Course
Outline  Mat 70300  Fall 2013
Course meetings, Tues
Thurs 2:00 PM  3:30 PM pm Room TBA
Texts: To be decided. In the meantime, consult , Complex Analysis by L. Ahlfors, McGraw Hill.
Theory of Functions by Caratheodory, Chelsea, The five small volumes on Theory of
Functions by K. Knopp, dover
reprint. Some of the material can also be found in Hyperbolic Geometry from a
local viewpoint, L. Keen, N. Lakic, Cambridge_ Other
texts
Instructor:
Prof. Keen. Office Room 4208. Phone 212
817 8531 or email lkeen@gc.cuny.edu Office hours: By
appointment
Outline
I plan to give a
standard beginning complex analysis course. I will assign exercises from the text to be determined. I will assume the material in the
first chapter of Ahlfors on the algebraic properties of complex numbers and their
geometric representation.
Contents of the course will include:
Complex numbers, Topology of
the complex plane, power series and analytic functions.
Properties of analytic functions: Conformal mapping. Linear
fractional transformations. The exponential and
logarithmic functions.
Complex Integration: Cauchy’s theorem, Cauchy’s integral formulas, Liouville theorem, principle of analytic continuation, Morera’s theorem,
Singularites:
Poles, zeros, residues, the argument principle, Rouche’s
theorem
Maximum modulus
theorem, Schwarz’s Lemma
Riemann Mapping
Theorem, Schwarz reflection principle, Harmonic functions, Poisson
kernel,
Entire functions, MittagLeffler theorem, Jensen’s theorem
Homework assignments will appear on this page
approximately every week. Students are strongly advised to work on all the
homework problems to make sure they are keeping pace with the class.
The final grade will be based only on the homework grades.


Date 
Class
Topic 
Reading
and Assignment 
Aug 29  Introduction  Complex numbers  Ahlfors, Chap 1, sec 1.4/ 3,4,5, sec 1.5/1,4 due 9/10 
Sep 3  Riemann sphere, Complex functions  Chap 1, sec 2.3/ 3,5 sec.2.4/1,2 due 9/17 
Sep 10  Analytic functions, Cauchy Riemann equations  Chap 2, sec 1.2/3,4,7 due 9/26 
Sep 12  Rational functions, Power Series  Chap 2, sec. 1.4/1,2 due 9/26 
Sep 17  Power Series, Exponential  Chap 2, sec 2.3/4,5 sec 2.4/3,4,7 due 9/26 
Sep 19  Rational functions, Power Series  Chap 2, sec 3.4/1,6,8 due 10/2 
Sep 24  Logarithm, Analytic and Conformal maps  Chap 2, sec 3.4/ 7, 8, 10 due 10/9 
Sep 26  Linear transformations, Cross ratios  Chap 3, sec 2.2/1,4, sec 3.1/3,4 due 10/11 
Oct 1  Cross ratios, symmetry  
Oct 3  Circles and conformal maps  Chap 3, 4.2/1,4,6 Due 10/ 17 
Oct 8  Class will start at 2:30 Riemann surfaces, Complex Integration  
Oct 10  Complex Integration, Cauchy's theorem I  Chap 4. sec 1.3/ 2,3,5,8 due 10/22 
Oct 15  No class  Monday classes  
Oct 17  Cauchy's theorem II, Integral formula  Chap 4, sec 2.1/3 due 10/29 
Oct 22  Cauchy's Integral formula cont'd.  Chap 4. sec.2.2/2,3 sec 2.3/2,4,5 due 10/31 
Oct 24  Higher derivatives.  Chap 4. sec 2.3/2,4,5 due 11/ 5 
Oct 29  Local Properties, Maximum Principle, Schwarz Lemma  
Nov 5  General Form of Cauchy Theorem, Chains and Cycles  Chap 4. sec 3.4/2,4,5 due Nov 14 
Nov 7  Homotopic curves and connectivity  see e.g. Conway Chap IV/6 
Nov 12  Multiply connected regions, Residue Theorem  Ahlfors Chap 4 sec 4.4/1,2,3 due Nov 21 
Nov 14  The argument principle, evaluation of integrals  Ahlfors Chap 4 sec 5.3/2,4,6,8 due Nov 26 
Nov 19  Evaluation of integrals, cont'd. Harmonic functions  
Nov 21  Harmonic functions, Schwarz Theorem, Poisson integral, class will start at 2:15  
Nov 26  Schwarz Reflection Weierstrass Theorem, Taylor's Theorem, Laurent Series  Chap 5 sec 1/2,5 Due Dec 5 
Dec 3  Partial Fractions, MittagLeffler Theorem, Infinite Products  Chap 5 Sec 2.1/2,3 Due Dec 10 
Dec 5  Canonical Products, Gamma Functions  Chap 5 Sec 2.2/ 1,2 sec 2.3/ 2,3 Due Dec 12 
Dec 10  Entire Functions, Jensen's Formula  Chap 5 Sec 2.4/1,2 Due Dec 17 
Dec 12  Normal Families and compactness 