Course Syllabus
Course Information
Complex Analysis
MATH 6300
Section 1
RPI Fall 2013
4 cr
Lecture
MR
2:00PM-3:50PM
DCC 235
Course Website:
http://homepages.rpi.edu/~kramep/ComplexAnalysis/ca2013.html
Prerequisites or Other Requirements:
Undergraduate level experience with mathematical analysis and complex
variables
Instructor
Dr. Peter Kramer
Office Location: EATON 310
Office Hours: TBA
kramep@rpi.edu
(518) 276-6896
Teaching Assistant(s)
Name
Office
Office Hours
Email Address
Course Description
A basic graduate course covering Cauchy’s Theorem, residues, infinite series and
products, partial fractions, conformal mapping and the Riemann mapping
theorem, analytic continuation, zeros and growth of analytic functions,
approximation by rational functions, Phragmen-Lindelof Theorems, inversescattering theory, elliptic functions, and Riemann Surfaces.
Course Text(s)
All textbooks are optional. I may suggest readings in one or the other during the
lectures, but many topics can be found in all three texts. For those instances in
which I cover a topic not treated in all the books, I'll have the relevant material
scanned into a PDF file and posted here. So you should have at least one graduate
level complex analysis text, but you can choose the one (or several) that suit your
taste:
 Ahlfors, Complex Analysis: standard graduate textbook on complex
analysis. Theoretically inclined (analysis and algebra).
 Ablowitz and Fokas, Complex Analysis: advanced applied
mathematical treatment of complex analysis.
 Dettman, Applied Complex Variables: a relatively inexpensive
textbook with a concrete, application-oriented style
Syllabus
1 of 5
2.10.2016
Course Goals / Objectives
1. Provide a systematic, mathematical development of the essentials of
complex variable and function theory.
2. Describe some advanced applications of complex variables to problems in
physics, etc., other than those which are covered in other classes. In
particular, there is a ``Complex Variables and Integral Transforms with
Applications'' (MATH 6640, Spring 2015) class which doesn't emphasize
proofs but elaborates upon the use of contour integration in asymptotics
and evaluating integrals. This ``Complex Analysis'' class will develop the
mathematical foundations for contour integration (why it works) but not
repeat in detail the applications which are covered in the other class.
Rather, I will try to spend some time on other applications of complex
variable theory, primarily to physics.
3. Develop students' capacity for developing careful arguments and proofs.
Actually complex analysis is a rather friendly field, and the proofs are not
as technical as in other areas (such as real analysis, believe it or not). So if
you learn the concepts and know what constitutes a sound argument, and
can think about how to link the key concepts together, you can do the
proofs.
4. Cover some of the classical mathematically elegant aspects of complex
variables theory which do not necessarily have much application.
Course Content
Algebra of Complex Numbers
Geometry of Complex Numbers (including Stereographic Projection)
Linear Fractional Transformations
Analytic Functions
Branch Cuts and Riemann Surfaces
Complex Integrals
Cauchy-Goursat Theorem, Cauchy Integral Formula, and Corollaries
Taylor and Laurent Series in Complex Plane
Singularities of Analytic Functions
Analytic Continuation
Pade Approximants
Residue Theory for Integration
Physical Resonances Represented in Complex Plane
Landau Damping
Rouche''s Theorem and Applications
Conformal Mapping, with Application to Fluid Flow
Student Learning Outcomes
1. A student successfully completing the course will
learn the essentials of complex variable and function theory.
2. A student successfully completing the course will be able to deploy complex
variable techniques to practical applications.
Syllabus
2 of 5
2.10.2016
3. A student successfully completing the course will develop their capacity for
careful arguments and proofs.
Course Assessment Measures
Assessment
Due Date
Learning Outcome #s
Homework
Exam
4 installments
TBA
1, 2, 3
1, 2, 3
Grading Criteria
The course grade will be determined by a 70% weighting of homework (4
assignments) and a 30% weighting of the final exam. Students whose homework
shows clear positive evidence of representing their own thinking will be allowed
to skip the final exam and have their course grade determined completely (100%)
by their homework scores.
The course grade will be determined by 4 homework assignments. The first
homework will be due on or about September 27.
Each assignment will be scored out of 100 points, though usually more than 100
points are available so that students have some choice in which problems to invest
their effort in. I certainly do not expect every student to work on every problem,
but rather expect students to work out some subset of the homework problems
with care, diligence, and clarity of presentation. The grading standard will
correspond to this expectation. That is, the full points for a problem are generally
only awarded for a solution which approaches the problem with the elegance and
efficiency which should be expected from a proper understanding of the lectures
and the readings. Moreover, all nontrivial steps must be explained, particularly
those involving the concepts and techniques covered in this course. Routine
calculations involving lower-level mathematical manipulations such as matrix
algebra and calculus can be summarized without providing details. If you use a
numerical software package such as MATLAB or Maple to assist your
calculations, please attach a copy of your code or worksheet in order to receive
credit for that work.
Average Score (rounded)
9690-95
83-89
76-82
70-75
63-69
56-62
50-55
0-49
Syllabus
Grade
A
AB+
B
BC+
C
CF
3 of 5
2.10.2016
Late homework will be penalized 10 points per business day, and no credit will be
awarded once solutions are posted (which can be as soon as the next class). A
homework submitted on the due date but after the time specified will be
penalized 5 points.
Grade Appeals
First of all, you are always welcome to ask me during office hours for an
explanation for why a problem solution was deemed incorrect or incomplete. I
certainly would like all students to understand how to solve the problems, and to
resolve any confusion about what constitutes a proper solution. The following
applies only to situations in which the student is asking for a change in the score.
The only circumstance under which an appeal of a homework score will be
entertained is a demonstrable factual error in grading, meaning either that scores
were incorrectly totaled, or a correct response was marked incorrect. To
determine whether your response met the criteria for being deemed correct, you
should first consult the homework solutions, when they are posted. Uniform
standards for partial credit are applied for the class, so I will not revisit the
amount of points awarded for an incorrect or incomplete solution just because you
think or feel you deserved more points. Any request for a grade correction must
be made within one week of the date the solutions are posted for that homework.
If you think you have not been meted due justice by me, your next step is to
present your concern to the head of the Department of Mathematical Sciences.
If any grade appeal is deemed to be frivolous (meaning it falls outside the
guidelines of a legitimate appeal as described above), the student making the
frivolous appeal will be warned. Any future frivolous appeal will be penalized by
a deduction from the homework score equal to the number of points concerned in
the frivolous appeal.
Midterm Assessment
When your second homework is returned, you will receive a projection of your
course grade based on your performance to that point. At least if I remember.
(Feel free to ask if I forget.)
Attendance Policy
You don't have to tell me if you miss a class. But don't expect me to spend much
time giving you help with homework if you're not attending class.
Syllabus
4 of 5
2.10.2016
Academic Integrity
Student-teacher relationships are built on trust. For example, students must trust
that teachers have made appropriate decisions about the structure and content of
the courses they teach, and teachers must trust that the assignments that students
turn in are their own. Acts that violate this trust undermine the educational
process. The Rensselaer Handbook of Student Rights and Responsibilities defines
various forms of Academic Dishonesty and you should make yourself familiar
with these. In this class, all assignments that are turned in for a grade must
represent the student’s own work. In cases where help was received, or teamwork
was allowed, a notation on the assignment should indicate your collaboration.
If you obtained assistance from anyone outside of the course or any written
material beyond the lecture notes and the recommended texts for the course, you
must explicitly acknowledge the source.
If the solutions of two or more students do not demonstrate sufficient
independence of thought, but do not rise to the level of academic dishonesty, then
I may either simply split the points earned among all parties whose collective
mind produced the solution or impose an oral final exam on all parties. Flagrantly
corrupt homework will earn no credit, and clear violations of academic integrity
will also be reported to the Dean of Students' Office. The distinction between
"insufficient independence of thought" and "academic dishonesty" is primarily a
matter of whether the work demonstrates an intent to misrepresent one's own
work. If you are not clear on the concept of academic dishonesty, you might
consult the Rensselaer Handbook of Students Rights and Responsibilities or ask
me directly about my expectations for integrity.
If you have any question concerning this policy before submitting an assignment,
please ask for clarification.
You are encouraged to work in small groups on the homework assignments,
indicating on your submitted homework those other students with whom you had
significant interaction. Your actual solutions should be your own work. That is,
you should feel free to discuss how to approach the problems, to consult on how
to do certain calculations, or to check your results. But you should never be
copying from other students. I will only give credit for work that demonstrates
that you understand what you are doing. Therefore, be sure to explain all major
steps, especially how you are setting up the problem. It is not necessary to
provide detailed reports on routine calculations, but do at least explain in words
what you are doing.
Syllabus
5 of 5
2.10.2016