Complex Analysis 1 - Fall 2007

The students of Complex Analysis 1, Fall 2007

COURSE: MATH 5510
TIME: 12:10-1:30 MW

PLACE: Room 477 of Brown Hall
CALL # 34750

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.

The necessary background material in real variables can be found in:

  1. An Introduction to Analysis, by J. R. Kirkwood.
  2. The Elements of Real Analysis, by Robert Gardner Bartle.
  3. A Primer of Real Functions, by R. P. Boas.
Some references for introductory complex variables are:
  1. Complex Variables and Applications, by R. Churchill and J. Brown.
  2. Complex Variables, Schaum's Outline Series.

OUTLINE: Our tentative (and ambitious) 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.
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.
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).

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.

IMPORTANT DATES:
Monday, September 3 = Labor Day Holiday.
Friday, September 7 = Last day to drop without a grade of "W."
Monday, September 10 = Last day for 75% refund.
Monday, October 15 = Fall Break Holiday.
Monday, October 22 = Last day to drop without dean's permission.
Thursday and Friday, November 22 and 23 = Thanksgiving Holiday.
Tuesday, December 5 = Last day to withdraw from the university.
Friday, December 7 = Last day of class.
Wednesday, December 12 = Final, 1:20 p.m. to 3:20 p.m.

OTHER RESOURCES. The following were mentioned in class:

  1. The Meaning of Mathematics (Lecture notes from the September 5, 2007 class).

HOMEWORK.The following homework is assigned:

Assignment
Problems
Due Date
Credit
Cumulative Credit
HW1
1.2.1c, 1.2.1d, 1.2.4a
Friday August 31
3+3+3=9
9
HW2
1.4.2c, 1.4.3, 1.4.5, BONUS: 1.4.6
Monday September 17
3+3+3+(3)=9+(3)
18+(3)
HW3
2.1.1c or d, 2.1.3, 2.1.10d, BONUS 2.1.7
Wednesday September 26
3+3+3+(3)=9+(3)
27+(6)
HW4
2.2.1b, 2.2.3c
Wednesday October 3
3+3=6
33+(6)
HW5
2.3.3, 2.3.8, BONUS 2.3.4
Wednesday October 10
3+3+(3)=6+(3)
39+(9)
HW6
2.4.1, 2.4.4, BONUS 2.4.6
Wednesday October 17
3+3+(3)=6+(3)
45+(12)
HW7
3.1.6a, 3.1.6b, 3.1.7, BONUS 3.1.5
Wednesday October 17
3+3+6+(3)=12+(3)
57+(15)
HW8
3.2.1, 3.2.11, 3.2.14, BONUS 3.2.19
Monday November 19
3+3+3+(3)=9+(3)
66+(18)
HW9
4.1.6, 4.1.8, 4.1.11, BONUS 4.1.19
Wednesday November 28
3+3+3+(3)=9+(3)
75+(21)
HW10
4.2.7a, 4.2.9a, 4.2.9c, BONUS 4.2.6
Wednesday December 5
3+3+3+(3)=9+(3)
84+(24)
Bonus points are in parentheses.

Return to Bob Gardner's webpage