COURSE: MATH 5520
TIME: 3:45-5:05 TR; PLACE: Sam Wilson Hall, Room 322; CALL# 13372
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 study complex integration, 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 spaces of analytic functions, the Riemann Zeta Function, analytic continuation, and Riemann surfaces.

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: Spaces of analytic functions (Sections VII.1-3), factorization (Sections VII.5 and 6), the Riemann zeta function (Section VII.8), analytic continuation and Riemann surfaces (Chapter IX). We may also consider research results on the location of zeros of a polynomial in terms of coefficients (so called "Enestrom-Kakeya Theorem" type results).

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).

A NOTE ABOUT HOMEWORK: You must show all details on the homework problems!!! Justify every step and claim you make - this is how you convince me that you know what you are doing. While I suspect that you may work with each other on the homework problems (in fact, I encourage you to), I expect that the work you turn in is your own and that you understand it. Some of the homework problems are fairly standard for this class, and you may find proofs online or in an online version of the solutions manual. The online proofs may not be done with the notation, definitions, and specific methods which we are developing and, therefore, are not acceptable for this class. If I get homework from two (or more) of you that is virtually identical, then neither of you will get any credit. If you copy homework solutions from an online source, then you will get no credit. These are examples of plagiarism and I will have to act on this as spelled out on ETSU's "Academic Integrity @ ETSU" webpage: http://www.etsu.edu/academicintegrity/faculty.aspx (last accessed 1/11/2014). To avoid this, do not copy homework and turn it in as your own!!! Even if you collaborate with someone, if you write the homework problems out in such a way that you understand all of the little steps and details, then it will be unique and your own work.

IMPORTANT DATES (see http://www.etsu.edu/etsu/academicdates.aspx for the official ETSU calendar; last accessed 1/11/2014):
Monday, January 20 = Martin Luther King, Jr. Holiday.
Wednesday, January 29 = Last day to drop without a grade of "W."
Monday through Friday, March 10 to 15 = Spring Break Holiday.
Wednesday, March 12 = Last day to drop without dean's permission.
Wednesday, April 30 = Last day to withdraw from the university.
Friday, May 2 = Last day of class.

OTHER RESOURCES. The following may be useful in Complex Analysis 2.

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.
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. "Dr. Bob's Favorite Results on Polynomials," presented to the Math Department, Fall 2011: PowerPoint.
7. "Bernstein Inequalities for Polynomials," a 20th anniversary presentation of my interview talk at ETSU; March 8, 2013: PDF.

HOMEWORK.The following homework is assigned:

Return to Bob Gardner's webpage
Last Updated: April 26, 2014.