Complex Analysis 1 - Fall 2011

COURSE: MATH 5510
TIME: 9:45--11:05 TR; PLACE: Room 112C of Wilson-Wallace Hall; CALL# 80433
INSTRUCTOR: Dr. Robert Gardner; OFFICE: Room 308F of Gilbreath Hall
OFFICE HOURS: 10:15-11:15 MWF; 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: An undergraduate real analysis class or advanced calculus class.

ABOUT THE COURSE: It will be assumed that the student has been exposed to (and has a reasonable recollection of) the topology of R (open and closed sets, limit points, connectedness, compactness, completeness, lub and sup, glb and inf, sequences and series of real numbers, convergence, uniform convergence, comparison tests, Cauchy sequences), and properties of functions of a real variable (continuity, differentiability, power series representation). It is also assumed that the student has been exposed to some elementary properties of the complex numbers (algebra, geometry, roots of unity, modulus) and functions of complex variables.

OUTLINE: Our tentative outline is:
Chapter 1. The Complex Number System: Introduction to the complex plane, real and imaginary parts, modulus, polar representation, extended complex plane, Riemann sphere.
Chapter 2. Metric Spaces and Topology of C: Extensions of several ideas from R to C and other metric spaces, open and closed sets, connectedness, sequences, completeness, compact sets, continuity, convergence, uniform convergence.
Chapter 3. Elementary Properties and Examples of Analytic Functions: Series, convergence of series, differentiability, analytic functions, mappings, Mobius transformations.
Chapter 4. Complex Integration: Riemann-Stieltjes integrals, power series, zeros of analytic functions, Fundamental Theorem of Algebra, Maximum Modulus Theorem, winding number, Cauchy's Integral Formula, properties of path integrals, Open Mapping Theorem.

GRADING: Homework (H) to be turned in will be assigned regularly. We will have two tests (T1 and T2) and your average will be computed as follows:

AVERAGE = (2H + T1 + T2)/4.
Grades will be assigned based on a 10 point scale with "plus" and "minus" grades being assigned as appropriate.

IMPORTANT DATES:
Monday, September 5 = Labor Day Holiday.
Friday, September 9 = Last day for 75% refund of fees.
Friday, September 9 = Last day to drop without a grade of "W."
Friday, September 23 = Last day for 25% refund of fees.
Monday and Tuesday, October 17 and 18 = Fall Break Holiday.
Monday, October 24 = Last day to drop without dean's permission.
Tuesday, November 1 = Midterm Exam.
Thursday and Friday, November 24 and 25 = Thanksgiving Holiday.
Wednesday, December 7 = Last day to withdraw from the university.
Friday, December 9 = Last day of class.
Thursday, December 15 = Comprehensive final, 8:00 a.m. to 10:00 a.m.

OTHER RESOURCES. The following were mentioned in class:

  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. Study Guide for Midterm Exam in PDF.
  9. "Dr. Bob's Favorite Results on Polynomials," presented to the Math Department, Fall 2011: PowerPoint.
  10. Study Guide for Final Exam in PDF.

HOMEWORK.The following homework is assigned:

Assignment
Problems
Due Date
Credit
Cumulative Credit
HW1
I.2.1a, I.2.1c, I.2.1d, I.2.1g, I.2.2d, I.2.4
Tuesday September 13
1+1+1+1+2+6=12
12
HW2
I.3.3, I.4.2c, I.4.5, BONUS: I.4.6
Thursday September 22
3+3+3+(3)=9+(3)
21+(3)
HW3
I.6.2, I.6.4a, III.1.6a, III.1.7 BONUS: I.6.4b
Tuesday October 11
6+3+2+4+(3)=15+(3)
36+(6)
HW4
III.2.6a,d; III.2.12, III.2.14, III.2.17; BONUS: III.2.18a,b,c
Thursday October 27
4+3+3+3+(9)=13+(9)
49+(15)
HW5
IV.1.6; IV.1.8, IV.1.11
Wednesday November 16
3+3+3=9
58+(15)
HW6
IV.1.19; IV.1.20, IV.1.22
Thursday December 1
3+3+3=9
67+(15)
HW7
IV.2.5; IV.2.7a, IV.2.9a
Thursday December 9
3+3+3=9
76+(15)
Bonus points are in parentheses.

Return to Bob Gardner's webpage
Last Updated: January 2, 2012.