20 points)
A viable approach earns 8 points (for example, some integral formula). The balance is earned by carrying it out.

Problem 10 (20 points)
a) (10 points) 7 points for the non-zero coefficients (1 each for each coefficient and term, and 5 for the process); 3 points for an answer to the convergence question.
b) (10 points) 7 points for the non-zero coefficients (1 each for each coefficient and term, and 5 for the process); 3 points for an answer to the convergence question (-1 if the distance is not to Π/2).

Course grades