COURSE: MATH 5520
TIME: TBA TR; PLACE: TBA; CALL# 12228
INSTRUCTOR: Dr. Robert Gardner; OFFICE: Room 308F of Gilbreath Hall
OFFICE HOURS: TBA TR; PHONE: 439-6979 (Math Office 439-4349)

E-MAIL: gardnerr@etsu.edu
WEBPAGE: www.etsu.edu/math/gardner/gardner.htm (see my webpage for a copy of this course syllabus and updates for the course).

TEXT: Functions of One Complex Variable, Second Edition, by John Conway.

PREREQUISITE: Complex Analysis 1 (MATH 5510).

ABOUT THE COURSE: We continue our exploration of functions of a complex variable. We'll finish the results on complex integration, study singularities and Laurent series, and look at the various versions of the Maximum Modulus Theorem and some of its implications (not all of which are in the Conway book). If time permits, we will look at additional topics, such as metric spaces, Riemann surfaces, or applications.

OUTLINE: Our tentative outline is:
Chapter 4. Complex Integration (continued): Zeros of analytic functions, Fundamental Theorem of Algebra, Maximum Modulus Theorem, winding number, Cauchy's Integral Formula, properties of path integrals, Open Mapping Theorem.
Chapter 5. Singularities: Classification of singularities, Laurent series, residues, integrals, meromorphic functions, argument principle, Rouche's Theorem.
Chapter 6. Maximum Modulus Theorem: Versions of Max Mod Theorem, Schwarz's Lemma, Hadamard's Three Circles Theorem (maybe), Pragmen-Lindelof Theorem (maybe).
Other Possible Topics: Metric spaces (Chapter 2), Riemann zeta function (Section VII.8), Riemann surfaces (Chapter IX), applications of complex analysis results, research results on the location of zeros of a polynomial in terms of coefficients.

GRADING: Homework will be assigned and collected regularly. Grades will be assigned based on a 10 point scale with "plus" and "minus" grades being assigned as appropriate (based on grade points assigned by the university, on a plus/minus 3 point system). I expect that you will work together on the homework, but make sure that the work you turn in is your's and that you understand it. Your homework grade will also reflect how clearly you write up your solution and document your claims.

IMPORTANT DATES:
Monday, January 16 = Martin Luther King, Jr. Day, no class.
Monday, March 5 to Friday March 9 = Spring Break, no class.
Friday, April 27 = Last day of class.

OTHER RESOURCES. The following were mentioned in the Complex Analysis sequence:

1. The Meaning of Mathematics (Lecture notes from the September 1, 2009 class).
2. Ordering the Complex Numbers. (Also available in PostScript.)
3. Complex Polynomials on GoogleBooks by Terence Sheil-Small, Cambridge University Press, 2002. See Chapter 6, "The Illief-Sendov Conjecture."
4. Geometry of Polynomials, Mathematical Monographs and Surveys #3, on GoogleBooks by Morris Marden, AMS, 1986. See Chapter II "The Critical Points of a Polynomial." For a study of the location of the zeros of a polynomial in terms of the coefficients, see Chapter VII.
5. To access the Mathematical Reviews: Go to the Sherrod Library online catalog. Click the "Title" tab and enter "Mathematical Reviews." Select "MathSciNet [Electronic Resource]" and follow the links. You will be asked to enter your user ID and password (the same you use for your ETSU e-mail). You are then redirected to MathSciNet and can freely use it and even download PDF versions of some of the papers you find! Of course, this protocol will work for any electronic journal available through the Sherrod Library.
6. Notes on "The Ilieff-Sendov Conjecture" in PDF and PostScript.
7. Handout "A Primer on Lipschitz Functions" in PDF and PostScript.
8. "Dr. Bob's Favorite Results on Polynomials," presented to the Math Department, Fall 2011: PowerPoint.

HOMEWORK.The following homework is assigned:

Return to Bob Gardner's webpage
Last Updated: April 26, 2012.