Penn State University

PSU Mathematics DepartmentEberly College of Science

Math  502,  Spring 2008

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Math 502 

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.

Course Materials:
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 major.   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 they may have about the course or about mathematics more generally .

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, up 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 plane.

The 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 questions).  

 

 

 

 

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