Course Outline - Mat 70300 - Fall 2008
Course meetings, Tues -Thurs 11:45 AM - 1:15 PM pm Room 5417
Text: Complex Analysis by L. Ahlfors, McGraw Hill.
Other classical texts you may want to consult are: Theory of Functions by
Caratheldory,
Instructor: Prof. Keen. Office Room 4302 Phone
212 817 8558 or 718 960 8867 email linda.keen@lehman.cuny.edu
Office hours: By appointment
Outline
I plan to give a very standard beginning complex analysis course. I expect to
cover the material in Ahlfors in the first semester. I will assume the material
in the first chapter on the algebraic properties of complex numbers and their
geometric representation.
Contents of the course:
Complex numbers, holomorphic
functions, Cauchy-Riemann equations, power series, complex integration.
Conformal mapping. Linear fractional transformations. The exponential and
logarithmic functions.
Cauchy’s theorem, applications to integrals, Cauchy’s integral formulas,
Liouville theorem, principle of analytic continuation, Morera’s theorem,
theorems of Weierstraß and Hurwitz on uniform convergence.
Schwarz reflection principle, zeros, poles and residues, Calculus of residues,
applications to definite integrals.
Casorati-Weierstraß theorem, argument principle, Rouch´e’s theorem, maximum
modulus theorem, open mapping theorem, fundamental theorem of algebra, Schwarz
lemma, automorphisms of the unit disc and the upper half plane.
Harmonic functions, Poisson kernel. Mittag-Leffler 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.
Homework is accepted until the solutions have been posted.
The final grade will be based on the
midterm and final exam and the homework grades.
The date for the final exam will be announced on this page well in
advance. The final will be in class on
Dec 16
Class and Homework
Assignments (Exercises from Ahlfors 1979 ed unless otherwise noted - due date on selected problems is not
relevant:
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Date |
Class Topic |
Assignment |
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Aug 31 |
Introduction |
Read Chapter 1
and Chapter 2 sec 1 , ex: sec 1.1/1,2 sec1.2/2,3
sec 1.3/2, sec 1.4/3,5 sec 1.5/1,3 sec 2.1 /1,2
sec 2.2/1,4 sec 2.3/1,3,5 sec 2.4/ 1,2,5 first solution set page 1, page 2, page 3 solutions to selected problems ignore the date |
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Sept 2 |
Analytic
functions |
Chapter 2, sec
1.2/1,4,6 |
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Sept 4 |
Polynomial and
Rational Functions |
Chapter 2 sec
1.4 /1,2,4 |
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Sept 9 |
Power Series and Exponential Functions |
Chapter 2 sections 2,3, selected problems 2 ignore the date |
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Sept 11 |
Exponential and Trigonometric functions |
Chapter 2 section 3 |
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Sept 16 |
Quick review of elementary point set topology |
Chapter 3 section 1 |
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Sept 18 |
Conformality |
Chapter 3 section 2 |
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Sept 23 |
Conformality, analytic functions, conformal mapping |
Chapter 3 section 2 |
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Sept 25 |
Linear Fractional transformations |
Chapter 3 section 3 hand in problem set 2 |
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Sept 30 |
No Class Rosh Hashana |
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Oct 2 |
Linear Fractional transformations, elementary mappings |
Chapter 3 sections 3, 4 selected problems 3 |
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Oct 7 |
Elementary mappings |
Chapter 3 section 4 Solutions to selected problems 2 |
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Oct 9 |
No class Yom Kippur |
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Oct 14 |
Monday classes – no meeting |
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Oct 16 |
Complex Integration 4.1 Fundamental theorems |
Chapter 4, 4.1 selected problems 4 |
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Oct 21 |
Complex integration 4.2 Cauchy’s Integral Formula |
Chapter 4, 4.2 Solutions to selected problems 3 |
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Oct 23 |
Complex integration 4.3 Local Properties of Analytic functions |
Chapter 4, 4.3 |
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Oct 28 |
Catch up and review |
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Oct 30 |
Midterm Exam |
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Nov 4 |
Local properties |
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Nov 6 |
Maximum principle |
Chapter 4.3 keep working on problem 4 set |
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Nov 11 |
General form of Cauchy |
Chapter 4.4 |
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Nov 13 |
General form of Cauchy |
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Nov 18 |
Rouche’s theorem and argument principle |
Chapter 4.5 selected problems 5 – Solutions |
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Nov 20 |
Residues |
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Nov 25 |
Harmonic Functions: Poisson Integral, Schwarz reflection |
Chapter 4.6 selected problems 6 - Solutions |
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Dec 2 |
Power series: Weierstrass theorem , Mittag-Leffler Theorem |
Chap 5.1.1,
5.2.1 |
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Dec 4 |
Normal Families Equicontinuity and Compactness |
Chapter 5.5.1, 5.5.2 |
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Dec 9 |
Families of Analytic Functions |
Chap 5.5.3, 5.5.4 |
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Dec 11 |
Review |
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Dec 16 |
Final Exam in class |
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