Complex Analysis I

Fall Semester 2016



Instructor: Matt Kerr
Office: Cupples I, Room 114
e-mail: matkerr [at] math.wustl.edu
Office Hours: Tu/Th 12-1, and by appointment

Course Outline:

I. Basics on complex functions and topology
II. Formal and convergent power series
III. Conformal maps and fractional linear transformations
IV. Complex integration and Cauchy's theorem
V. Properties of analytic and harmonic functions

This is the first half of a year-long course which forms the basis for the Ph.D. qualifying examination in complex variables. This semester we will cover the first four to five chapters of the classic book by Ahlfors (3rd edition), with various embellishments from other points of view, especially the systematic use of power and Laurent series as in the books by Cartan and Lang (see below). The central result is the homology version of Cauchy's theorem.

In the second semester we will finish Ahlfors, covering (among other things) elliptic functions, the Riemann mapping theorem, the big Picard theorem, and the prime number theorem.

Prerequisites: Math 4111, 4171 and 4181, or permission of instructor.

Class Schedule:

Lectures are on Monday, Wednesday and Friday from 2-3 in Cupples I Rm. 207. First class is Monday Aug. 29 and last class is Friday Dec. 9, with holidays on Monday Sep. 5, Monday Oct. 17, Wednesday Nov. 23, and Friday Nov. 25.

Midterm Exam: Wednesday Oct. 26 (in class) [solutions]
Final Exam: Monday Dec. 19, 3:30-5:30, in the same classroom. [study guide]

Both exams are in Rm. 207. The midterm will cover all of Chap. 1-3 and some of Chap. 4 (in Alhfors).

Assignments:

Posted here each Friday, collected 1 week later (at beginning of class on Friday), and returned the following Monday or Wednesday. Solutions will also be posted and will include students' work. Please feel free to come to office hours to discuss problem sets. Make sure you can do the problems I don't assign.

Problem Set 1: hand in #3, 4, 9, 10, 11 (due Friday Sep. 9) (solutions)
Problem Set 2: hand in #3, 4(a), 5(b), 7, 9(a) (due Friday Sep. 16) (solutions)
Problem Set 3: hand in #1, 4, 5, 6, 7 (due Friday Sep. 23) (solutions)
Problem Set 4: hand in #2, 3, 5, 6, 8 (due Friday Sep. 30) (solutions)
Problem Set 5: hand in #2, 4, 5, 7, 8 (due Friday Oct. 7) (solutions)
Problem Set 6: hand in #1, 2, 3, 7, 8 (due Friday Oct. 14) (solutions)
Problem Set 7: hand in #1, 2, 3, 4, 6 (due Friday Oct. 21) (solutions)
Problem Set 8: hand in #1, 3, 4 (due Friday Oct. 28) (solutions)
Problem Set 9: hand in all (due Friday Nov. 4) (solutions)
Problem Set 10: hand in all (due Friday Nov. 11) (solutions)
Problem Set 11: hand in #1,3,4(a,b,d,g,h),5 (due Friday Nov. 18) (solutions)
Problem Set 12: hand in all, but pick (a),(b), or (c) for #3 and #5 (solutions)
Problem Set 13: hand in #1-6 (due Friday Dec. 9)

Grader: Xiaoyu Dai
Office: Cupples I, Rm. 213
e-mail: daixygogogo@math.wustl.edu

Lecture Notes:

Will be scanned and posted here as I write them. The hope is that this makes taking notes optional.

Lecture 1: Complex numbers
Lecture 2: Complex functions
Lecture 3: More on Cauchy-Riemann
Lecture 4: Topology of the complex plane
Lecture 5: Power series
Lecture 6: The analyst's nightmare
Lecture 7: Variations on Abel's theorem
Lecture 8: Analytic functions
Lecture 9: Continuation and multivaluedness
Lecture 10: Fractional linear transformations
Lecture 11: Conformal mappings
Lecture 12: Constructing conformal equivalencies
Lecture 13: Complex integration
Lecture 14: Prelude to Cauchy
Lecture 15: Cauchy's Theorem (I)
Lecture 16: Some interesting functions
Lecture 17: Cauchy integral formula
Lecture 18: Liouville's theorem; homology classes
Lecture 19: Cauchy's Theorem (II)
Lecture 20: Applications of Cauchy
Lecture 21: The Schwarz awakens
Lecture 22: Poincare metric
Lecture 23: Function series
Lecture 24: Isolated singularities
Lecture 25: Residue calculus
Lecture 26: Rouche's theorem
Lecture 27: An algebro-geometric detour
Lecture 28: Computing real integrals
Lecture 29: Integral transforms
Lecture 30: Harmonic functions
Lecture 31: Poisson formula
Lecture 32: More on harmonic functions
Lecture 33: Extensions and boundary values
Lecture 34: Functions with prescribed principal parts
Lecture 35: Functions with prescribed zeroes
Lecture 36: Jensen's formula
Lecture 37: Rank, genus, and order
Lecture 38: Applications of product theorems
Lecture 39: Blaschke products
Lecture 40: Little Picard


Books:

Lars Ahlfors, Complex Analysis (3rd Ed.); McGraw-Hill

is the recommended textbook, which means I will follow it at least half the time and some of the problems I assign will come from it. Buy, check out, or borrow a copy.

If you would like to read more adventurously than Ahlfors and/or my lecture notes, here are some suggestions. First, there are many other excellent standard texts, including

John B. Conway, Functions of One Complex Variable; Springer

Robert Greene and Steven Krantz, Function Theory of One Complex Variable; AMS

and the second half of

Walter Rudin, Real and Complex Analysis (3rd Ed.); McGraw-Hill.

For a point of view based in formal and convergent power series (convenient for locally computing composition inverses and solutions of differential equations) you can consult

Henri Cartan, Elementary Theory of Analytic Functions of One and Several Complex Variables; Addison-Wesley

Serge Lang, Complex Analysis (3rd Ed.); Springer.

For a view toward several complex variables there is

Raghavan Narasimhan and Yves Nievergelt, Complex Analysis in One Variable; Birkhauser ;

and the beautiful expository monograph

Steven Krantz, Complex Analysis: the Geometric Viewpoint; MAA

treats theorems in complex analysis through the prism of differential geometry. Finally,

Harvey Cohn, Conformal Mapping on Riemann Surfaces; Dover

leads (with lots of beautiful pictures and physical intuition) into Riemann surfaces and complex algebraic geometry.

I will place one copy of each of these books on reserve at the Olin Library.

Grading Policy:

Your final grade for the semester is determined as follows: HW 40%, midterm 20%, final exam 40%. I will drop the lowest two grades you receive on homework.

Homework and examination grades will be regularly updated on blackboard.