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 email@example.com Office hours: By appointment
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.
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|