Math 440/508 - Complex Analysis - Fall 2016
Instructor: Malabika Pramanik
Office: 214 Mathematics Building
E-mail: malabika at math dot ubc dot ca
Lectures: Mon,Wed,Fri 11:00 AM to 12:00 noon in Room 105 of Mathematics Building.
Office hours: Wed 10-11AM, Fri 12-1PM or by appointment.
Course information
Homework
All the homework problems are from the textbook. Homework assignments will be collected at the beginning of class on the indicated day. Late homework assignments will not be accepted.
- Homework 1, due on Monday September 19
- Chapter 1 exercises: 4, 5, 7, 21, 22.
- Homework 2, due on Monday October 3
- Chapter 2 exercises: 1, 2, 5, 7, 8, 9, 10, 1, 12, 13.
- Homework 3, due on Friday October 21
- Chapter 2 exercises: 14, 15
- Chapter 2 problems: 1, 3, 5.
- Midterm, due on Wednesday November 2
- Chapter 3 exercises: 14, 15, 16, 21
- Chapter 3 problems: 3.
- Homework 4, due on Wednesday November 23
- Chapter 8 exercises: 5, 13, 14.
- Chapter 8 problems: 3, 4.
- Take-home final, due by noon on Monday December 5
- Chapter 8 exercises: 19.
- Chapter 8 problems: 5, 7, 8,
- Please leave your final exam in the instructor's mailbox or slide under the office door.
Week-by-week course outline
Here is a rough guideline of the course structure, arranged by week. The textbook pages are mentioned as a reference and as a reading guide. The treatment of these topics in lecture may vary somewhat from that of the text.
- Week 1: (pages 1-18)
- Definition of holomorphy
- Cauchy-Riemann equations
- Power series
- Week 2: (pages 21-36)
- Integration along curves
- Goursat's theorem
- Week 3: (pages 37-50)
- Cauchy's theorem on the disc
- Toy contours
- Cauchy integral formulae
- Week 4: (pages 50-53, 41-45)
- Power series representation of holomorphic functions
- Liouville's theorem
- Fundamental theorem of algebra
- Principle of analytic continuation
- Morera's theorem
- Evaluation of some integrals
- Morera's theorem
- Week 5: (pages 71-87)
- Morera's theorem
- Extended complex plane
- Isolated singularities
- Removable, pole and essential singularities
- Week 6: (pages 89-93)
- Residue theorem
- Argument principle
- Homotopies and simply connected domains
- Week 7: (pages 93-101)
- Cauchy's theorem on homotopies
- The complex logarithm
- Week 8: (pages 205-212, 218)
- Conformal mappings
- Schwarz lemma
- Automorphisms of the disc
- Week 9:
- Mobius transformations
- Cross-ratio
- Preservation of generalized circles