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 straigthforward "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.

**
Maintained by
greenfie@math.rutgers.edu and last modified 4/22/2008.
**