30243 Math 561 Func of Cmplx Var I

MWF 1400-1450 HUM 428 

                                                                 

Instructor: Dimiter Vassilev     Office :  Humanities Bldg, Office 447  Email: vassilev@unm.edu  Phone Number: 505 277 2136

 

Office Hours: MF 10:00-11:00; W 15:00-16:00 or drop-in anytime if you have a quick question

                                                                                                                           

 

Final exam: Wednesday, December 15 3:00–5:00 p.m. Please see Final Examination Schedule   .

Students having conflicts with this examination schedule must notify the appropriate instructor before Friday, April 9, 2010. Any student having more than three examinations scheduled in any one day may notify the instructor of the last examination listed. If notified before April 9, 2010, the instructor shall make arrangements to give a special examination. Conflicts arising as a result of scheduling out of normal hours-pattern or day sequences must be resolved by the instructor of the off-pattern courses. Changes in this examination schedule are not permitted except by formal approval of the instructor’s College Dean.       

Description: Analyticity, Cauchy theorem and formulas, Taylor and Laurent series, singularities and residues, conformal mapping, selected topics.

Please note the following guidelines for the course:

Prerequisite: Math 311 or 402

Textbook: Function Theory of One Complex Variable: Third Edition (Graduate Studies in Mathematics) (Hardcover) Robert E. Greene and Steven G. Krantz

Grades: The final grade will be determined by homework (100 points) and a final exam (200 points).  All grades will be posted on WebCT.

 

Homework:  You can work together on the homework, but you do need to write up your own solutions in your own words. To help the grader, please write your solutions up neatly and clearly and staple the sheets.   Each problem is worth 1 point. I will determine the number of points you have to do in order to receive the possible maximum of 100 points. You will have a chance to turn in more problems at the end of the semester in order to get closer to the desirable number of solved problems.

 

Missed Exams:  Make-up exams can be arranged for exams missed with a VALID excuse (illness, family emergency, active participation in scholarly or athletic activities), and ONLY if prior notice is given. 

 

Disability Statement: We will accommodate students with documented disabilities. During the first two weeks of the semester, those students should inform the instructor of their particular needs and they should also contact Accessibility Services in Mesa Vista Hall, Room 2021, phone 277-3506.  In addition, they should see CATS- Counseling and Therapy Services; Student Health Center (277-4537). (They can help if you suffer from exam anxiety).    

 

 

 

                                            Syllabus and Homework – Fall 2010. You should also work-out all past qualifying exams

 

Week

Topics Covered

Due Homework. Turn in as many problems as you can do from any homework any time.

1. Aug. 23

1. Fundamental Concepts.

§         Complex numbers and functions –  examples and geometric picture.

§         Holomorphic Functions, the Cauchy-Riemann Equations, and Harmonic Functions.

§         Real and Holomorphic Antiderivatives.

HW 1

2. Aug. 30

 

HW 2

3. Sep. 6

Labor Day, Monday, Sep 6

2. Complex line integrals - The Cauchy formula, Morera’s theorem.

Evaluation of integrals.

Integrals depending holomorphically on a parameter.

Last day to drop without a grade: Friday, Sep 10

HW 3

4. Sep. 13

Last day to change grading options: Friday, Sep 17

 

HW 4

5. Sep. 20

 

HW 5

6. Sep. 27

3. Power series - radius of convergence, uniform convergence and properties.

4. Uniform convergence of holomorphic functions. Weierstrass’ theorem.

HW 6

7. Oct. 4

5. Generalization’s of Cauchy’s estimate – estimates on derivatives by max of the modulus of the function. Mean value inequality.

 

HW 7

8. Oct. 11

6. The power series of a holomorphic function.- zeros, uniqueness and order of vanishing of a holomorphic function.

7. The open mapping theorem.

8. Inverse of a holomorphic function.

 

Fall Break, Oct 14 – 15

HW 8

9. Oct. 18

9. The maximum principle. Hurwitz (preservation of zeros) theorem.

10. The Schwarz lemma.

HW 9

10. Oct. 25

11. Schwarz-Pick lemmas. Linear fractional transformations.The automorphism group of the unit disc.

HW 10

11. Nov. 1

 

12. Isolated singularities. Laurent series. Residues.

HW 11

12. Nov. 8

13. The calculus of residues. Applications.

 

Last day to withdraw without the Dean’s approval: Friday, Nov 12 (WP/WF required)

HW 12

13. Nov. 15

14. The Riemann sphere. Singularities at infinity. Meromorphic functions.

 

HW 13

14. Nov. 22

15. The argument principle. Multiple points. Rouche’s theorem.

 

Thanksgiving Holiday, Nov 25 – 28

HW 14

15. Nov. 29

 

 

HW 15

16. Dec. 6

 

 

Dec. 13 -18, Finals week

 

 Wednesday, December 15 3:00–5:00 p.m.