Spring 2006 : MAT 542 Complex Analysis I

Instructor : Justin Sawon

Tuesday and Thursday 11:20am-12:40pm in PHY P128

My office is Math Tower 4-104 and you can email me at sawon@math.sunysb.edu to make an appointment.

Course syllabus

1. The field of complex numbers, geometric representation of complex numbers
2. Analytic functions
   • Definition, Cauchy-Riemann equations
   • Elementary theory of power series, uniform convergence
   • Elementary functions: rational, exponential and trigonometric functions
   • The logarithm
3. Analytic functions as mappings
   • Conformality
   • Linear fractional transformations
   • Elementary conformal mappings
4. Complex integration
   • Line integrals and Cauchy's theorem for disk and rectangle
   • Cauchy's integral formula
   • Cauchy's inequalities
   • Morera's theorem, Liouville's theorem and fundamental theorem of algebra
   • The general form of Cauchy's theorem
5. Local properties of analytic functions
   • Removable singularities, Taylor's theorem
   • Zeros and poles, classification of isolated singularities
   • The local mapping theorem
   • The maximum modulus principle, Schwarz's lemma
6. The calculus of residues
   • The residue theorem
   • The argument principle
   • Rouche's theorem
   • Evaluation of definite integrals
7. Power series
   • Weierstrass theorem
   • The Taylor and Laurent series
   • Partial fractions and infinite products
   • Normal families
8. The Riemann mapping theorem
9. Harmonic functions
   • The mean-value property
   • Harnack's inequality
   • The Dirichlet problem
10. Picard's theorem

Prerequisites : you should have taken MAT ???.

The text-book for this course is :

John B. Conway "Functions of one complex variable I" 2nd edition, Springer 1978.

It will be available from the campus bookstore. A copy will also be placed on two-hour loan in the Math/Physics/Astronomy Library, along with the following two reference books :

Lars V. Ahlfors "Complex analysis: an introduction to the theory of analytic functions of one complex variable" 3rd edition, McGraw-Hill 1979.

Walter Rudin "Real and complex analysis" 2nd edition, McGraw-Hill 1974.

There will be a take-home exam for this course and weekly problem sheets. The latter will be designed to supplement what is covered in class and will consist of both examples and general results. I strongly urge you to have a look at all of the problems : some of them may contain results which will be important in later lectures, and of course a wide collection of examples is essential to a good understand of the subject.

Students with Disabilities: If you have a physical, psychological, medical, or learning disability that may impact on your ability to carry out assigned course work, you are strongly urged to contact the staff in the Disabled Student Services (DSS) office: Room 133 in the Humanities Building; 632-6748v/TDD. The DSS office will review your concerns and determine, with you, what accommodations are necessary and appropriate. A written DSS recommendation should be brought to your lecturer who will make a decision on what special arrangements will be made. All information and documentation of disability is confidential. Arrangements should be made early in the semester so that your needs can be accommodated.

Exercise sheet one: postscript, pdf.

Please attempt at least ten problems, and submit your work during the class on Tuesday 7th February.

Exercise sheet two: postscript, pdf.

Please attempt at least seven problems, and submit your work during the class on Tuesday 14th February.

Exercise sheet three: postscript, pdf.

Please attempt at least eight problems, and submit your work during the class on Tuesday 28th February.

Exercise sheet four: postscript, pdf.

Please attempt at least eight problems, and submit your work during the class on Thursday 9th March.

Exercise sheet five: postscript, pdf.

Please attempt at least fourteen problems, and submit your work during the class on Tuesday 28th March.

Exercise sheet six: postscript, pdf.

Please attempt at least ten problems, and submit your work during the class on Thursday 6th April.

For your amusement, problems from past comprehensive exams: postscript, pdf.

You may replace any homework problems with the equivalent number of comps problems. Please take a look at the first ten problems during the Spring Break - we will go through these in the class on 18th April.

Exercise sheet seven: postscript, pdf.

Please attempt at least eight problems, and submit your work during the class on Tuesday 25th April.

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This page last modified by Justin Sawon
Friday, 14-Apr-2006 15:32:46 EDT
Email corrections and comments to sawon@math.sunysb.edu