30243
Math 561 Func
of Cmplx Var I
MWF 14001450
Instructor: Dimiter
Vassilev Office :
Humanities
Bldg, Office 447 Email: vassilev@unm.edu
Phone Number: 505 277 2136
Description: Analyticity,
Cauchy theorem and formulas,
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: Makeup
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 2773506. In addition, they should see CATS
Counseling and Therapy Services;
Syllabus
and Homework – Fall 2010. You should also workout 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 CauchyRiemann 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.
SchwarzPick 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 
HW
14 
15. Nov. 29 

HW
15 
16. Dec. 6 


Dec. 13 18, Finals week 
Wednesday, December 15 3:00–5:00
p.m. 
