This is the second part of the
basic graduate sequences course in analysis. It covers complex analysis together with selected topic in functional analysis.
We will be
using a number of online and electronic resources. Each day's
lecture notes will be available on the web in advance, and you are
expected to study these notes before you come to the day's lecture.
We will use ANGEL (Penn State's course management software)
for online quizzes and for other purposes; you will need to log on to ANGEL regularly throughout
the semester. You can connect to the ANGEL main page by
pressing the button below. You will find the course syllabus and
other relevant information on the ANGEL page for the course.
Some course items will be linked
directly from this page; they will all be available through ANGEL also.
facebook: There is a Facebook group for the course at http://psu.facebook.com/group.php?gid=7178676443
Instructor: John Roe
Teaching Assistant: Rufus Willett
Link to the Lecture Notes here, and link to solutions to in-class problems here.
Textbook: "Complex Analysis" by Stein and Shakarchi is the recommended text for the course. "Real and Complex Analysis" by Rudin may be used for supplementary reading.
Prerequisites: A knowledge of calculus, analysis and linear
algebra at the undergraduate level, such as is provided by the Graduate
Study option in the Penn State mathematics
No formal acquaintance with complex analysis is required, but
an undergraduate course (like Penn State's Math 421) will
probably help by making you more familiar with some of the material.
The functional analysis component of the course will build on the
functional analysis from last semester's Math 501 course.
Meeting Times: The class meets three times a week, on Mondays,
Wednesdays, and Fridays from 11.15 to 12.05 in 113 McAllister Building.
My office hours will take place Wednesday 2.30 -
3.30 in 107J McAllister. Rufus' office hours are Thursdays 1.00 - 2.00.
Students are strongly encouraged to make use
of available office hours to discuss any questions or problems that
may have about the course or about mathematics more
Academic Integrity Statement All Penn State policies regarding
ethics and honorable behavior apply to this course. Academic integrity
is the pursuit of scholarly activity free from fraud and deception and
is an educational objective of this institution. Academic dishonesty
includes, but is not limited to, cheating, plagiarizing, fabricating of
information or citations, facilitating acts of academic dishonesty by others,
having unauthorized possession of examinations, submitting work of another
person or work previously used without informing the instructor, or tampering
with the academic work of other students. For any material or ideas obtained
from other sources, such as the text or things you see on the web, in the
library, etc., a source reference must be given. Direct quotes from any
source must be identified as such. All exam answers must be your own, and
you must not provide any assistance to other students during exams. Any
instances of academic dishonesty will be pursued under the University and
Eberly College of Science regulations
concerning academic integrity.
Grading Your grades for this course will be computed on
the basis of twelve weekly homework assignments and a final exam. Homework
assignments will be posted on this web site and handed out in class.
They will be due on Fridays (starting Friday January 25th) and I will
aim to return graded homework on the following Mondays. Each assignment
will contain three questions. You will be graded on the quality and coherence of your exposition as well
as on whether you have the "right answer".
Grades will be calculated as follows:
- Each homework assignment will be graded out of 20
points (six points per question, plus two for turning in the
assignment). The best
ten of your homework assignments will contribute to your total score,
to a maximum of 200 points.
- The midterm exam will consist of four questions
each of which will be graded out of 25 points. You may submit at
most two questions for grading, which will therefore contribute up to
50 points to your total score.
- The final exam will consist of six questions (four
on complex analysis, two on functional analysis) each of which will be
graded out of 40 points. You may submit at most three
questions for grading, which will therefore contribute up to 120 points
to your total score.
- Finally, 30 points will be allocated based on the
results of quizzes and problems that you are expected to solve after
each lecture. Many of these you will grade yourselves on the
honor system and enter the results into ANGEL.
Grades will be assigned on the basis of your total score (maximum 400). I anticipate
that around 330 points will suffice for an A grade, but this may be changed
as the course progresses; a more detailed grading scale will be posted
by midsemester. Please note that this cutoff is lower than you may
be used to in an undergraduate course; this reflects the fact there is
usually room for improvement in an extended-answer homework solution.
No late homework will be accepted under any circumstances.
Course Description: In the main part of this
course we will study the behavior of differentiable complex-valued
functions f(z) of a complex variable z. The key idea in
the course is that complex differentiability is a much more restrictive
condition than real differentiability. In fact,
complex-differentiable functions are so rigid that the entire behavior of
such a function is completely determined if you know its values even on a
tiny open set. One understands these rigidity properties by making
use of contour integration - integration along a path in the complex
theory gains its force because there are plenty of interesting functions to
which it applies. All the usual functions - polynomials, rational
functions, exponential, trigonometric functions, and so on - are
differentiable in the complex sense. Very often, complex analysis
provides the solution to "real variable" problems involving these
functions; as someone said, "The shortest path between two real points
often passes through the complex domain." Moreover, complex analysis
is a key tool for understanding other "higher transcendental
functions" such as the Gamma function, the Zeta function, and the
elliptic functions, which are important in number theory and many other
parts of mathematics. A secondary aim of this course is to introduce
you to some of these functions.
One of the surprises of complex
analysis is the role that topology plays. Simple
questions like "do I choose the positive or negative sign with the
square root" turn out to have surprisingly subtle answers, rooted in
the notion of the fundamental group of a topological space (which some of you will
be looking at in the Topology and Geometry course parallel to this). These
topological notions eventually culminate in the notion of a Riemann
surface as the correct global context for complex analysis. We
will not develop this idea fully, but we will discuss `multiple-valued
functions' and their branch points; again, we will try to illustrate how
these exotic-sounding concepts help in doing practical calculations.
In the second part of the course we will discuss some classical theorems in functional analysis,
involving the concepts of Banach and Hilbert spaces. The
motivation here is to apply tools originating in linear algebra to
analytical problems (such as differential equations or approximation