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, Mittag-Leffler 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.

 

The learning goals for this course are to master the basic ideas and tools of complex function theory: the ability to understand basic properties of analytic functions, Cauchy's theorem, conformal mapping properties, integration theory for analytic functions, and basic theory of harmonic functions.

 

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, Mittag-Leffler 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