Fall 2023 MATH W4065 Honors Complex Variables

Instructor

Name: Eric Urban (Professor)
Email: urban @ math.columbia.edu
Office: 608 Mathematics
Tel: (212) 854-6362
Fax: (212) 854-8962

Office hours

Mondays, 10:00 am - 12:00 pm (room 608)
or by appointment.

Meeting time and location

Day/Time: Tuesdays and Thursdays, 11:40 pm- 12:55 pm.
Location: Math 407

Prerequisites

Prerequisites: MATH V1202. Students should be comfortable with rigorous mathematical proofs and elementary topics such as real numbers, limits, series, continuous and differentiable functions, Riemann integration, change of variables.

Textbooks

Stein, Elias M.; Shakarchi, Rami: Complex analysis. Princeton Lectures in Analysis, II. Princeton University Press, Princeton, NJ, 2003.

About the course

This is a theoretical introduction to complex analysis exposing fundamental facts about analytic functions, Cauchy-Riemann equations, Cauchy's integral formula and conformal mappings. Topics covered include holomorphic functions, harmonic functions, power series, Cauchy-Riemann equations, Cauchy's integral formula, poles, meromorphic functions, Laurent series, residue theorem, entire functions, conformal mapping and if time permits the Zeta function and the prime number theorem and/or Elliptic functions.

Teaching Assistant

Roy Magen

Office hours: TBA

Homework

There will be weekly written homework assignments to be handed in. Late homework will not be accepted. You are encouraged to discuss the homework with other fellow students. However, you must write up the final solutions by yourself. You might also use any resources you like to solve the problems. However, any resource you use other than Stein-Shakarchi's textbook must be cited in your homework.
To receive credit you must
1. Write you name clearly on the front.
2. Staple your work.
3. Justify your work (not just write the answer) and simplify your answers as far as possible.
4. Hand in your work in the designated mail box outside room 410 with my name on it on the 4th flour of the math department.
You can see your grades and the homework assignment in the Courseworks Grade Book. Your graded homeworks will be returned in class one week later. Uncollected homework should be retrieved and left in front of 604/605 Math in the receptacle provided for a period of two weeks. It will be discarded after that period of time.

Exams

Midterm Exams

There will be two midterm exams during class, which are scheduled on :
- Midterm 1: Oct 5th
- Midterm 2: Nov 12th

Final exam
The final exam is projected to be on Dec 21st from 4:10pm to 7pm. All students must take the final exam at the time scheduled by the university. If you have a conflict with the final exam, please contact me as soon as possible.

Make-up exams will not be given. Students can only be excused from the exams because of serious illness: you will need a note from your doctor and your dean.

Grading policy

Homework: 30%

Midterms: 20% each

Final exam: 30 %

Help Room

406 Mathematics Building, Columbia Help Room (on Columbia campus). There is more information here.

Academic honesty

Copying your written work from somebody else or from any other source is considered cheating and will be dealt with severely. Permitting someone else to copy your work is also considered cheating. Any cheating in homeworks, midterms or finals will result in you failing the course and the matter being reported to your dean.