This is the second part of the basic graduate sequences course in analysis. It covers complex analysis which is the study of holomorphic functions of a complex variable. Complex analysis is one of the central topics in mathematics with applications that range from number theory to electrical engineering.
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 here. You will find the course syllabus and other relevant information on the ANGEL page for the course.
Facebook: There is a Facebook group for the course at http://www.facebook.com/home.php?sk=group_189283981087238
Instructor: John Roe
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.
Meeting Times: The class meets three times a week, on Mondays, Wednesdays, and Fridays from 11.15 to 12.05 in 106 McAllister Building.
My office hours will take place Thursdays 1:30-2:30, or at other times by appointment, in 107J McAllister. 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:
- There will be weekly homework assignments in class; probably a total of 12 assignments, due on Fridays. Of these assignments I will select the best 9 scores, each of which will count for 5% of your grade.
- Your written work will be graded on the logic and coherence of the exposition as well as on whether you reach the "right" answer.
- Submitted homework materials must be handwritten. Typed or TeXed materials will not be accepted.
- Late homework will not be accepted for grading.
- The final exam will count for 30% of the grade, the midterm for 15%, and 10% of the grade will be awarded for in-class activities which we will schedule as the course progresses.
Course Description: In 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. 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.
Course Materials Please follow links below (or on ANGEL) for lecture notes and homework assignments.