Problem | #1 | #2 | #3 | #4 | #5 | #6 | #7 | #8 | Total |
---|---|---|---|---|---|---|---|---|---|

Max grade | 9 | 14 | 12 | 13 | 16 | 12 | 12 | 12 | 96 |

Min grade | 3 | 0 | 2 | 2 | 3 | 0 | 1 | 5 | 19 |

Mean grade | 6.67 | 11.57 | 7.95 | 9.86 | 12.52 | 9.62 | 8.57 | 9.71 | 76.48 |

Median grade | 8 | 13 | 8 | 11 | 16 | 10 | 9 | 10 | 82 |

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:

Letter equivalent | A | B+ | B | C+ | C | D | F |
---|---|---|---|---|---|---|---|

Range | [85,100] | [80,84] | [70,79] | [65,69] | [55,64] | [50,54] | [0,49] |

**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.

Problem | #1 | #2 | #3 | #4 | #5 | #6 | #7 | Total |
---|---|---|---|---|---|---|---|---|

Max grade | 18 | 18 | 14 | 12 | 12 | 16 | 10 | 87 |

Min grade | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 15 |

Mean grade | 13.5 | 11.55 | 10.5 | 10.4 | 4.9 | 7.85 | 2.9 | 61.6 |

Median grade | 15.5 | 15.5 | 13 | 12 | 3 | 7 | 3 | 68 |

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:

Letter equivalent | A | B+ | B | C+ | C | D | F |
---|---|---|---|---|---|---|---|

Range | [75,100] | [65,74] | [55,64] | [50,54] | [40,49] | [35,39] | [0,34] |

**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 z^{z} 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 i^{i}, 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 x^{x} 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.

Problem | #1 | #2 | #3 | #4 | #5 | #6 | #7 | #8 | #9 | #10 | Total |
---|---|---|---|---|---|---|---|---|---|---|---|

Max grade | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 198 |

Min grade | 0 | 0 | 1 | 13 | 3 | 4 | 5 | 0 | 0 | 0 | 86 |

Mean grade | 16.11 | 16.11 | 18.21 | 18.58 | 13.42 | 17.79 | 14.84 | 17.95 | 14.37 | 16.05 | 160.63 |

Median grade | 20 | 20 | 16 | 20 | 16 | 19 | 17 | 19 | 20 | 20 | 180 |

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:

Letter equivalent | A | B+ | B | C+ | C | D | F |
---|---|---|---|---|---|---|---|

Range | [170,200] | [160,169] | [140,159] | [130,139] | [110,134] | [100,109] | [0,99] |

**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* z^{1/3} 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/z^{2}!]). 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.

**Course grading**

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 everhappen to havestudents in this course in some other course, I will be sure to request or evenrequirethat 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
greenfie@math.rutgers.edu and last modified 5/12/2008.
**